#set document(title: "3.6 Graphs of Functions", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 3.6#h(0.6em)Graphs of Functions #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Before you get started, take this readiness quiz. Evaluate: ⓐ #math.equation(block: false, alt: "2 cubed")[$2^(3)$] ⓑ #math.equation(block: false, alt: "3 squared .")[$3^(2) .$] #linebreak() If you missed this problem, review . #solutionbox[ ⓐ #math.equation(block: false, alt: "8")[$8$]; ⓑ #math.equation(block: false, alt: "9")[$9$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate: ⓐ #math.equation(block: false, alt: "| 7 |")[$| 7 |$] ⓑ #math.equation(block: false, alt: "| −3 | .")[$| −3 | .$] #linebreak() If you missed this problem, review . #solutionbox[ ⓐ #math.equation(block: false, alt: "7")[$7$]; ⓑ #math.equation(block: false, alt: "3")[$3$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate: ⓐ #math.equation(block: false, alt: "the square root of 4")[$sqrt(4)$] ⓑ #math.equation(block: false, alt: "the square root of 16 .")[$sqrt(16) .$] #linebreak() If you missed this problem, review . #solutionbox[ ⓐ #math.equation(block: false, alt: "2")[$2$]; ⓑ #math.equation(block: false, alt: "4")[$4$] ] ] === Use the Vertical Line Test In the last section we learned how to determine if a relation is a function. The relations we looked at were expressed as a set of ordered pairs, a mapping or an equation. We will now look at how to tell if a graph is that of a function. An ordered pair #math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$] is a solution of a linear equation, if the equation is a true statement when the #emph[x]- and #emph[y]-values of the ordered pair are substituted into the equation. The graph of a linear equation is a straight line where every point on the line is a solution of the equation and every solution of this equation is a point on this line. In , we can see that, in graph of the equation #math.equation(block: false, alt: "y equals 2 x minus 3 ,")[$y = 2 x − 3 ,$] for every #emph[x]-value there is only one #emph[y]-value, as shown in the accompanying table. #figure(figph[plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, negative 3), (1, negative 1), and (2, 1). The line is labeled y equals2 x minus 3. There are several vertical arrows that relate values on the x-axis to points on the line. The first arrow relates x equalsnegative 2 on the x-axis to the point (negative 2, negative 7) on the line. The second arrow relates x equalsnegative 1 on the x-axis to the point (negative 1, negative 5) on the line. The next arrow relates x equals0 on the x-axis to the point (0, negative 3) on the line. The next arrow relates x equals3 on the x-axis to the point (3, 3) on the line. The last arrow relates x equals4 on the x-axis to the point (4, 5) on the line. The table has 7 rows and 3 columns. The first row is a title row with the label y equals2 x minus 3. The second row is a header row with the headers x, y, and (x, y). The third row has the coordinates negative 2, negative 7, and (negative 2, negative 7). The fourth row has the coordinates negative 1, negative 5, and (negative 1, negative 5). The fifth row has the coordinates 0, negative 3, and (0, negative 3). The sixth row has the coordinates 3, 3, and (3, 3). The seventh row has the coordinates 4, 5, and (4, 5).], alt: "plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, negative 3), (1, negative 1), and (2, 1). The line is labeled y equals2 x minus 3. There are several vertical arrows that relate values on the x-axis to points on the line. The first arrow relates x equalsnegative 2 on the x-axis to the point (negative 2, negative 7) on the line. The second arrow relates x equalsnegative 1 on the x-axis to the point (negative 1, negative 5) on the line. The next arrow relates x equals0 on the x-axis to the point (0, negative 3) on the line. The next arrow relates x equals3 on the x-axis to the point (3, 3) on the line. The last arrow relates x equals4 on the x-axis to the point (4, 5) on the line. The table has 7 rows and 3 columns. The first row is a title row with the label y equals2 x minus 3. The second row is a header row with the headers x, y, and (x, y). The third row has the coordinates negative 2, negative 7, and (negative 2, negative 7). The fourth row has the coordinates negative 1, negative 5, and (negative 1, negative 5). The fifth row has the coordinates 0, negative 3, and (0, negative 3). The sixth row has the coordinates 3, 3, and (3, 3). The seventh row has the coordinates 4, 5, and (4, 5).", caption: none) A relation is a function if every element of the domain has exactly one value in the range. So the relation defined by the equation #math.equation(block: false, alt: "y equals 2 x minus 3")[$y = 2 x − 3$] is a function. If we look at the graph, each vertical dashed line only intersects the line at one point. This makes sense as in a function, for every #emph[x]-value there is only one #emph[y]-value. If the vertical line hit the graph twice, the #emph[x]-value would be mapped to two #emph[y]-values, and so the graph would not represent a function. This leads us to the vertical line test. A set of points in a rectangular coordinate system is the graph of a function if every vertical line intersects the graph in at most one point. If any vertical line intersects the graph in more than one point, the graph does not represent a function. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Vertical Line Test] A set of points in a rectangular coordinate system is the graph of a function if every vertical line intersects the graph in at most one point. If any vertical line intersects the graph in more than one point, the graph does not represent a function. ] #examplebox("Example 1")[][ Determine whether each graph is the graph of a function. #figure(figph[The figure has two graphs. In graph a there is a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, 2), (3, 0), and (6, negative 2). In graph b there is a parabola opening to the right graphed on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The parabola goes through the points (negative 1, 0), (0, 1), (0, negative 1), (3, 2), and (3, negative 2).], alt: "The figure has two graphs. In graph a there is a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, 2), (3, 0), and (6, negative 2). In graph b there is a parabola opening to the right graphed on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The parabola goes through the points (negative 1, 0), (0, 1), (0, negative 1), (3, 2), and (3, negative 2).", caption: none) #solutionbox[ ⓐ Since any vertical line intersects the graph in at most one point, the graph is the graph of a function. #linebreak() #figure(figph[The figure has a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, 2), (3, 0), and (6, negative 2). Three dashed vertical straight lines are drawn at x equalsnegative 5, x equalsnegative 3, and x equals3. Each line intersects the slanted line at exactly one point.], alt: "The figure has a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, 2), (3, 0), and (6, negative 2). Three dashed vertical straight lines are drawn at x equalsnegative 5, x equalsnegative 3, and x equals3. Each line intersects the slanted line at exactly one point.", caption: none) #linebreak() #linebreak() ⓑ One of the vertical lines shown on the graph, intersects it in two points. This graph does not represent a function. #linebreak() #figure(figph[The figure has a parabola opening to the right graphed on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The parabola goes through the points (negative 1, 0), (0, 1), (0, negative 1), (3, 2), and (3, negative 2). Three dashed vertical straight lines are drawn at x equalsnegative 2, x equalsnegative 1, and x equals2. The vertical line x – negative 2 does not intersect the parabola. The vertical line x equalsnegative 1 intersects the parabola at exactly one point. The vertical line x equals3 intersects the parabola at two separate points.], alt: "The figure has a parabola opening to the right graphed on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The parabola goes through the points (negative 1, 0), (0, 1), (0, negative 1), (3, 2), and (3, negative 2). Three dashed vertical straight lines are drawn at x equalsnegative 2, x equalsnegative 1, and x equals2. The vertical line x – negative 2 does not intersect the parabola. The vertical line x equalsnegative 1 intersects the parabola at exactly one point. The vertical line x equals3 intersects the parabola at two separate points.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Determine whether each graph is the graph of a function. #figure(figph[The figure has two graphs. In graph a there is a parabola opening up graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The parabola goes through the points (0, negative 1), (negative 1, 0), (1, 0), (negative 2, 3), and (2, 3). In graph b there is a circle graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The circle goes through the points (negative 2, 0), (2, 0), (0, negative 2), and (0, 2).], alt: "The figure has two graphs. In graph a there is a parabola opening up graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The parabola goes through the points (0, negative 1), (negative 1, 0), (1, 0), (negative 2, 3), and (2, 3). In graph b there is a circle graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The circle goes through the points (negative 2, 0), (2, 0), (0, negative 2), and (0, 2).", caption: none) #solutionbox[ ⓐ yes ⓑ no ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Determine whether each graph is the graph of a function. #figure(figph[The figure has two graphs. In graph a there is an ellipse graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The ellipse goes through the points (0, negative 3), (negative 2, 0), (2, 0), and (0, 3). In graph b there is a straight line graphed on the x y-coordinate plane. The x-axis runs from negative 12 to 12. The y-axis runs from negative 12 to 12. The line goes through the points (0, negative 2), (2, 0), and (4, 2).], alt: "The figure has two graphs. In graph a there is an ellipse graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The ellipse goes through the points (0, negative 3), (negative 2, 0), (2, 0), and (0, 3). In graph b there is a straight line graphed on the x y-coordinate plane. The x-axis runs from negative 12 to 12. The y-axis runs from negative 12 to 12. The line goes through the points (0, negative 2), (2, 0), and (4, 2).", caption: none) #solutionbox[ ⓐ no ⓑ yes ] ] === Identify Graphs of Basic Functions We used the equation #math.equation(block: false, alt: "y equals 2 x minus 3")[$y = 2 x − 3$] and its graph as we developed the vertical line test. We said that the relation defined by the equation #math.equation(block: false, alt: "y equals 2 x minus 3")[$y = 2 x − 3$] is a function. We can write this as in function notation as #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 2 x minus 3 .")[$f ( x ) = 2 x − 3 .$] It still means the same thing. The graph of the function is the graph of all ordered pairs #math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$] where #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis .")[$y = f ( x ) .$] So we can write the ordered pairs as #math.equation(block: false, alt: "open parenthesis x , f open parenthesis x close parenthesis close parenthesis .")[$( x , f ( x ) ) .$] It looks different but the graph will be the same. Compare the graph of #math.equation(block: false, alt: "y equals 2 x minus 3")[$y = 2 x − 3$] previously shown in with the graph of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 2 x minus 3")[$f ( x ) = 2 x − 3$] shown. Nothing has changed but the notation. #figure(figph[This figure has a graph next to a table. The graph has a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, negative 3), (1, negative 1), and (2, 1). The line is labeled f of x equals2 x minus 3. There are several vertical arrows that relate values on the x-axis to points on the line. The first arrow relates x equalsnegative 2 on the x-axis to the point (negative 2, negative 7) on the line. The second arrow relates x equalsnegative 1 on the x-axis to the point (negative 1, negative 5) on the line. The next arrow relates x equals0 on the x-axis to the point (0, negative 3) on the line. The next arrow relates x equals3 on the x-axis to the point (3, 3) on the line. The last arrow relates x equals4 on the x-axis to the point (4, 5) on the line. The table has 7 rows and 3 columns. The first row is a title row with the label f of x equals2 x minus 3. The second row is a header row with the headers x, f of x, and (x, f of x). The third row has the coordinates negative 2, negative 7, and (negative 2, negative 7). The fourth row has the coordinates negative 1, negative 5, and (negative 1, negative 5). The fifth row has the coordinates 0, negative 3, and (0, negative 3). The sixth row has the coordinates 3, 3, and (3, 3). The seventh row has the coordinates 4, 5, and (4, 5).], alt: "This figure has a graph next to a table. The graph has a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, negative 3), (1, negative 1), and (2, 1). The line is labeled f of x equals2 x minus 3. There are several vertical arrows that relate values on the x-axis to points on the line. The first arrow relates x equalsnegative 2 on the x-axis to the point (negative 2, negative 7) on the line. The second arrow relates x equalsnegative 1 on the x-axis to the point (negative 1, negative 5) on the line. The next arrow relates x equals0 on the x-axis to the point (0, negative 3) on the line. The next arrow relates x equals3 on the x-axis to the point (3, 3) on the line. The last arrow relates x equals4 on the x-axis to the point (4, 5) on the line. The table has 7 rows and 3 columns. The first row is a title row with the label f of x equals2 x minus 3. The second row is a header row with the headers x, f of x, and (x, f of x). The third row has the coordinates negative 2, negative 7, and (negative 2, negative 7). The fourth row has the coordinates negative 1, negative 5, and (negative 1, negative 5). The fifth row has the coordinates 0, negative 3, and (0, negative 3). The sixth row has the coordinates 3, 3, and (3, 3). The seventh row has the coordinates 4, 5, and (4, 5).", caption: none) #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Graph of a Function] The graph of a function is the graph of all its ordered pairs, #math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$] or using function notation, #math.equation(block: false, alt: "open parenthesis x , f open parenthesis x close parenthesis close parenthesis")[$( x , f ( x ) )$] where #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis .")[$y = f ( x ) .$] #math.equation(block: true, alt: "f, name of function; x, x -coordinate of the ordered pair; f open parenthesis x close parenthesis, y -coordinate of the ordered pair")[$f & & & "name of function" \ x & & & x "-coordinate of the ordered pair" \ f ( x ) & & & y "-coordinate of the ordered pair"$] ] As we move forward in our study, it is helpful to be familiar with the graphs of several basic functions and be able to identify them. Through our earlier work, we are familiar with the graphs of linear equations. The process we used to decide if #math.equation(block: false, alt: "y equals 2 x minus 3")[$y = 2 x − 3$] is a function would apply to all linear equations. All non-vertical linear equations are functions. Vertical lines are not functions as the #emph[x]-value has infinitely many #emph[y]-values. We wrote linear equations in several forms, but it will be most helpful for us here to use the slope-intercept form of the linear equation. The slope-intercept form of a linear equation is #math.equation(block: false, alt: "y equals m x plus b .")[$y = m x + b .$] In function notation, this linear function becomes #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals m x plus b")[$f ( x ) = m x + b$] where #emph[m] is the slope of the line and #emph[b] is the #emph[y]-intercept. The domain is the set of all real numbers, and the range is also the set of all real numbers. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Linear Function] #figure(figph[This figure has a graph of a straight line on the x y-coordinate plane. The line goes through the point (0, b). Next to the graph are the following: “f of x equalsm x plus b”, “m, b: all real numbers”, “m: slope of the line”, “b: y-intercept”, “Domain: (negative infinity, infinity)”, and “Range: (negative infinity, infinity)”.], alt: "This figure has a graph of a straight line on the x y-coordinate plane. The line goes through the point (0, b). Next to the graph are the following: “f of x equalsm x plus b”, “m, b: all real numbers”, “m: slope of the line”, “b: y-intercept”, “Domain: (negative infinity, infinity)”, and “Range: (negative infinity, infinity)”.", caption: none) ] We will use the graphing techniques we used earlier, to graph the basic functions. #examplebox("Example 2")[][ Graph: #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals −2 x minus 4 .")[$f ( x ) = −2 x − 4 .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "f open parenthesis x close parenthesis equals −2 x minus 4")[$#h(4.5em) f ( x ) = −2 x − 4$]]), [We recognize this as a linear function.], [], [Find the slope and #emph[y]-intercept.], [#math.equation(block: false, alt: "m equals −2")[$#h(5.5em) m = −2$] #linebreak() #math.equation(block: false, alt: "b equals −4")[$#h(5.65em) b = −4$]], [Graph using the slope intercept.], [#figure(figph[A coordinate plane with a straight line passing through (-2, 0) and (0, -4).], alt: "A coordinate plane with a straight line passing through (-2, 0) and (0, -4).", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph: #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals −3 x minus 1")[$f ( x ) = −3 x − 1$] #solutionbox[ #figure(figph[The figure has the graph of a linear function on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The line goes through the points (1, negative 4), (0, negative 1), and (negative 1, 2).], alt: "The figure has the graph of a linear function on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The line goes through the points (1, negative 4), (0, negative 1), and (negative 1, 2).", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph: #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals −4 x minus 5")[$f ( x ) = −4 x − 5$] #solutionbox[ #figure(figph[The figure has the graph of a linear function on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The line goes through the points (negative 2, 3), (0, negative 5), and (negative 1, negative 1).], alt: "The figure has the graph of a linear function on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The line goes through the points (negative 2, 3), (0, negative 5), and (negative 1, negative 1).", caption: none) ] ] The next function whose graph we will look at is called the constant function and its equation is of the form #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals b ,")[$f ( x ) = b ,$] where #emph[b] is any real number. If we replace the #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] with y, we get #math.equation(block: false, alt: "y equals b .")[$y = b .$] We recognize this as the horizontal line whose #emph[y]-intercept is #emph[b]. The graph of the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals b ,")[$f ( x ) = b ,$] is also the horizontal line whose #emph[y]-intercept is #emph[b]. Notice that for any real number we put in the function, the function value will be #emph[b]. This tells us the range has only one value, #emph[b]. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Constant Function] #figure(figph[This figure has a graph of a straight horizontal line on the x y-coordinate plane. The line goes through the point (0, b). Next to the graph are the following: “f of x equalsb”, “b: any real number”, “b: y-intercept”, “Domain: (negative infinity, infinity)”, and “Range: b”.], alt: "This figure has a graph of a straight horizontal line on the x y-coordinate plane. The line goes through the point (0, b). Next to the graph are the following: “f of x equalsb”, “b: any real number”, “b: y-intercept”, “Domain: (negative infinity, infinity)”, and “Range: b”.", caption: none) ] #examplebox("Example 3")[][ Graph: #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 4 .")[$f ( x ) = 4 .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 4")[$f ( x ) = 4$]]), [We recognize this as a constant function.], [], [The graph will be a horizontal line through #math.equation(block: false, alt: "open parenthesis 0 , 4 close parenthesis .")[$( 0 , 4 ) .$]], [#figure(figph[A graph displays a horizontal line at y=4, extending across the x-axis, on a Cartesian coordinate plane with labeled axes from -7 to 7 for x, and -1 to 11 for y, marked with grid lines.], alt: "A graph displays a horizontal line at y=4, extending across the x-axis, on a Cartesian coordinate plane with labeled axes from -7 to 7 for x, and -1 to 11 for y, marked with grid lines.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph: #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals −2 .")[$f ( x ) = −2 .$] #solutionbox[ #figure(figph[The figure has the graph of a constant function on the x y-coordinate plane. The x-axis runs from negative 12 to 12. The y-axis runs from negative 12 to 12. The line goes through the points (0, negative 2), (1, negative 2), and (2, negative 2).], alt: "The figure has the graph of a constant function on the x y-coordinate plane. The x-axis runs from negative 12 to 12. The y-axis runs from negative 12 to 12. The line goes through the points (0, negative 2), (1, negative 2), and (2, negative 2).", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph: #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 3 .")[$f ( x ) = 3 .$] #solutionbox[ #figure(figph[The figure has the graph of a constant function on the x y-coordinate plane. The x-axis runs from negative 12 to 12. The y-axis runs from negative 12 to 12. The line goes through the points (0, 3), (1, 3), and (2, 3).], alt: "The figure has the graph of a constant function on the x y-coordinate plane. The x-axis runs from negative 12 to 12. The y-axis runs from negative 12 to 12. The line goes through the points (0, 3), (1, 3), and (2, 3).", caption: none) ] ] The identity function, #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x")[$f ( x ) = x$] is a special case of the linear function. If we write it in linear function form, #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 1 x plus 0 ,")[$f ( x ) = 1 x + 0 ,$] we see the slope is 1 and the #emph[y]-intercept is 0. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Identity Function] #figure(figph[This figure has a graph of a straight line on the x y-coordinate plane. The line goes through the points (0, 0), (1, 1), and (2, 2). Next to the graph are the following: “f of x equalsx”, “m: 1”, “b: 0”, “Domain: (negative infinity, infinity)”, and “Range: (negative infinity, infinity)”.], alt: "This figure has a graph of a straight line on the x y-coordinate plane. The line goes through the points (0, 0), (1, 1), and (2, 2). Next to the graph are the following: “f of x equalsx”, “m: 1”, “b: 0”, “Domain: (negative infinity, infinity)”, and “Range: (negative infinity, infinity)”.", caption: none) ] The next function we will look at is not a linear function. So the graph will not be a line. The only method we have to graph this function is point plotting. Because this is an unfamiliar function, we make sure to choose several positive and negative values as well as 0 for our x-values. #examplebox("Example 4")[][ Graph: #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x squared .")[$f ( x ) = x^(2) .$] #solutionbox[ We choose #emph[x]-values. We substitute them in and then create a chart as shown. #linebreak() #figure(figph[This figure has a graph next to a table. In the graph there is a parabola opening up graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 2 to 6. The parabola goes through the points (negative 3, 9), (negative 2, 4), (negative 1, 1), (0, 0), (1, 1), (2, 4), and (3, 9). The table has 8 rows and 3 columns. The first row is a header row with the headers x, f of x equalsx squared, and (x, f of x). The second row has the coordinates negative 3, 9, and (negative 3, 9). The third row has the coordinates negative 2, 4, and (negative 2, 4). The fourth row has the coordinates negative 1, 1, and (negative 1, 1). The fifth row has the coordinates 0, 0, and (0, 0). The sixth row has the coordinates 1, 1, and (1, 1). The seventh row has the coordinates 2, 4, and (2, 4). The seventh row has the coordinates 3, 9, and (3, 9).], alt: "This figure has a graph next to a table. In the graph there is a parabola opening up graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 2 to 6. The parabola goes through the points (negative 3, 9), (negative 2, 4), (negative 1, 1), (0, 0), (1, 1), (2, 4), and (3, 9). The table has 8 rows and 3 columns. The first row is a header row with the headers x, f of x equalsx squared, and (x, f of x). The second row has the coordinates negative 3, 9, and (negative 3, 9). The third row has the coordinates negative 2, 4, and (negative 2, 4). The fourth row has the coordinates negative 1, 1, and (negative 1, 1). The fifth row has the coordinates 0, 0, and (0, 0). The sixth row has the coordinates 1, 1, and (1, 1). The seventh row has the coordinates 2, 4, and (2, 4). The seventh row has the coordinates 3, 9, and (3, 9).", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph: #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x squared .")[$f ( x ) = x^(2) .$] #solutionbox[ #figure(figph[This figure has a graph next to a table. In the graph there is a parabola opening up graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 4 to 8. The parabola goes through the points (negative 2, 4), (negative 1, 1), (0, 0), (1, 1), and (2, 4).], alt: "This figure has a graph next to a table. In the graph there is a parabola opening up graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 4 to 8. The parabola goes through the points (negative 2, 4), (negative 1, 1), (0, 0), (1, 1), and (2, 4).", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals − x squared")[$f ( x ) = "−" x^(2)$] #solutionbox[ #figure(figph[This figure has a graph next to a table. In the graph there is a parabola opening up graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 4 to 8. The parabola goes through the points (negative 2, negative 4), (negative 1, negative 1), (0, 0), (1, negative 1), and (2, negative 4).], alt: "This figure has a graph next to a table. In the graph there is a parabola opening up graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 4 to 8. The parabola goes through the points (negative 2, negative 4), (negative 1, negative 1), (0, 0), (1, negative 1), and (2, negative 4).", caption: none) ] ] Looking at the result in , we can summarize the features of the square function. We call this graph a parabola. As we consider the domain, notice any real number can be used as an #emph[x]-value. The domain is all real numbers. The range is not all real numbers. Notice the graph consists of values of #emph[y] never go below zero. This makes sense as the square of any number cannot be negative. So, the range of the square function is all non-negative real numbers. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Square Function] #figure(figph[This figure has a graph of a parabola opening up graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 2 to 6. The parabola goes through the points (negative 2, 4), (negative 1, 1), (0, 0), (1, 1), and (2, 4). Next to the graph are the following: “f of x equalsx squared”, “Domain: (negative infinity, infinity)”, and “Range: \[0, infinity)”.], alt: "This figure has a graph of a parabola opening up graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 2 to 6. The parabola goes through the points (negative 2, 4), (negative 1, 1), (0, 0), (1, 1), and (2, 4). Next to the graph are the following: “f of x equalsx squared”, “Domain: (negative infinity, infinity)”, and “Range: [0, infinity)”.", caption: none) ] The next function we will look at is also not a linear function so the graph will not be a line. Again we will use point plotting, and make sure to choose several positive and negative values as well as 0 for our #emph[x]-values. #examplebox("Example 5")[][ Graph: #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x cubed .")[$f ( x ) = x^(3) .$] #solutionbox[ We choose #emph[x]-values. We substitute them in and then create a chart. #linebreak() #figure(figph[This figure has a curved line graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 4 to 4. The curved line goes through the points (negative 2, negative 8), (negative 1, negative 1), (0, 0), (1, 1), and (2, 8). Next to the graph is a table. The table has 6 rows and 3 columns. The first row is a header row with the headers x, f of x equalsx cubed, and (x, f of x). The second row has the coordinates negative 2, negative 8, and (negative 2, negative 8). The third row has the coordinates negative 1, negative 1, and (negative 1, negative 1). The fourth row has the coordinates 0, 0, and (0, 0). The fifth row has the coordinates 1, 1, and (1, 1). The sixth row has the coordinates 2, 8, and (2, 8).], alt: "This figure has a curved line graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 4 to 4. The curved line goes through the points (negative 2, negative 8), (negative 1, negative 1), (0, 0), (1, 1), and (2, 8). Next to the graph is a table. The table has 6 rows and 3 columns. The first row is a header row with the headers x, f of x equalsx cubed, and (x, f of x). The second row has the coordinates negative 2, negative 8, and (negative 2, negative 8). The third row has the coordinates negative 1, negative 1, and (negative 1, negative 1). The fourth row has the coordinates 0, 0, and (0, 0). The fifth row has the coordinates 1, 1, and (1, 1). The sixth row has the coordinates 2, 8, and (2, 8).", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph: #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x cubed .")[$f ( x ) = x^(3) .$] #solutionbox[ #figure(figph[This figure has a curved line graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line goes through the points (negative 2, negative 8), (negative 1, negative 1), (0, 0), (1, 1), and (2, 8).], alt: "This figure has a curved line graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line goes through the points (negative 2, negative 8), (negative 1, negative 1), (0, 0), (1, 1), and (2, 8).", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph: #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals − x cubed .")[$f ( x ) = "−" x^(3) .$] #solutionbox[ #figure(figph[This figure has a curved line graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line goes through the points (negative 2, 8), (negative 1, 1), (0, 0), (1, negative 1), and (2, negative 8).], alt: "This figure has a curved line graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line goes through the points (negative 2, 8), (negative 1, 1), (0, 0), (1, negative 1), and (2, negative 8).", caption: none) ] ] Looking at the result in , we can summarize the features of the cube function. As we consider the domain, notice any real number can be used as an #emph[x]-value. The domain is all real numbers. The range is all real numbers. This makes sense as the cube of any non-zero number can be positive or negative. So, the range of the cube function is all real numbers. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Cube Function] #figure(figph[This figure has a curved line graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 4 to 4. The curved line goes through the points (negative 2, negative 8), (negative 1, negative 1), (0, 0), (1, 1), and (2, 8).). Next to the graph are the following: “f of x equalsx cubed”, “Domain: (negative infinity, infinity)”, and “Range: (negative infinity, infinity)”.], alt: "This figure has a curved line graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 4 to 4. The curved line goes through the points (negative 2, negative 8), (negative 1, negative 1), (0, 0), (1, 1), and (2, 8).). Next to the graph are the following: “f of x equalsx cubed”, “Domain: (negative infinity, infinity)”, and “Range: (negative infinity, infinity)”.", caption: none) ] The next function we will look at does not square or cube the input values, but rather takes the square root of those values. Let’s graph the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the square root of x")[$f ( x ) = sqrt(x)$] and then summarize the features of the function. Remember, we can only take the square root of non-negative real numbers, so our domain will be the non-negative real numbers. #examplebox("Example 6")[][ #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the square root of x")[$f ( x ) = sqrt(x)$] #solutionbox[ We choose #emph[x]-values. Since we will be taking the square root, we choose numbers that are perfect squares, to make our work easier. We substitute them in and then create a chart. #linebreak() #figure(figph[This figure has a curved half-line graphed on the x y-coordinate plane. The x-axis runs from 0 to 8. The y-axis runs from 0 to 8. The curved half-line starts at the point (0, 0) and then goes up and to the right. The curved half line goes through the points (1, 1) and (4, 2). Next to the graph is a table. The table has 5 rows and 3 columns. The first row is a header row with the headers x, f of x equalssquare root of x, and (x, f of x). The second row has the coordinates 0, 0, and (0, 0). The third row has the coordinates 1, 1, and (1, 1). The fourth row has the coordinates 4, 2, and (4, 2). The fifth row has the coordinates 9, 3, and (9, 3).], alt: "This figure has a curved half-line graphed on the x y-coordinate plane. The x-axis runs from 0 to 8. The y-axis runs from 0 to 8. The curved half-line starts at the point (0, 0) and then goes up and to the right. The curved half line goes through the points (1, 1) and (4, 2). Next to the graph is a table. The table has 5 rows and 3 columns. The first row is a header row with the headers x, f of x equalssquare root of x, and (x, f of x). The second row has the coordinates 0, 0, and (0, 0). The third row has the coordinates 1, 1, and (1, 1). The fourth row has the coordinates 4, 2, and (4, 2). The fifth row has the coordinates 9, 3, and (9, 3).", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph: #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the square root of x .")[$f ( x ) = sqrt(x) .$] #solutionbox[ #figure(figph[This figure has a curved half-line graphed on the x y-coordinate plane. The x-axis runs from 0 to 10. The y-axis runs from 0 to 10. The curved half-line starts at the point (0, 0) and then goes up and to the right. The curved half line goes through the points (1, 1), (4, 2), and (9, 3).], alt: "This figure has a curved half-line graphed on the x y-coordinate plane. The x-axis runs from 0 to 10. The y-axis runs from 0 to 10. The curved half-line starts at the point (0, 0) and then goes up and to the right. The curved half line goes through the points (1, 1), (4, 2), and (9, 3).", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph: #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals − the square root of x .")[$f ( x ) = "−" sqrt(x) .$] #solutionbox[ #figure(figph[This figure has a curved half-line graphed on the x y-coordinate plane. The x-axis runs from 0 to 10. The y-axis runs from negative 10 to 0. The curved half-line starts at the point (0, 0) and then goes down and to the right. The curved half line goes through the points (1, negative 1), (4, negative 2), and (9, negative 3).], alt: "This figure has a curved half-line graphed on the x y-coordinate plane. The x-axis runs from 0 to 10. The y-axis runs from negative 10 to 0. The curved half-line starts at the point (0, 0) and then goes down and to the right. The curved half line goes through the points (1, negative 1), (4, negative 2), and (9, negative 3).", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Square Root Function] #figure(figph[This figure has a curved half-line graphed on the x y-coordinate plane. The x-axis runs from 0 to 8. The y-axis runs from 0 to 8. The curved half-line starts at the point (0, 0) and then goes up and to the right. The curved half line goes through the points (1, 1) and (4, 2). Next to the graph are the following: “f of x equalssquare root of x”, “Domain: \[0, infinity)”, and “Range: \[0, infinity)”.], alt: "This figure has a curved half-line graphed on the x y-coordinate plane. The x-axis runs from 0 to 8. The y-axis runs from 0 to 8. The curved half-line starts at the point (0, 0) and then goes up and to the right. The curved half line goes through the points (1, 1) and (4, 2). Next to the graph are the following: “f of x equalssquare root of x”, “Domain: [0, infinity)”, and “Range: [0, infinity)”.", caption: none) ] Our last basic function is the absolute value function, #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals | x | .")[$f ( x ) = | x | .$] Keep in mind that the absolute value of a number is its distance from zero. Since we never measure distance as a negative number, we will never get a negative number in the range. #examplebox("Example 7")[][ Graph: #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals | x | .")[$f ( x ) = | x | .$] #solutionbox[ We choose #emph[x]-values. We substitute them in and then create a chart. #linebreak() #figure(figph[This figure has a v-shaped line graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 1 to 6. The v-shaped line goes through the points (negative 3, 3), (negative 2, 2), (negative 1, 1), (0, 0), (1, 1), (2, 2), and (3, 3). Next to the graph is a table. The table has 8 rows and 3 columns. The first row is a header row with the headers x, f of x equalsabsolute value of x, and (x, f of x). The second row has the coordinates negative 3, 3, and (negative 3, 3). The third row has the coordinates negative 2, 2, and (negative 2, 2). The fourth row has the coordinates negative 1, 1, and (negative 1, 1). The fifth row has the coordinates 0, 0, and (0, 0). The sixth row has the coordinates 1, 1, and (1, 1). The seventh row has the coordinates 2, 2, and (2, 2). The eighth row has the coordinates 3, 3, and (3, 3).], alt: "This figure has a v-shaped line graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 1 to 6. The v-shaped line goes through the points (negative 3, 3), (negative 2, 2), (negative 1, 1), (0, 0), (1, 1), (2, 2), and (3, 3). Next to the graph is a table. The table has 8 rows and 3 columns. The first row is a header row with the headers x, f of x equalsabsolute value of x, and (x, f of x). The second row has the coordinates negative 3, 3, and (negative 3, 3). The third row has the coordinates negative 2, 2, and (negative 2, 2). The fourth row has the coordinates negative 1, 1, and (negative 1, 1). The fifth row has the coordinates 0, 0, and (0, 0). The sixth row has the coordinates 1, 1, and (1, 1). The seventh row has the coordinates 2, 2, and (2, 2). The eighth row has the coordinates 3, 3, and (3, 3).", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph: #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals | x | .")[$f ( x ) = | x | .$] #solutionbox[ #figure(figph[This figure has a v-shaped line graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The v-shaped line goes through the points (negative 3, 3), (negative 2, 2), (negative 1, 1), (0, 0), (1, 1), (2, 2), and (3, 3).], alt: "This figure has a v-shaped line graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The v-shaped line goes through the points (negative 3, 3), (negative 2, 2), (negative 1, 1), (0, 0), (1, 1), (2, 2), and (3, 3).", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph: #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals − | x | .")[$f ( x ) = "−" | x | .$] #solutionbox[ #figure(figph[This figure has a v-shaped line graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 8 to 4. The v-shaped line goes through the points (negative 3, negative 3), (negative 2, negative 2), (negative 1, negative 1), (0, 0), (1, negative 1), (2, negative 2), and (3, negative 3).], alt: "This figure has a v-shaped line graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 8 to 4. The v-shaped line goes through the points (negative 3, negative 3), (negative 2, negative 2), (negative 1, negative 1), (0, 0), (1, negative 1), (2, negative 2), and (3, negative 3).", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Absolute Value Function] #figure(figph[This figure has a v-shaped line graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 1 to 6. The v-shaped line goes through the points (negative 3, 3), (negative 2, 2), (negative 1, 1), (0, 0), (1, 1), (2, 2), and (3, 3). The point (0, 0) where the line changes slope is called the vertex. Next to the graph are the following: “f of x equalsabsolute value of x”, “Domain: (negative infinity, infinity)”, and “Range: \[0, infinity)”.], alt: "This figure has a v-shaped line graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 1 to 6. The v-shaped line goes through the points (negative 3, 3), (negative 2, 2), (negative 1, 1), (0, 0), (1, 1), (2, 2), and (3, 3). The point (0, 0) where the line changes slope is called the vertex. Next to the graph are the following: “f of x equalsabsolute value of x”, “Domain: (negative infinity, infinity)”, and “Range: [0, infinity)”.", caption: none) ] === Read Information from a Graph of a Function In the sciences and business, data is often collected and then graphed. The graph is analyzed, information is obtained from the graph and then often predictions are made from the data. We will start by reading the domain and range of a function from its graph. Remember the domain is the set of all the #emph[x]-values in the ordered pairs in the function. To find the domain we look at the graph and find all the values of #emph[x] that have a corresponding value on the graph. Follow the value #emph[x] up or down vertically. If you hit the graph of the function then #emph[x] is in the domain. Remember the range is the set of all the #emph[y]-values in the ordered pairs in the function. To find the range we look at the graph and find all the values of #emph[y] that have a corresponding value on the graph. Follow the value #emph[y] left or right horizontally. If you hit the graph of the function then #emph[y] is in the range. #examplebox("Example 8")[][ Use the graph of the function to find its domain and range. Write the domain and range in interval notation. #figure(figph[This figure has a curved line segment graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 4 to 4. The curved line segment goes through the points (negative 3, negative 1), (1.5, 3), and (3, 1). The interval \[negative 3, 3\] is marked on the horizontal axis. The interval \[negative 1, 3\] is marked on the vertical axis.], alt: "This figure has a curved line segment graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 4 to 4. The curved line segment goes through the points (negative 3, negative 1), (1.5, 3), and (3, 1). The interval [negative 3, 3] is marked on the horizontal axis. The interval [negative 1, 3] is marked on the vertical axis.", caption: none) #solutionbox[ To find the domain we look at the graph and find all the values of #emph[x] that correspond to a point on the graph. The domain is highlighted in red on the graph. The domain is #math.equation(block: false, alt: "[ −3 , 3 ] .")[$[ −3 , 3 ] .$] To find the range we look at the graph and find all the values of #emph[y] that correspond to a point on the graph. The range is highlighted in blue on the graph. The range is #math.equation(block: false, alt: "[ −1 , 3 ] .")[$[ −1 , 3 ] .$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Use the graph of the function to find its domain and range. Write the domain and range in interval notation. #figure(figph[This figure has a curved line segment graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line segment goes through the points (negative 5, negative 4), (0, negative 3), and (1, 2). The interval \[negative 5, 1\] is marked on the horizontal axis. The interval \[negative 4, 2\] is marked on the vertical axis.], alt: "This figure has a curved line segment graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line segment goes through the points (negative 5, negative 4), (0, negative 3), and (1, 2). The interval [negative 5, 1] is marked on the horizontal axis. The interval [negative 4, 2] is marked on the vertical axis.", caption: none) #solutionbox[ The domain is #math.equation(block: false, alt: "[ −5 , 1 ] .")[$[ −5 , 1 ] .$] The range is #math.equation(block: false, alt: "[ −4 , 2 ] .")[$[ −4 , 2 ] .$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Use the graph of the function to find its domain and range. Write the domain and range in interval notation. #figure(figph[This figure has a curved line segment graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 5. The y-axis runs from negative 6 to 4. The curved line segment goes through the points (negative 2, 1), (0, 3), and (4, negative 5). The interval \[negative 2, 4\] is marked on the horizontal axis. The interval \[negative 5, 3\] is marked on the vertical axis.], alt: "This figure has a curved line segment graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 5. The y-axis runs from negative 6 to 4. The curved line segment goes through the points (negative 2, 1), (0, 3), and (4, negative 5). The interval [negative 2, 4] is marked on the horizontal axis. The interval [negative 5, 3] is marked on the vertical axis.", caption: none) #solutionbox[ The domain is #math.equation(block: false, alt: "[ −2 , 4 ] .")[$[ −2 , 4 ] .$] The range is #math.equation(block: false, alt: "[ −5 , 3 ] .")[$[ −5 , 3 ] .$] ] ] We are now going to read information from the graph that you may see in future math classes. #examplebox("Example 9")[][ Use the graph of the function to find the indicated values. #figure(figph[This figure has a wavy curved line graphed on the x y-coordinate plane. The x-axis runs from negative 2 times pi to 2 times pi. The y-axis runs from negative 4 to 4. The curved line segment goes through the points (negative 2 times pi, 0), (negative 3 divided by 2 times pi, 1), (negative pi, 0), (negative 1 divided by 2 times pi, negative 1), (0, 0), (1 divided by 2 times pi, 1), (pi, 0), (3 divided by 2 times pi, negative 1), and (2 times pi, 0). The points (negative 3 divided by 2 times pi, 1) and (1 divided by 2 times pi, 1) are the highest points on the graph. The points (negative 1 divided by 2 times pi, negative 1) and (3 divided by 2 times pi, negative 1) are the lowest points on the graph. The pattern extends infinitely to the left and right.], alt: "This figure has a wavy curved line graphed on the x y-coordinate plane. The x-axis runs from negative 2 times pi to 2 times pi. The y-axis runs from negative 4 to 4. The curved line segment goes through the points (negative 2 times pi, 0), (negative 3 divided by 2 times pi, 1), (negative pi, 0), (negative 1 divided by 2 times pi, negative 1), (0, 0), (1 divided by 2 times pi, 1), (pi, 0), (3 divided by 2 times pi, negative 1), and (2 times pi, 0). The points (negative 3 divided by 2 times pi, 1) and (1 divided by 2 times pi, 1) are the highest points on the graph. The points (negative 1 divided by 2 times pi, negative 1) and (3 divided by 2 times pi, negative 1) are the lowest points on the graph. The pattern extends infinitely to the left and right.", caption: none) ⓐ Find: #math.equation(block: false, alt: "f open parenthesis 0 close parenthesis .")[$f ( 0 ) .$] #linebreak() ⓑ Find: #math.equation(block: false, alt: "f open parenthesis the fraction 3 over 2 π close parenthesis .")[$f ( frac(3, 2) π ) .$] #linebreak() ⓒ Find: #math.equation(block: false, alt: "f open parenthesis minus the fraction 1 over 2 π close parenthesis .")[$f ( − frac(1, 2) π ) .$] #linebreak() ⓓ Find the values for #emph[x] when #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 0 .")[$f ( x ) = 0 .$] #linebreak() ⓔ Find the #emph[x]-intercepts. #linebreak() ⓕ Find the #emph[y]-intercepts. #linebreak() ⓖ Find the domain. Write it in interval notation. #linebreak() ⓗ Find the range. Write it in interval notation. #solutionbox[ ⓐ When #math.equation(block: false, alt: "x equals 0 ,")[$x = 0 ,$] the function crosses the #emph[y]-axis at 0. So, #math.equation(block: false, alt: "f open parenthesis 0 close parenthesis equals 0 .")[$f ( 0 ) = 0 .$] #linebreak() ⓑ When #math.equation(block: false, alt: "x equals the fraction 3 over 2 π ,")[$x = frac(3, 2) π ,$] the #emph[y]-value of the function is #math.equation(block: false, alt: "−1 .")[$−1 .$] So, #math.equation(block: false, alt: "f open parenthesis the fraction 3 over 2 π close parenthesis equals −1 .")[$f ( frac(3, 2) π ) = −1 .$] #linebreak() ⓒ When #math.equation(block: false, alt: "x equals minus the fraction 1 over 2 π ,")[$x = − frac(1, 2) π ,$] the #emph[y]-value of the function is #math.equation(block: false, alt: "−1 .")[$−1 .$] So, #math.equation(block: false, alt: "f open parenthesis minus the fraction 1 over 2 π close parenthesis equals −1 .")[$f ( − frac(1, 2) π ) = −1 .$] #linebreak() ⓓ The function is 0 at the points, #math.equation(block: false, alt: "open parenthesis −2 π , 0 close parenthesis , open parenthesis − π , 0 close parenthesis , open parenthesis 0 , 0 close parenthesis , open parenthesis π , 0 close parenthesis , open parenthesis 2 π , 0 close parenthesis .")[$( −2 π , 0 ) , ( "−" π , 0 ) , ( 0 , 0 ) , ( π , 0 ) , ( 2 π , 0 ) .$] The #emph[x]-values when #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 0")[$f ( x ) = 0$] are #math.equation(block: false, alt: "−2 π , − π , 0 , π , 2 π .")[$−2 π , "−" π , 0 , π , 2 π .$] #linebreak() ⓔ The #emph[x]-intercepts occur when #math.equation(block: false, alt: "y equals 0 .")[$y = 0 .$] So the #emph[x]-intercepts occur when #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 0 .")[$f ( x ) = 0 .$] The #emph[x]-intercepts are #math.equation(block: false, alt: "open parenthesis −2 π , 0 close parenthesis , open parenthesis − π , 0 close parenthesis , open parenthesis 0 , 0 close parenthesis , open parenthesis π , 0 close parenthesis , open parenthesis 2 π , 0 close parenthesis .")[$( −2 π , 0 ) , ( "−" π , 0 ) , ( 0 , 0 ) , ( π , 0 ) , ( 2 π , 0 ) .$] #linebreak() ⓕ The #emph[y]-intercepts occur when #math.equation(block: false, alt: "x equals 0 .")[$x = 0 .$] So the #emph[y]-intercepts occur at #math.equation(block: false, alt: "f open parenthesis 0 close parenthesis .")[$f ( 0 ) .$] The #emph[y]-intercept is #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis .")[$( 0 , 0 ) .$] #linebreak() ⓖ This function has a value for all values of #emph[x]. Therefore, the domain in interval notation is #math.equation(block: false, alt: "open parenthesis minus ∞ , ∞ close parenthesis")[$( − ∞ , #h(0.2em) ∞ )$] #linebreak() ⓗ This function values, or #emph[y]-values go from #math.equation(block: false, alt: "−1")[$−1$] to 1. Therefore, the range, in interval notation, is #math.equation(block: false, alt: "[ −1 , 1 ] .")[$[ −1 , 1 ] .$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Use the graph of the function to find the indicated values. #figure(figph[This figure has a wavy curved line graphed on the x y-coordinate plane. The x-axis runs from negative 2 times pi to 2 times pi. The y-axis runs from negative 6 to 6. The curved line segment goes through the points (negative 2 times pi, 0), (negative 3 divided by 2 times pi, 2), (negative pi, 0), (negative 1 divided by 2 times pi, negative 2), (0, 0), (1 divided by 2 times pi, 2), (pi, 0), (3 divided by 2 times pi, negative 2), and (2 times pi, 0). The points (negative 3 divided by 2 times pi, 2) and (1 divided by 2 times pi, 2) are the highest points on the graph. The points (negative 1 divided by 2 times pi, negative 2) and (3 divided by 2 times pi, negative 2) are the lowest points on the graph. The line extends infinitely to the left and right.], alt: "This figure has a wavy curved line graphed on the x y-coordinate plane. The x-axis runs from negative 2 times pi to 2 times pi. The y-axis runs from negative 6 to 6. The curved line segment goes through the points (negative 2 times pi, 0), (negative 3 divided by 2 times pi, 2), (negative pi, 0), (negative 1 divided by 2 times pi, negative 2), (0, 0), (1 divided by 2 times pi, 2), (pi, 0), (3 divided by 2 times pi, negative 2), and (2 times pi, 0). The points (negative 3 divided by 2 times pi, 2) and (1 divided by 2 times pi, 2) are the highest points on the graph. The points (negative 1 divided by 2 times pi, negative 2) and (3 divided by 2 times pi, negative 2) are the lowest points on the graph. The line extends infinitely to the left and right.", caption: none) ⓐ Find: #math.equation(block: false, alt: "f open parenthesis 0 close parenthesis .")[$f ( 0 ) .$] #linebreak() ⓑ Find: #math.equation(block: false, alt: "f open parenthesis the fraction 1 over 2 π close parenthesis .")[$f ( frac(1, 2) π ) .$] #linebreak() ⓒ Find: #math.equation(block: false, alt: "f open parenthesis minus the fraction 3 over 2 π close parenthesis .")[$f ( − frac(3, 2) π ) .$] #linebreak() ⓓ Find the values for #emph[x] in \[#math.equation(block: false, alt: "minus 2 π , 2 π")[$− 2 π , 2 π$]\] when #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 0 .")[$f ( x ) = 0 .$] #linebreak() ⓔ Find the #emph[x]-intercepts. #linebreak() ⓕ Find the #emph[y]-intercepts. #linebreak() ⓖ Find the domain. Write it in interval notation. #linebreak() ⓗ Find the range. Write it in interval notation. #solutionbox[ ⓐ #math.equation(block: false, alt: "f open parenthesis 0 close parenthesis equals 0")[$f ( 0 ) = 0$] ⓑ #math.equation(block: false, alt: "f equals open parenthesis the fraction π over 2 close parenthesis equals 2")[$f = ( frac(π, 2) ) = 2$] ⓒ #math.equation(block: false, alt: "f equals open parenthesis the fraction −3 π over 2 close parenthesis equals 2")[$f = ( frac(−3 π, 2) ) = 2$] ⓓ #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 0")[$f ( x ) = 0$] for #math.equation(block: false, alt: "x equals −2 π , − π , 0 , π , 2 π")[$x = −2 π , "−" π , 0 , π , 2 π$] ⓔ #math.equation(block: false, alt: "open parenthesis −2 π , 0 close parenthesis , open parenthesis − π , 0 close parenthesis , open parenthesis 0 , 0 close parenthesis , open parenthesis π , 0 close parenthesis , open parenthesis 2 π , 0 close parenthesis")[$( −2 π , 0 ) , ( "−" π , 0 ) , ( 0 , 0 ) , ( π , 0 ) , ( 2 π , 0 )$] ⓕ #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] ⓖ #math.equation(block: false, alt: "open parenthesis minus ∞ , ∞ close parenthesis")[$( − ∞ , #h(0.2em) ∞ )$] ⓗ #math.equation(block: false, alt: "[ −2 , 2 ]")[$[ −2 , 2 ]$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Use the graph of the function to find the indicated values. #figure(figph[This figure has a wavy curved line graphed on the x y-coordinate plane. The x-axis runs from negative 2 times pi to 2 times pi. The y-axis runs from negative 6 to 6. The curved line segment goes through the points (negative 2 times pi, 1), (negative 3 divided by 2 times pi, 0), (negative pi, negative 1), (negative 1 divided by 2 times pi, 0), (0, 1), (1 divided by 2 times pi, 0), (pi, negative 1), (3 divided by 2 times pi, 0), and (2 times pi, 1). The points (negative 2 times pi, 1), (0, 1), and (2 times pi, 1) are the highest points on the graph. The points (negative pi, negative 1) and (pi, negative 1) are the lowest points on the graph. The pattern extends infinitely to the left and right.], alt: "This figure has a wavy curved line graphed on the x y-coordinate plane. The x-axis runs from negative 2 times pi to 2 times pi. The y-axis runs from negative 6 to 6. The curved line segment goes through the points (negative 2 times pi, 1), (negative 3 divided by 2 times pi, 0), (negative pi, negative 1), (negative 1 divided by 2 times pi, 0), (0, 1), (1 divided by 2 times pi, 0), (pi, negative 1), (3 divided by 2 times pi, 0), and (2 times pi, 1). The points (negative 2 times pi, 1), (0, 1), and (2 times pi, 1) are the highest points on the graph. The points (negative pi, negative 1) and (pi, negative 1) are the lowest points on the graph. The pattern extends infinitely to the left and right.", caption: none) ⓐ Find: #math.equation(block: false, alt: "f open parenthesis 0 close parenthesis .")[$f ( 0 ) .$] #linebreak() ⓑ Find: #math.equation(block: false, alt: "f open parenthesis π close parenthesis .")[$f ( π ) .$] #linebreak() ⓒ Find: #math.equation(block: false, alt: "f open parenthesis − π close parenthesis .")[$f ( "−" π ) .$] #linebreak() ⓓ Find the values for #emph[x] in \[#math.equation(block: false, alt: "minus 2 π , 2 π")[$− 2 π , 2 π$]\] when #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 0 .")[$f ( x ) = 0 .$] #linebreak() ⓔ Find the #emph[x]-intercepts. #linebreak() ⓕ Find the #emph[y]-intercepts. #linebreak() ⓖ Find the domain. Write it in interval notation. #linebreak() ⓗ Find the range. Write it in interval notation. #solutionbox[ ⓐ #math.equation(block: false, alt: "f open parenthesis 0 close parenthesis equals 1")[$f ( 0 ) = 1$] ⓑ #math.equation(block: false, alt: "f open parenthesis π close parenthesis equals −1")[$f ( π ) = −1$] ⓒ #math.equation(block: false, alt: "f open parenthesis − π close parenthesis equals −1")[$f ( "−" π ) = −1$] ⓓ #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 0")[$f ( x ) = 0$] for #math.equation(block: false, alt: "x equals minus the fraction 3 π over 2 , minus the fraction π over 2 , the fraction π over 2 , the fraction 3 π over 2")[$x = − frac(3 π, 2) , − frac(π, 2) , frac(π, 2) , frac(3 π, 2)$] ⓔ #math.equation(block: false, alt: "open parenthesis minus the fraction 3 π over 2 , 0 close parenthesis , open parenthesis minus the fraction π over 2 , 0 close parenthesis , open parenthesis the fraction π over 2 , 0 close parenthesis , open parenthesis the fraction 3 π over 2 , 0 close parenthesis")[$( − frac(3 π, 2) , 0 ) , ( − frac(π, 2) , 0 ) , ( frac(π, 2) , 0 ) , ( frac(3 π, 2) , 0 )$] ⓕ #math.equation(block: false, alt: "open parenthesis 0 , 1 close parenthesis")[$( 0 , 1 )$] ⓖ #math.equation(block: false, alt: "open parenthesis minus ∞ , ∞ close parenthesis")[$( − ∞ , #h(0.2em) ∞ )$] ⓗ #math.equation(block: false, alt: "[ −1 , 1 ]")[$[ −1 , 1 ]$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Access this online resource for additional instruction and practice with graphs of functions. - #link("https://openstax.org/l/37domainrange")[Find Domain and Range] ] === Key Concepts - #strong[Vertical Line Test] - A set of points in a rectangular coordinate system is the graph of a function if every vertical line intersects the graph in at most one point. - If any vertical line intersects the graph in more than one point, the graph does not represent a function. - #strong[Graph of a Function] - The graph of a function is the graph of all its ordered pairs, #math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$] or using function notation, #math.equation(block: false, alt: "open parenthesis x , f open parenthesis x close parenthesis close parenthesis")[$( x , f ( x ) )$] where #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis .")[$y = f ( x ) .$] #linebreak() #math.equation(block: true, alt: "f, name of function; x, x -coordinate of the ordered pair; f open parenthesis x close parenthesis, y -coordinate of the ordered pair")[$f & & & "name of function" \ x & & & x "-coordinate of the ordered pair" \ f ( x ) & & & y "-coordinate of the ordered pair"$] - #strong[Linear Function] #linebreak() #figure(figph[This figure has a graph of a straight line on the x y-coordinate plane. The line goes through the point (0, b). Next to the graph are the following: “f of x equalsm x plus b”, “m, b: all real numbers”, “m: slope of the line”, “b: y-intercept”, “Domain: (negative infinity, infinity)”, and “Range: (negative infinity, infinity)”.], alt: "This figure has a graph of a straight line on the x y-coordinate plane. The line goes through the point (0, b). Next to the graph are the following: “f of x equalsm x plus b”, “m, b: all real numbers”, “m: slope of the line”, “b: y-intercept”, “Domain: (negative infinity, infinity)”, and “Range: (negative infinity, infinity)”.", caption: none) - #strong[Constant Function] #linebreak() #figure(figph[This figure has a graph of a straight horizontal line on the x y-coordinate plane. The line goes through the point (0, b). Next to the graph are the following: “f of x equalsb”, “b: any real number”, “b: y-intercept”, “Domain: (negative infinity, infinity)”, and “Range: b”.], alt: "This figure has a graph of a straight horizontal line on the x y-coordinate plane. The line goes through the point (0, b). Next to the graph are the following: “f of x equalsb”, “b: any real number”, “b: y-intercept”, “Domain: (negative infinity, infinity)”, and “Range: b”.", caption: none) - #strong[Identity Function] #linebreak() #figure(figph[This figure has a graph of a straight line on the x y-coordinate plane. The line goes through the points (0, 0), (1, 1), and (2, 2). Next to the graph are the following: “f of x equalsx”, “m: 1”, “b: 0”, “Domain: (negative infinity, infinity)”, and “Range: (negative infinity, infinity)”.], alt: "This figure has a graph of a straight line on the x y-coordinate plane. The line goes through the points (0, 0), (1, 1), and (2, 2). Next to the graph are the following: “f of x equalsx”, “m: 1”, “b: 0”, “Domain: (negative infinity, infinity)”, and “Range: (negative infinity, infinity)”.", caption: none) - #strong[Square Function] #linebreak() #figure(figph[This figure has a graph of a parabola opening up graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 2 to 6. The parabola goes through the points (negative 2, 4), (negative 1, 1), (0, 0), (1, 1), and (2, 4). Next to the graph are the following: “f of x equalsx squared”, “Domain: (negative infinity, infinity)”, and “Range: \[0, infinity)”.], alt: "This figure has a graph of a parabola opening up graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 2 to 6. The parabola goes through the points (negative 2, 4), (negative 1, 1), (0, 0), (1, 1), and (2, 4). Next to the graph are the following: “f of x equalsx squared”, “Domain: (negative infinity, infinity)”, and “Range: [0, infinity)”.", caption: none) - #strong[Cube Function] #linebreak() #figure(figph[This figure has a curved line graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 4 to 4. The curved line goes through the points (negative 2, negative 8), (negative 1, negative 1), (0, 0), (1, 1), and (2, 8).). Next to the graph are the following: “f of x equalsx cubed”, “Domain: (negative infinity, infinity)”, and “Range: (negative infinity, infinity)”.], alt: "This figure has a curved line graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 4 to 4. The curved line goes through the points (negative 2, negative 8), (negative 1, negative 1), (0, 0), (1, 1), and (2, 8).). Next to the graph are the following: “f of x equalsx cubed”, “Domain: (negative infinity, infinity)”, and “Range: (negative infinity, infinity)”.", caption: none) - #strong[Square Root Function] #linebreak() #figure(figph[This figure has a curved half-line graphed on the x y-coordinate plane. The x-axis runs from 0 to 8. The y-axis runs from 0 to 8. The curved half-line starts at the point (0, 0) and then goes up and to the right. The curved half line goes through the points (1, 1) and (4, 2). Next to the graph are the following: “f of x equalssquare root of x”, “Domain: \[0, infinity)”, and “Range: \[0, infinity)”.], alt: "This figure has a curved half-line graphed on the x y-coordinate plane. The x-axis runs from 0 to 8. The y-axis runs from 0 to 8. The curved half-line starts at the point (0, 0) and then goes up and to the right. The curved half line goes through the points (1, 1) and (4, 2). Next to the graph are the following: “f of x equalssquare root of x”, “Domain: [0, infinity)”, and “Range: [0, infinity)”.", caption: none) - #strong[Absolute Value Function] #linebreak() #figure(figph[This figure has a v-shaped line graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 1 to 6. The v-shaped line goes through the points (negative 3, 3), (negative 2, 2), (negative 1, 1), (0, 0), (1, 1), (2, 2), and (3, 3). The point (0, 0) where the line changes slope is called the vertex. Next to the graph are the following: “f of x equalsabsolute value of x”, “Domain: (negative infinity, infinity)”, and “Range: \[0, infinity)”.], alt: "This figure has a v-shaped line graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 1 to 6. The v-shaped line goes through the points (negative 3, 3), (negative 2, 2), (negative 1, 1), (0, 0), (1, 1), (2, 2), and (3, 3). The point (0, 0) where the line changes slope is called the vertex. Next to the graph are the following: “f of x equalsabsolute value of x”, “Domain: (negative infinity, infinity)”, and “Range: [0, infinity)”.", caption: none) === Section Exercises ==== Practice Makes Perfect #strong[Use the Vertical Line Test] In the following exercises, determine whether each graph is the graph of a function. ⓐ #linebreak() #figure(figph[The figure has a circle graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The circle goes through the points (negative 3, 0), (3, 0), (0, negative 3), and (0, 3).], alt: "The figure has a circle graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The circle goes through the points (negative 3, 0), (3, 0), (0, negative 3), and (0, 3).", caption: none) #linebreak() ⓑ #linebreak() #figure(figph[The figure has a parabola opening up graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 4 to 8. The parabola goes through the points (negative 2, 6), (1, 3), (0, 2), (1, 3), and (2, 6).], alt: "The figure has a parabola opening up graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 4 to 8. The parabola goes through the points (negative 2, 6), (1, 3), (0, 2), (1, 3), and (2, 6).", caption: none) #solutionbox[ ⓐ no ⓑ yes ] ⓐ #linebreak() #figure(figph[The figure has an s-shaped curved line graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The s-shaped curved line goes through the points (negative 1, 1), (0, 0), and (1, 1).], alt: "The figure has an s-shaped curved line graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The s-shaped curved line goes through the points (negative 1, 1), (0, 0), and (1, 1).", caption: none) #linebreak() ⓑ #linebreak() #figure(figph[The figure has a circle graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The circle goes through the points (negative 4, 0), (4, 0), (0, negative 4), and (0, 4).], alt: "The figure has a circle graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The circle goes through the points (negative 4, 0), (4, 0), (0, negative 4), and (0, 4).", caption: none) ⓐ #linebreak() #figure(figph[The figure has a parabola opening right graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The parabola goes through the points (negative 2, 0), (negative 1, 1), (negative 1, negative 1), (negative 2, 2), and (2, 2).], alt: "The figure has a parabola opening right graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The parabola goes through the points (negative 2, 0), (negative 1, 1), (negative 1, negative 1), (negative 2, 2), and (2, 2).", caption: none) #linebreak() ⓑ #linebreak() #figure(figph[The figure has a cube function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line goes through the points (negative 1, negative 1), (0, 0), and (1, 1).], alt: "The figure has a cube function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line goes through the points (negative 1, negative 1), (0, 0), and (1, 1).", caption: none) #solutionbox[ ⓐ no ⓑ yes ] ⓐ #linebreak() #figure(figph[The figure has two curved lines graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line on the left goes through the points (negative 2, 0), (negative 4, 5), and (negative 4, negative 5). The curved line on the right goes through the points (2, 0), (4, 5), and (4, negative 5).], alt: "The figure has two curved lines graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line on the left goes through the points (negative 2, 0), (negative 4, 5), and (negative 4, negative 5). The curved line on the right goes through the points (2, 0), (4, 5), and (4, negative 5).", caption: none) #linebreak() ⓑ #linebreak() #figure(figph[The figure has a sideways absolute value function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The line bends at the point (0, 2) and goes to the right. The line goes through the points (1, 3), (2, 4), (1, 1), and (2, 0).], alt: "The figure has a sideways absolute value function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The line bends at the point (0, 2) and goes to the right. The line goes through the points (1, 3), (2, 4), (1, 1), and (2, 0).", caption: none) #strong[Identify Graphs of Basic Functions] In the following exercises, ⓐ graph each function ⓑ state its domain and range. Write the domain and range in interval notation. #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals 3 x plus 4")[$f ( x ) = 3 x + 4$] #solutionbox[ ⓐ #linebreak() #figure(figph[The figure has a linear function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The line goes through the points (negative 2, negative 2), (negative 1, 1), and (0, 4).], alt: "The figure has a linear function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The line goes through the points (negative 2, negative 2), (negative 1, 1), and (0, 4).", caption: none) #linebreak() #linebreak() ⓑ D:(-∞,∞), R:(-∞,∞) ] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals 2 x plus 5")[$f ( x ) = 2 x + 5$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals − x minus 2")[$f ( x ) = "−" x − 2$] #solutionbox[ ⓐ #linebreak() #figure(figph[The figure has a linear function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The line goes through the points (negative 2, 0), (0, negative 2), and (2, negative 4).], alt: "The figure has a linear function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The line goes through the points (negative 2, 0), (0, negative 2), and (2, negative 4).", caption: none) #linebreak() #linebreak() ⓑ D:(-∞,∞), R:(-∞,∞) ] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals −4 x minus 3")[$f ( x ) = −4 x − 3$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals −2 x plus 2")[$f ( x ) = −2 x + 2$] #solutionbox[ ⓐ #linebreak() #figure(figph[The figure has a linear function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The line goes through the points (negative 2, 2), (negative 1, 0), and (0, negative 2).], alt: "The figure has a linear function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The line goes through the points (negative 2, 2), (negative 1, 0), and (0, negative 2).", caption: none) #linebreak() #linebreak() ⓑ D:(-∞,∞), R:(-∞,∞) ] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals −3 x plus 3")[$f ( x ) = −3 x + 3$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the fraction 1 over 2 x plus 1")[$f ( x ) = frac(1, 2) x + 1$] #solutionbox[ ⓐ #linebreak() #figure(figph[The figure has a linear function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The line goes through the points (negative 2, 0), (0, 1), and (2, 2).], alt: "The figure has a linear function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The line goes through the points (negative 2, 0), (0, 1), and (2, 2).", caption: none) #linebreak() #linebreak() ⓑ D:(-∞,∞), R:(-∞,∞) ] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the fraction 2 over 3 x minus 2")[$f ( x ) = frac(2, 3) x − 2$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals 5")[$f ( x ) = 5$] #solutionbox[ ⓐ #linebreak() #figure(figph[The figure has a constant function graphed on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (negative 2, 5), (negative 1, 5), and (0, 5).], alt: "The figure has a constant function graphed on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (negative 2, 5), (negative 1, 5), and (0, 5).", caption: none) #linebreak() #linebreak() ⓑ D:(-∞,∞), R:{5} ] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals 2")[$f ( x ) = 2$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals −3")[$f ( x ) = −3$] #solutionbox[ ⓐ #linebreak() #figure(figph[The figure has a constant function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The line goes through the points (0, negative 3), (1, negative 3), and (2, negative 3).], alt: "The figure has a constant function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The line goes through the points (0, negative 3), (1, negative 3), and (2, negative 3).", caption: none) #linebreak() #linebreak() ⓑ D:(-∞,∞), R: #math.equation(block: false, alt: "{ −3 }")[$\{ −3 \}$] ] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals −1")[$f ( x ) = −1$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals 2 x")[$f ( x ) = 2 x$] #solutionbox[ ⓐ #linebreak() #figure(figph[The figure has a linear function graphed on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (0, 0), (2, 4), and (negative 2, negative 4).], alt: "The figure has a linear function graphed on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (0, 0), (2, 4), and (negative 2, negative 4).", caption: none) #linebreak() #linebreak() ⓑ D:(-∞,∞), R:(-∞,∞) ] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals 3 x")[$f ( x ) = 3 x$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals −2 x")[$f ( x ) = −2 x$] #solutionbox[ ⓐ #linebreak() #figure(figph[The figure has a linear function graphed on the x y-coordinate plane. The x-axis runs from negative 12 to 12. The y-axis runs from negative 12 to 12. The line goes through the points (0, 0), (1, negative 2), and (negative 1, 2).], alt: "The figure has a linear function graphed on the x y-coordinate plane. The x-axis runs from negative 12 to 12. The y-axis runs from negative 12 to 12. The line goes through the points (0, 0), (1, negative 2), and (negative 1, 2).", caption: none) #linebreak() #linebreak() ⓑ D:(-∞,∞), R:(-∞,∞) ] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals −3 x")[$f ( x ) = −3 x$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals 3 x squared")[$f ( x ) = 3 x^(2)$] #solutionbox[ ⓐ #linebreak() #figure(figph[The figure has a square function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The parabola goes through the points (negative 1, 3), (0, 0), and (1, 3). The lowest point on the graph is (0, 0).], alt: "The figure has a square function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The parabola goes through the points (negative 1, 3), (0, 0), and (1, 3). The lowest point on the graph is (0, 0).", caption: none) #linebreak() #linebreak() ⓑ D:(-∞,∞), R:\[0,∞) ] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals 2 x squared")[$f ( x ) = 2 x^(2)$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals −3 x squared")[$f ( x ) = −3 x^(2)$] #solutionbox[ ⓐ #linebreak() #figure(figph[The figure has a square function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 10 to 2. The parabola goes through the points (negative 1, negative 3), (0, 0), and (1, negative 3). The highest point on the graph is (0, 0).], alt: "The figure has a square function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 10 to 2. The parabola goes through the points (negative 1, negative 3), (0, 0), and (1, negative 3). The highest point on the graph is (0, 0).", caption: none) #linebreak() #linebreak() ⓑ (-∞,∞), R:(-∞,0\] ] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals −2 x squared")[$f ( x ) = −2 x^(2)$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the fraction 1 over 2 x squared")[$f ( x ) = frac(1, 2) x^(2)$] #solutionbox[ ⓐ #linebreak() #figure(figph[The figure has a square function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The parabola goes through the points (negative 4, 8), (negative 2, 2), (0, 0), (2, 2), and (4, 8). The lowest point on the graph is (0, 0).], alt: "The figure has a square function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The parabola goes through the points (negative 4, 8), (negative 2, 2), (0, 0), (2, 2), and (4, 8). The lowest point on the graph is (0, 0).", caption: none) #linebreak() #linebreak() ⓑ (-∞,∞), R:\[0,∞) ] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the fraction 1 over 3 x squared")[$f ( x ) = frac(1, 3) x^(2)$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals x squared minus 1")[$f ( x ) = x^(2) − 1$] #solutionbox[ ⓐ #linebreak() #figure(figph[The figure has a square function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The parabola goes through the points (negative 2, 3), (negative 1, 0), (0, negative 1), (1, 0), and (2, 3). The lowest point on the graph is (0, negative 1).], alt: "The figure has a square function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The parabola goes through the points (negative 2, 3), (negative 1, 0), (0, negative 1), (1, 0), and (2, 3). The lowest point on the graph is (0, negative 1).", caption: none) #linebreak() #linebreak() ⓑ (-∞,∞), R:\[#math.equation(block: false, alt: "−1 ,")[$−1 ,$] ∞) ] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals x squared plus 1")[$f ( x ) = x^(2) + 1$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals −2 x cubed")[$f ( x ) = −2 x^(3)$] #solutionbox[ ⓐ #linebreak() #figure(figph[The figure has a cube function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line goes through the points (negative 1, 2), (0, 0), and (1, negative 2).], alt: "The figure has a cube function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line goes through the points (negative 1, 2), (0, 0), and (1, negative 2).", caption: none) #linebreak() #linebreak() ⓑ D:(-∞,∞), R:(-∞,∞) ] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals 2 x cubed")[$f ( x ) = 2 x^(3)$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals x cubed plus 2")[$f ( x ) = x^(3) + 2$] #solutionbox[ ⓐ #linebreak() #figure(figph[The figure has a cube function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line goes through the points (negative 1, 1), (0, 2), and (1, 3).], alt: "The figure has a cube function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line goes through the points (negative 1, 1), (0, 2), and (1, 3).", caption: none) #linebreak() #linebreak() ⓑ D:(-∞,∞), R:(-∞,∞) ] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals x cubed minus 2")[$f ( x ) = x^(3) − 2$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals 2 the square root of x")[$f ( x ) = 2 sqrt(x)$] #solutionbox[ ⓐ #linebreak() #figure(figph[The figure has a square root function graphed on the x y-coordinate plane. The x-axis runs from 0 to 10. The y-axis runs from 0 to 10. The half-line starts at the point (0, 0) and goes through the points (1, 2) and (4, 4).], alt: "The figure has a square root function graphed on the x y-coordinate plane. The x-axis runs from 0 to 10. The y-axis runs from 0 to 10. The half-line starts at the point (0, 0) and goes through the points (1, 2) and (4, 4).", caption: none) #linebreak() #linebreak() ⓑ D:\[0,∞), R:\[0,∞) ] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals −2 the square root of x")[$f ( x ) = −2 sqrt(x)$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the square root of x minus 1")[$f ( x ) = sqrt(x − 1)$] #solutionbox[ ⓐ #linebreak() #figure(figph[The figure has a square root function graphed on the x y-coordinate plane. The x-axis runs from 0 to 10. The y-axis runs from 0 to 10. The half-line starts at the point (1, 0) and goes through the points (2, 1) and (5, 2).], alt: "The figure has a square root function graphed on the x y-coordinate plane. The x-axis runs from 0 to 10. The y-axis runs from 0 to 10. The half-line starts at the point (1, 0) and goes through the points (2, 1) and (5, 2).", caption: none) #linebreak() #linebreak() ⓑ D:\[1,∞), R:\[0,∞) ] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the square root of x plus 1")[$f ( x ) = sqrt(x + 1)$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals 3 | x |")[$f ( x ) = 3 | x |$] #solutionbox[ ⓐ #linebreak() #figure(figph[The figure has an absolute value function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The vertex is at the point (0, 0). The line goes through the points (negative 1, 3) and (1, 3).], alt: "The figure has an absolute value function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The vertex is at the point (0, 0). The line goes through the points (negative 1, 3) and (1, 3).", caption: none) #linebreak() #linebreak() ⓑ #math.equation(block: false, alt: "D: minus ∞ , ∞ , R : [ 0 , ∞ close parenthesis")[$"D:" #h(0.2em) − ∞ , #h(0.2em) ∞ , #h(0.2em) "R" : [ 0 , #h(0.2em) ∞ )$] ] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals −2 | x |")[$f ( x ) = −2 | x |$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals | x | plus 1")[$f ( x ) = | x | + 1$] #solutionbox[ ⓐ #linebreak() #figure(figph[The figure has an absolute value function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The vertex is at the point (0, 1). The line goes through the points (negative 1, 2) and (1, 2).], alt: "The figure has an absolute value function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The vertex is at the point (0, 1). The line goes through the points (negative 1, 2) and (1, 2).", caption: none) #linebreak() #linebreak() ⓑ D:(-∞,∞), R:\[1,∞) ] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals | x | minus 1")[$f ( x ) = | x | − 1$] #strong[Read Information from a Graph of a Function] In the following exercises, use the graph of the function to find its domain and range. Write the domain and range in interval notation. #figure(figph[The figure has a square root function graphed on the x y-coordinate plane. The x-axis runs from negative 2 to 8. The y-axis runs from negative 2 to 8. The half-line starts at the point (2, 0) and goes through the points (3, 1) and (6, 2).], alt: "The figure has a square root function graphed on the x y-coordinate plane. The x-axis runs from negative 2 to 8. The y-axis runs from negative 2 to 8. The half-line starts at the point (2, 0) and goes through the points (3, 1) and (6, 2).", caption: none) #solutionbox[ D: \[2,∞), R: \[0,∞) ] #figure(figph[The figure has a square root function graphed on the x y-coordinate plane. The x-axis runs from negative 2 to 8. The y-axis runs from negative 2 to 10. The half-line starts at the point (negative 3, 0) and goes through the points (negative 2, 1) and (1, 2).], alt: "The figure has a square root function graphed on the x y-coordinate plane. The x-axis runs from negative 2 to 8. The y-axis runs from negative 2 to 10. The half-line starts at the point (negative 3, 0) and goes through the points (negative 2, 1) and (1, 2).", caption: none) #figure(figph[The figure has an absolute value function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from 0 to 12. The vertex is at the point (0, 4). The line goes through the points (negative 2, 6) and (2, 6).], alt: "The figure has an absolute value function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from 0 to 12. The vertex is at the point (0, 4). The line goes through the points (negative 2, 6) and (2, 6).", caption: none) #solutionbox[ D: (-∞,∞), R: \[4,∞) ] #figure(figph[The figure has an absolute value function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 4 to 8. The vertex is at the point (0, negative 1). The line goes through the points (negative 1, 0) and (1, 0).], alt: "The figure has an absolute value function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 4 to 8. The vertex is at the point (0, negative 1). The line goes through the points (negative 1, 0) and (1, 0).", caption: none) #figure(figph[The figure has a half-circle graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line segment starts at the point (negative 2, 0). The line goes through the point (0, 2) and ends at the point (2, 0).], alt: "The figure has a half-circle graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line segment starts at the point (negative 2, 0). The line goes through the point (0, 2) and ends at the point (2, 0).", caption: none) #solutionbox[ D: #math.equation(block: false, alt: "[ −2 , 2 ] ,")[$[ −2 , 2 ] ,$] R: \[0, 2\] ] #figure(figph[The figure has a half-circle graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The curved line segment starts at the point (negative 3, 3). The line goes through the point (0, 6) and ends at the point (3, 3).], alt: "The figure has a half-circle graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The curved line segment starts at the point (negative 3, 3). The line goes through the point (0, 6) and ends at the point (3, 3).", caption: none) In the following exercises, use the graph of the function to find the indicated values. #figure(figph[This figure has a wavy curved line graphed on the x y-coordinate plane. The x-axis runs from negative 2 times pi to 2 times pi. The y-axis runs from negative 6 to 6. The curved line segment goes through the points (negative 2 times pi, 0), (negative 3 divided by 2 times pi, negative 1), (negative pi, 0), (negative 1 divided by 2 times pi, 1), (0, 0), (1 divided by 2 times pi, negative 1), (pi, 0), (3 divided by 2 times pi, 1), and (2 times pi, 0). The points (negative 3 divided by 2 times pi, negative 1) and (1 divided by 2 times pi, negative 1) are the lowest points on the graph. The points (negative 1 divided by 2 times pi, 1) and (3 divided by 2 times pi, 1) are the highest points on the graph. The pattern extends infinitely to the left and right.], alt: "This figure has a wavy curved line graphed on the x y-coordinate plane. The x-axis runs from negative 2 times pi to 2 times pi. The y-axis runs from negative 6 to 6. The curved line segment goes through the points (negative 2 times pi, 0), (negative 3 divided by 2 times pi, negative 1), (negative pi, 0), (negative 1 divided by 2 times pi, 1), (0, 0), (1 divided by 2 times pi, negative 1), (pi, 0), (3 divided by 2 times pi, 1), and (2 times pi, 0). The points (negative 3 divided by 2 times pi, negative 1) and (1 divided by 2 times pi, negative 1) are the lowest points on the graph. The points (negative 1 divided by 2 times pi, 1) and (3 divided by 2 times pi, 1) are the highest points on the graph. The pattern extends infinitely to the left and right.", caption: none) ⓐ Find: #math.equation(block: false, alt: "f open parenthesis 0 close parenthesis .")[$f ( 0 ) .$] #linebreak() ⓑ Find: #math.equation(block: false, alt: "f open parenthesis the fraction 1 over 2 π close parenthesis .")[$f ( frac(1, 2) π ) .$] #linebreak() ⓒ Find: #math.equation(block: false, alt: "f open parenthesis minus the fraction 3 over 2 π close parenthesis .")[$f ( − frac(3, 2) π ) .$] #linebreak() ⓓ Find the values for #emph[x] when #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 0 .")[$f ( x ) = 0 .$] #linebreak() ⓔ Find the #emph[x]-intercepts. #linebreak() ⓕ Find the #emph[y]-intercepts. #linebreak() ⓖ Find the domain. Write it in interval notation. #linebreak() ⓗ Find the range. Write it in interval notation. #solutionbox[ ⓐ #math.equation(block: false, alt: "f open parenthesis 0 close parenthesis equals 0")[$f ( 0 ) = 0$] ⓑ #math.equation(block: false, alt: "f the fraction π over 2 equals minus 1")[$f frac(π, 2) = − 1$] #linebreak() ⓒ #math.equation(block: false, alt: "f minus the fraction 3 π over 2 equals minus 1")[$f − frac(3 π, 2) = − 1$] ⓓ #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 0")[$f ( x ) = 0$] for #math.equation(block: false, alt: "x equals −2 π , − π , 0 , π , 2 π")[$x = −2 π , "−" π , 0 , π , 2 π$] #linebreak() ⓔ #math.equation(block: false, alt: "open parenthesis −2 π , 0 close parenthesis , open parenthesis − π , 0 close parenthesis ,")[$( −2 π , 0 ) , ( "−" π , 0 ) ,$] #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis , open parenthesis π , 0 close parenthesis , open parenthesis 2 π , 0 close parenthesis")[$( 0 , 0 ) , ( π , 0 ) , ( 2 π , 0 )$] #linebreak() ⓕ #math.equation(block: false, alt: "0 , 0")[$0 , 0$] ⓖ #math.equation(block: false, alt: "open parenthesis minus ∞ , ∞ close parenthesis")[$( − ∞ , #h(0.2em) ∞ )$] #linebreak() ⓗ #math.equation(block: false, alt: "[ −1 , 1 ]")[$[ −1 , 1 ]$] ] #figure(figph[This figure has a wavy curved line graphed on the x y-coordinate plane. The x-axis runs from negative 2 times pi to 2 times pi. The y-axis runs from negative 6 to 6. The curved line segment goes through the points (negative 2 times pi, negative 1), (negative 3 divided by 2 times pi, 0), (negative pi, 1), (negative 1 divided by 2 times pi, 0), (0, negative 1), (1 divided by 2 times pi, 0), (pi, 1), (3 divided by 2 times pi, 0), and (2 times pi, negative 1). The points (negative 2 times pi, negative 1) and (2 times pi, negative 1) are the lowest points on the graph. The points (negative pi, 1) and (pi, 1) are the highest points on the graph. The pattern extends infinitely to the left and right.], alt: "This figure has a wavy curved line graphed on the x y-coordinate plane. The x-axis runs from negative 2 times pi to 2 times pi. The y-axis runs from negative 6 to 6. The curved line segment goes through the points (negative 2 times pi, negative 1), (negative 3 divided by 2 times pi, 0), (negative pi, 1), (negative 1 divided by 2 times pi, 0), (0, negative 1), (1 divided by 2 times pi, 0), (pi, 1), (3 divided by 2 times pi, 0), and (2 times pi, negative 1). The points (negative 2 times pi, negative 1) and (2 times pi, negative 1) are the lowest points on the graph. The points (negative pi, 1) and (pi, 1) are the highest points on the graph. The pattern extends infinitely to the left and right.", caption: none) ⓐ Find: #math.equation(block: false, alt: "f open parenthesis 0 close parenthesis .")[$f ( 0 ) .$] #linebreak() ⓑ Find: #math.equation(block: false, alt: "f open parenthesis π close parenthesis .")[$f ( π ) .$] #linebreak() ⓒ Find: #math.equation(block: false, alt: "f open parenthesis − π close parenthesis .")[$f ( "−" π ) .$] #linebreak() ⓓ Find the values for #emph[x] when #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 0 .")[$f ( x ) = 0 .$] #linebreak() ⓔ Find the #emph[x]-intercepts. #linebreak() ⓕ Find the #emph[y]-intercepts. #linebreak() ⓖ Find the domain. Write it in interval notation. #linebreak() ⓗ Find the range. Write it in interval notation #figure(figph[The figure has the top half of a circle graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 4 to 8. The curved line segment starts at the point (negative 3, 2). The line goes through the point (0, 5) and ends at the point (3, 2). The point (0, 5) is the highest point on the graph. The points (negative 3, 2) and (3, 2) are the lowest points on the graph.], alt: "The figure has the top half of a circle graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 4 to 8. The curved line segment starts at the point (negative 3, 2). The line goes through the point (0, 5) and ends at the point (3, 2). The point (0, 5) is the highest point on the graph. The points (negative 3, 2) and (3, 2) are the lowest points on the graph.", caption: none) ⓐ Find: #math.equation(block: false, alt: "f open parenthesis 0 close parenthesis .")[$f ( 0 ) .$] #linebreak() ⓑ Find: #math.equation(block: false, alt: "f open parenthesis −3 close parenthesis .")[$f ( −3 ) .$] #linebreak() ⓒ Find: #math.equation(block: false, alt: "f open parenthesis 3 close parenthesis .")[$f ( 3 ) .$] #linebreak() ⓓ Find the values for #emph[x] when #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 0 .")[$f ( x ) = 0 .$] #linebreak() ⓔ Find the #emph[x]-intercepts. #linebreak() ⓕ Find the #emph[y]-intercepts. #linebreak() ⓖ Find the domain. Write it in interval notation. #linebreak() ⓗ Find the range. Write it in interval notation. #solutionbox[ ⓐ #math.equation(block: false, alt: "5")[$5$] ⓑ #math.equation(block: false, alt: "2")[$2$] ⓒ #math.equation(block: false, alt: "2")[$2$] ⓓ #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 0")[$f ( x ) = 0$] for no #emph[x] ⓔ none ⓕ #math.equation(block: false, alt: "0 , 5")[$0 , #h(0.2em) 5$] ⓖ #math.equation(block: false, alt: "[ −3 , 3 ]")[$[ −3 , 3 ]$] #linebreak() ⓗ #math.equation(block: false, alt: "[ 2 , 5 ]")[$[ 2 , 5 ]$] ] #figure(figph[The figure has the top half of a circle graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 4 to 8. The curved line segment starts at the point (negative 4, 0). The line goes through the point (0, 4) and ends at the point (4, 0). The point (0, 4) is the highest point on the graph. The points (negative 4, 0) and (4, 0) are the lowest points on the graph.], alt: "The figure has the top half of a circle graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 4 to 8. The curved line segment starts at the point (negative 4, 0). The line goes through the point (0, 4) and ends at the point (4, 0). The point (0, 4) is the highest point on the graph. The points (negative 4, 0) and (4, 0) are the lowest points on the graph.", caption: none) ⓐ Find: #math.equation(block: false, alt: "f open parenthesis 0 close parenthesis .")[$f ( 0 ) .$] #linebreak() ⓑ Find the values for #emph[x] when #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 0 .")[$f ( x ) = 0 .$] #linebreak() ⓒ Find the #emph[x]-intercepts. #linebreak() ⓓ Find the #emph[y]-intercepts. #linebreak() ⓔ Find the domain. Write it in interval notation. #linebreak() ⓕ Find the range. Write it in interval notation ==== Writing Exercises Explain in your own words how to find the domain from a graph. Explain in your own words how to find the range from a graph. Explain in your own words how to use the vertical line test. Draw a sketch of the square and cube functions. What are the similarities and differences in the graphs? ==== Self Check ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. #figure(figph[The figure shows a table with four rows and four columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is "confidently", the third is “with some help”, “no minus I don’t get it!”. Under the first column are the phrases “use the vertical line test”, “identify graphs of basic functions”, and “read information from a graph”. Under the second, third, fourth columns are blank spaces where the learner can check what level of mastery they have achieved], alt: "The figure shows a table with four rows and four columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is \"confidently\", the third is “with some help”, “no minus I don’t get it!”. Under the first column are the phrases “use the vertical line test”, “identify graphs of basic functions”, and “read information from a graph”. Under the second, third, fourth columns are blank spaces where the learner can check what level of mastery they have achieved", caption: none) ⓑ After reviewing this checklist, what will you do to become confident for all objectives? === Chapter Review Exercises ==== Graph Linear Equations in Two Variables #strong[Plot Points in a Rectangular Coordinate System] In the following exercises, plot each point in a rectangular coordinate system. ⓐ #math.equation(block: false, alt: "open parenthesis −1 , −5 close parenthesis")[$( −1 , −5 )$] #linebreak() ⓑ #math.equation(block: false, alt: "open parenthesis −3 , 4 close parenthesis")[$( −3 , 4 )$] #linebreak() ⓒ #math.equation(block: false, alt: "open parenthesis 2 , −3 close parenthesis")[$( 2 , −3 )$] #linebreak() ⓓ #math.equation(block: false, alt: "open parenthesis 1 , the fraction 5 over 2 close parenthesis")[$( 1 , frac(5, 2) )$] #solutionbox[ #figure(figph[This figure shows points plotted on the x y-coordinate plane. The x and y axes run from negative 5 to 5. The point labeled a is 1 units to the left of the origin and 5 units below the origin and is located in quadrant III. The point labeled b is 3 units to the left of the origin and 4 units above the origin and is located in quadrant II. The point labeled c is 2 units to the right of the origin and 3 units below the origin and is located in quadrant IV. The point labeled d is 1 unit to the right of the origin and 2.5 units above the origin and is located in quadrant I.], alt: "This figure shows points plotted on the x y-coordinate plane. The x and y axes run from negative 5 to 5. The point labeled a is 1 units to the left of the origin and 5 units below the origin and is located in quadrant III. The point labeled b is 3 units to the left of the origin and 4 units above the origin and is located in quadrant II. The point labeled c is 2 units to the right of the origin and 3 units below the origin and is located in quadrant IV. The point labeled d is 1 unit to the right of the origin and 2.5 units above the origin and is located in quadrant I.", caption: none) ] ⓐ #math.equation(block: false, alt: "open parenthesis −2 , 0 close parenthesis")[$( −2 , 0 )$] #linebreak() ⓑ #math.equation(block: false, alt: "open parenthesis 0 , −4 close parenthesis")[$( 0 , −4 )$] #linebreak() ⓒ #math.equation(block: false, alt: "open parenthesis 0 , 5 close parenthesis")[$( 0 , 5 )$] #linebreak() ⓓ #math.equation(block: false, alt: "open parenthesis 3 , 0 close parenthesis")[$( 3 , 0 )$] In the following exercises, determine which ordered pairs are solutions to the given equations. #math.equation(block: false, alt: "5 x plus y equals 10 ;")[$5 x + y = 10 ;$] #linebreak() ⓐ #math.equation(block: false, alt: "open parenthesis 5 , 1 close parenthesis")[$( 5 , 1 )$] #linebreak() ⓑ #math.equation(block: false, alt: "open parenthesis 2 , 0 close parenthesis")[$( 2 , 0 )$] #linebreak() ⓒ #math.equation(block: false, alt: "open parenthesis 4 , −10 close parenthesis")[$( 4 , −10 )$] #solutionbox[ ⓑ , ⓒ ] #math.equation(block: false, alt: "y equals 6 x minus 2 ;")[$y = 6 x − 2 ;$] #linebreak() ⓐ #math.equation(block: false, alt: "open parenthesis 1 , 4 close parenthesis")[$( 1 , 4 )$] #linebreak() ⓑ #math.equation(block: false, alt: "open parenthesis the fraction 1 over 3 , 0 close parenthesis")[$( frac(1, 3) , 0 )$] #linebreak() ⓒ #math.equation(block: false, alt: "open parenthesis 6 , −2 close parenthesis")[$( 6 , −2 )$] #strong[Graph a Linear Equation by Plotting Points] In the following exercises, graph by plotting points. #math.equation(block: true, alt: "y equals 4 x minus 3")[$y = 4 x − 3$] #solutionbox[ #figure(figph[This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 1, negative 7), (0, negative 3), (1, negative 1), and (2, 3).], alt: "This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 1, negative 7), (0, negative 3), (1, negative 1), and (2, 3).", caption: none) ] #math.equation(block: true, alt: "y equals −3 x")[$y = −3 x$] #math.equation(block: true, alt: "y equals the fraction 1 over 2 x plus 3")[$y = frac(1, 2) x + 3$] #solutionbox[ #figure(figph[This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 6, 0), (0, 3), (2, 4), and (4, 5).], alt: "This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 6, 0), (0, 3), (2, 4), and (4, 5).", caption: none) ] #math.equation(block: true, alt: "y equals minus the fraction 4 over 5 x minus 1")[$y = − frac(4, 5) x − 1$] #math.equation(block: true, alt: "x minus y equals 6")[$x − y = 6$] #solutionbox[ #figure(figph[This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 1, negative 7), (0, negative 6), (3, negative 3), and (6, 0).], alt: "This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 1, negative 7), (0, negative 6), (3, negative 3), and (6, 0).", caption: none) ] #math.equation(block: true, alt: "2 x plus y equals 7")[$2 x + y = 7$] #math.equation(block: true, alt: "3 x minus 2 y equals 6")[$3 x − 2 y = 6$] #solutionbox[ #figure(figph[This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 2, negative 6), (0, negative 3), (2, 0), and (4, 3).], alt: "This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 2, negative 6), (0, negative 3), (2, 0), and (4, 3).", caption: none) ] #strong[Graph Vertical and Horizontal lines] In the following exercises, graph each equation. #math.equation(block: true, alt: "y equals −2")[$y = −2$] #math.equation(block: true, alt: "x equals 3")[$x = 3$] #solutionbox[ #figure(figph[This figure shows a vertical straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (3, negative 1), (3, 0), and (3, 1).], alt: "This figure shows a vertical straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (3, negative 1), (3, 0), and (3, 1).", caption: none) ] In the following exercises, graph each pair of equations in the same rectangular coordinate system. #math.equation(block: false, alt: "y equals −2 x")[$y = −2 x$] and #math.equation(block: false, alt: "y equals −2")[$y = −2$] #math.equation(block: false, alt: "y equals the fraction 4 over 3 x")[$y = frac(4, 3) x$] and #math.equation(block: false, alt: "y equals the fraction 4 over 3")[$y = frac(4, 3)$] #solutionbox[ #figure(figph[The figure shows the graphs of a straight horizontal line and a straight slanted line on the same x y-coordinate plane. The x and y axes run from negative 5 to 5. The horizontal line goes through the points (0, 4 divided by 3), (1, 4 divided by 3), and (2, 4 divided by 3). The slanted line goes through the points (0, 0), (1, 4 divided by 3), and (2, 8 divided by 3).], alt: "The figure shows the graphs of a straight horizontal line and a straight slanted line on the same x y-coordinate plane. The x and y axes run from negative 5 to 5. The horizontal line goes through the points (0, 4 divided by 3), (1, 4 divided by 3), and (2, 4 divided by 3). The slanted line goes through the points (0, 0), (1, 4 divided by 3), and (2, 8 divided by 3).", caption: none) ] #strong[Find #emph[x-] and #emph[y-]Intercepts] In the following exercises, find the #emph[x]- and #emph[y]-intercepts. #figure(figph[The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 6, negative 2), (negative 4, 0), (negative 2, 2), (0, 4), (2, 6), and (4, 8).], alt: "The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 6, negative 2), (negative 4, 0), (negative 2, 2), (0, 4), (2, 6), and (4, 8).", caption: none) #figure(figph[The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 2, 5), (negative 1, 4), (0, 3), (3, 0), and (6, negative 3).], alt: "The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 2, 5), (negative 1, 4), (0, 3), (3, 0), and (6, negative 3).", caption: none) #solutionbox[ #math.equation(block: true, alt: "open parenthesis 0 , 3 close parenthesis open parenthesis 3 , 0 close parenthesis")[$( 0 , 3 ) ( 3 , 0 )$] ] In the following exercises, find the intercepts of each equation. #math.equation(block: true, alt: "x minus y equals −1")[$x − y = −1$] #math.equation(block: true, alt: "x plus 2 y equals 6")[$x + 2 y = 6$] #solutionbox[ #math.equation(block: true, alt: "open parenthesis 6 , 0 close parenthesis , open parenthesis 0 , 3 close parenthesis")[$( 6 , 0 ) , ( 0 , 3 )$] ] #math.equation(block: true, alt: "2 x plus 3 y equals 12")[$2 x + 3 y = 12$] #math.equation(block: true, alt: "y equals the fraction 3 over 4 x minus 12")[$y = frac(3, 4) x − 12$] #solutionbox[ #math.equation(block: true, alt: "open parenthesis 16 , 0 close parenthesis , open parenthesis 0 , −12 close parenthesis")[$( 16 , 0 ) , ( 0 , −12 )$] ] #math.equation(block: true, alt: "y equals 3 x")[$y = 3 x$] #strong[Graph a Line Using the Intercepts] In the following exercises, graph using the intercepts. #math.equation(block: true, alt: "minus x plus 3 y equals 3")[$− x + 3 y = 3$] #solutionbox[ #figure(figph[The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 3, 0), (0, 1), (3, 2), and (6, 3).], alt: "The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 3, 0), (0, 1), (3, 2), and (6, 3).", caption: none) ] #math.equation(block: true, alt: "x minus y equals 4")[$x − y = 4$] #math.equation(block: true, alt: "2 x minus y equals 5")[$2 x − y = 5$] #solutionbox[ #figure(figph[The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (0, negative 5), (1, negative 3), (2, negative 1), and (3, 1).], alt: "The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (0, negative 5), (1, negative 3), (2, negative 1), and (3, 1).", caption: none) ] #math.equation(block: true, alt: "2 x minus 4 y equals 8")[$2 x − 4 y = 8$] #math.equation(block: true, alt: "y equals 4 x")[$y = 4 x$] #solutionbox[ #figure(figph[The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 1, 4), (0, 0), and (1, negative 4).], alt: "The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 1, 4), (0, 0), and (1, negative 4).", caption: none) ] ==== Slope of a Line #strong[Find the Slope of a Line] In the following exercises, find the slope of each line shown. #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The line goes through the points (0, 0) and (1, negative 3).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The line goes through the points (0, 0) and (1, negative 3).", caption: none) #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The line goes through the points (negative 4, 0) and (0, 4).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The line goes through the points (negative 4, 0) and (0, 4).", caption: none) #solutionbox[ 1 ] #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The line goes through the points (negative 4, negative 4) and (2, negative 2).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The line goes through the points (negative 4, negative 4) and (2, negative 2).", caption: none) #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The line goes through the points (1, 4) and (5, 2).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The line goes through the points (1, 4) and (5, 2).", caption: none) #solutionbox[ #math.equation(block: true, alt: "minus the fraction 1 over 2")[$− frac(1, 2)$] ] In the following exercises, find the slope of each line. #math.equation(block: true, alt: "y equals 2")[$y = 2$] #math.equation(block: true, alt: "x equals 5")[$x = 5$] #solutionbox[ undefined ] #math.equation(block: true, alt: "x equals −3")[$x = −3$] #math.equation(block: true, alt: "y equals −1")[$y = −1$] #solutionbox[ 0 ] #strong[Use the Slope Formula to find the Slope of a Line between Two Points] In the following exercises, use the slope formula to find the slope of the line between each pair of points. #math.equation(block: true, alt: "open parenthesis −1 , −1 close parenthesis , open parenthesis 0 , 5 close parenthesis")[$( −1 , −1 ) , ( 0 , 5 )$] #math.equation(block: true, alt: "open parenthesis 3,5 close parenthesis , open parenthesis 4 , −1 close parenthesis")[$( 3,5 ) , ( 4 , −1 )$] #solutionbox[ #math.equation(block: true, alt: "−6")[$−6$] ] #math.equation(block: true, alt: "open parenthesis −5 , −2 close parenthesis , open parenthesis 3 , 2 close parenthesis")[$( −5 , −2 ) , ( 3 , 2 )$] #math.equation(block: true, alt: "open parenthesis 2 , 1 close parenthesis , open parenthesis 4 , 6 close parenthesis")[$( 2 , 1 ) , ( 4 , 6 )$] #solutionbox[ #math.equation(block: true, alt: "the fraction 5 over 2")[$frac(5, 2)$] ] #strong[Graph a Line Given a Point and the Slope] In the following exercises, graph each line with the given point and slope. #math.equation(block: true, alt: "open parenthesis 2 , −2 close parenthesis ;")[$( 2 , −2 ) ;$] #math.equation(block: true, alt: "m equals the fraction 5 over 2")[$m = frac(5, 2)$] #math.equation(block: true, alt: "open parenthesis −3 , 4 close parenthesis ;")[$( −3 , 4 ) ;$] #math.equation(block: true, alt: "m equals minus the fraction 1 over 3")[$m = − frac(1, 3)$] #solutionbox[ #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (negative 3, 4) and (0, 3).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (negative 3, 4) and (0, 3).", caption: none) ] #emph[x]-intercept #math.equation(block: false, alt: "−4 ;")[$−4 ;$] #math.equation(block: false, alt: "m equals 3")[$m = 3$] #emph[y]-intercept 1; #math.equation(block: false, alt: "m equals minus the fraction 3 over 4")[$m = − frac(3, 4)$] #solutionbox[ #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (0, 1) and (4, negative 2).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (0, 1) and (4, negative 2).", caption: none) ] #strong[Graph a Line Using Its Slope and Intercept] In the following exercises, identify the slope and #emph[y]-intercept of each line. #math.equation(block: true, alt: "y equals −4 x plus 9")[$y = −4 x + 9$] #math.equation(block: true, alt: "y equals the fraction 5 over 3 x minus 6")[$y = frac(5, 3) x − 6$] #solutionbox[ #math.equation(block: true, alt: "m equals the fraction 5 over 3 ; open parenthesis 0 , −6 close parenthesis")[$m = frac(5, 3) ; ( 0 , −6 )$] ] #math.equation(block: true, alt: "5 x plus y equals 10")[$5 x + y = 10$] #math.equation(block: true, alt: "4 x minus 5 y equals 8")[$4 x − 5 y = 8$] #solutionbox[ #math.equation(block: true, alt: "m equals the fraction 4 over 5 ; open parenthesis 0 , minus the fraction 8 over 5 close parenthesis")[$m = frac(4, 5) ; ( 0 , − frac(8, 5) )$] ] In the following exercises, graph the line of each equation using its slope and #emph[y]-intercept. #math.equation(block: true, alt: "y equals 2 x plus 3")[$y = 2 x + 3$] #math.equation(block: true, alt: "y equals − x minus 1")[$y = "−" x − 1$] #solutionbox[ #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (0, negative 1) and (1, negative 2).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (0, negative 1) and (1, negative 2).", caption: none) ] #math.equation(block: true, alt: "y equals minus the fraction 2 over 5 x plus 3")[$y = − frac(2, 5) x + 3$] #math.equation(block: true, alt: "4 x minus 3 y equals 12")[$4 x − 3 y = 12$] #solutionbox[ #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (0, negative 4) and (3, 0).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (0, negative 4) and (3, 0).", caption: none) ] In the following exercises, determine the most convenient method to graph each line. #math.equation(block: true, alt: "x equals 5")[$x = 5$] #math.equation(block: true, alt: "y equals −3")[$y = −3$] #solutionbox[ horizontal line ] #math.equation(block: true, alt: "2 x plus y equals 5")[$2 x + y = 5$] #math.equation(block: true, alt: "x minus y equals 2")[$x − y = 2$] #solutionbox[ intercepts ] #math.equation(block: true, alt: "y equals the fraction 2 over 2 x plus 2")[$y = frac(2, 2) x + 2$] #math.equation(block: true, alt: "y equals the fraction 3 over 4 x minus 1")[$y = frac(3, 4) x − 1$] #solutionbox[ plotting points ] #strong[Graph and Interpret Applications of Slope-Intercept] Katherine is a private chef. The equation #math.equation(block: false, alt: "C equals 6.5 m plus 42")[$C = 6.5 m + 42$] models the relation between her weekly cost, #emph[C], in dollars and the number of meals, #emph[m], that she serves. ⓐ Find Katherine’s cost for a week when she serves no meals. #linebreak() ⓑ Find the cost for a week when she serves 14 meals. #linebreak() ⓒ Interpret the slope and #emph[C]-intercept of the equation. #linebreak() ⓓ Graph the equation. Marjorie teaches piano. The equation #math.equation(block: false, alt: "P equals 35 s minus 250")[$P = 35 s − 250$] models the relation between her weekly profit, #emph[P], in dollars and the number of student lessons, #emph[s], that she teaches. ⓐ Find Marjorie’s profit for a week when she teaches no student lessons. #linebreak() ⓑ Find the profit for a week when she teaches 20 student lessons. #linebreak() ⓒ Interpret the slope and #emph[P]-intercept of the equation. #linebreak() ⓓ Graph the equation. #solutionbox[ ⓐ #math.equation(block: false, alt: "− $ 250")[$"−" "$" 250$] #linebreak() ⓑ \$450 #linebreak() ⓒ The slope, 35, means that Marjorie’s weekly profit, #emph[P], increases by \$35 for each additional student lesson she teaches. #linebreak() The #emph[P]-intercept means that when the number of lessons is 0, Marjorie loses \$250. #linebreak() ⓓ #linebreak() #figure(figph[This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 4 to 28. The y-axis runs from negative 250 to 450. The line goes through the points (0, negative 250) and (20, 450).], alt: "This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 4 to 28. The y-axis runs from negative 250 to 450. The line goes through the points (0, negative 250) and (20, 450).", caption: none) ] #strong[Use Slopes to Identify Parallel and Perpendicular Lines] In the following exercises, use slopes and y-intercepts to determine if the lines are parallel, perpendicular, or neither. #math.equation(block: true, alt: "4 x minus 3 y equals −1 ; y equals the fraction 4 over 3 x minus 3")[$4 x − 3 y = −1 ; y = frac(4, 3) x − 3$] #math.equation(block: true, alt: "y equals 5 x minus 1 ; 10 x plus 2 y equals 0")[$y = 5 x − 1 ; 10 x + 2 y = 0$] #solutionbox[ neither ] #math.equation(block: true, alt: "3 x minus 2 y equals 5 ; 2 x plus 3 y equals 6")[$3 x − 2 y = 5 ; 2 x + 3 y = 6$] #math.equation(block: true, alt: "2 x minus y equals 8 ; x minus 2 y equals 4")[$2 x − y = 8 ; x − 2 y = 4$] #solutionbox[ neither ] ==== Find the Equation of a Line #strong[Find an Equation of the Line Given the Slope and #emph[y]-Intercept] In the following exercises, find the equation of a line with given slope and y-intercept. Write the equation in slope–intercept form. slope #math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$] and #math.equation(block: false, alt: "y")[$y$]-intercept #math.equation(block: false, alt: "open parenthesis 0 , −6 close parenthesis")[$( 0 , −6 )$] slope #math.equation(block: false, alt: "−5")[$−5$] and #math.equation(block: false, alt: "y")[$y$]-intercept #math.equation(block: false, alt: "open parenthesis 0 , −3 close parenthesis")[$( 0 , −3 )$] #solutionbox[ #math.equation(block: true, alt: "y equals −5 x minus 3")[$y = −5 x − 3$] ] slope 0 and #math.equation(block: false, alt: "y")[$y$]-intercept #math.equation(block: false, alt: "open parenthesis 0 , 4 close parenthesis")[$( 0 , 4 )$] slope #math.equation(block: false, alt: "−2")[$−2$] and #math.equation(block: false, alt: "y")[$y$]-intercept #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] #solutionbox[ #math.equation(block: true, alt: "y equals −2 x")[$y = −2 x$] ] In the following exercises, find the equation of the line shown in each graph. Write the equation in slope–intercept form. #figure(figph[This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, 1), (1, 3), and (2, 5).], alt: "This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, 1), (1, 3), and (2, 5).", caption: none) #figure(figph[This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, 5), (1, 2), and (2, negative 1).], alt: "This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, 5), (1, 2), and (2, negative 1).", caption: none) #solutionbox[ #math.equation(block: true, alt: "y equals −3 x plus 5")[$y = −3 x + 5$] ] #figure(figph[This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, negative 2), (4, 1), and (8, 4).], alt: "This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, negative 2), (4, 1), and (8, 4).", caption: none) #figure(figph[This figure has a graph of a horizontal straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, negative 4), (1, negative 4), and (2, negative 4).], alt: "This figure has a graph of a horizontal straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, negative 4), (1, negative 4), and (2, negative 4).", caption: none) #solutionbox[ #math.equation(block: true, alt: "y equals −4")[$y = −4$] ] #strong[Find an Equation of the Line Given the Slope and a Point] In the following exercises, find the equation of a line with given slope and containing the given point. Write the equation in slope–intercept form. #math.equation(block: false, alt: "m equals minus the fraction 1 over 4 ,")[$m = − frac(1, 4) ,$] point #math.equation(block: false, alt: "open parenthesis −8 , 3 close parenthesis")[$( −8 , 3 )$] #math.equation(block: false, alt: "m equals the fraction 3 over 5 ,")[$m = frac(3, 5) ,$] point #math.equation(block: false, alt: "open parenthesis 10 , 6 close parenthesis")[$( 10 , 6 )$] #solutionbox[ #math.equation(block: true, alt: "y equals the fraction 3 over 5 x")[$y = frac(3, 5) x$] ] Horizontal line containing #math.equation(block: false, alt: "open parenthesis −2 , 7 close parenthesis")[$( −2 , 7 )$] #math.equation(block: false, alt: "m equals −2 ,")[$m = −2 ,$] point #math.equation(block: false, alt: "open parenthesis −1 , −3 close parenthesis")[$( −1 , −3 )$] #solutionbox[ #math.equation(block: true, alt: "y equals −2 x minus 5")[$y = −2 x − 5$] ] #strong[Find an Equation of the Line Given Two Points] In the following exercises, find the equation of a line containing the given points. Write the equation in slope–intercept form. #math.equation(block: false, alt: "open parenthesis 2 , 10 close parenthesis")[$( 2 , 10 )$] and #math.equation(block: false, alt: "open parenthesis −2 , −2 close parenthesis")[$( −2 , −2 )$] #math.equation(block: false, alt: "open parenthesis 7 , 1 close parenthesis")[$( 7 , 1 )$] and #math.equation(block: false, alt: "open parenthesis 5 , 0 close parenthesis")[$( 5 , 0 )$] #solutionbox[ #math.equation(block: true, alt: "y equals the fraction 1 over 2 x minus the fraction 5 over 2")[$y = frac(1, 2) x − frac(5, 2)$] ] #math.equation(block: false, alt: "open parenthesis 3 , 8 close parenthesis")[$( 3 , 8 )$] and #math.equation(block: false, alt: "open parenthesis 3 , −4 close parenthesis")[$( 3 , −4 )$] #math.equation(block: false, alt: "open parenthesis 5 , 2 close parenthesis")[$( 5 , 2 )$] and #math.equation(block: false, alt: "open parenthesis −1 , 2 close parenthesis")[$( −1 , 2 )$] #solutionbox[ #math.equation(block: true, alt: "y equals 2")[$y = 2$] ] #strong[Find an Equation of a Line Parallel to a Given Line] In the following exercises, find an equation of a line parallel to the given line and contains the given point. Write the equation in slope–intercept form. line #math.equation(block: false, alt: "y equals −3 x plus 6 ,")[$y = −3 x + 6 ,$] point #math.equation(block: false, alt: "open parenthesis 1 , −5 close parenthesis")[$( 1 , −5 )$] line #math.equation(block: false, alt: "2 x plus 5 y equals −10 ,")[$2 x + 5 y = −10 ,$] point #math.equation(block: false, alt: "open parenthesis 10 , 4 close parenthesis")[$( 10 , 4 )$] #solutionbox[ #math.equation(block: true, alt: "y equals minus the fraction 2 over 5 x plus 8")[$y = − frac(2, 5) x + 8$] ] line #math.equation(block: false, alt: "x equals 4 ,")[$x = 4 ,$] point #math.equation(block: false, alt: "open parenthesis −2 , −1 close parenthesis")[$( −2 , −1 )$] line #math.equation(block: false, alt: "y equals −5 ,")[$y = −5 ,$] point #math.equation(block: false, alt: "open parenthesis −4 , 3 close parenthesis")[$( −4 , 3 )$] #solutionbox[ #math.equation(block: true, alt: "y equals 3")[$y = 3$] ] #strong[Find an Equation of a Line Perpendicular to a Given Line] In the following exercises, find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope–intercept form. line #math.equation(block: false, alt: "y equals minus the fraction 4 over 5 x plus 2 ,")[$y = − frac(4, 5) x + 2 ,$] point #math.equation(block: false, alt: "open parenthesis 8 , 9 close parenthesis")[$( 8 , 9 )$] line #math.equation(block: false, alt: "2 x minus 3 y equals 9 ,")[$2 x − 3 y = 9 ,$] point #math.equation(block: false, alt: "open parenthesis −4 , 0 close parenthesis")[$( −4 , 0 )$] #solutionbox[ #math.equation(block: true, alt: "y equals minus the fraction 3 over 2 x minus 6")[$y = − frac(3, 2) x − 6$] ] line #math.equation(block: false, alt: "y equals 3 ,")[$y = 3 ,$] point #math.equation(block: false, alt: "open parenthesis −1 , −3 close parenthesis")[$( −1 , −3 )$] line #math.equation(block: false, alt: "x equals −5")[$x = −5$] point #math.equation(block: false, alt: "open parenthesis 2 , 1 close parenthesis")[$( 2 , 1 )$] #solutionbox[ #math.equation(block: true, alt: "y equals 1")[$y = 1$] ] ==== Graph Linear Inequalities in Two Variables #strong[Verify Solutions to an Inequality in Two Variables] In the following exercises, determine whether each ordered pair is a solution to the given inequality. Determine whether each ordered pair is a solution to the inequality #math.equation(block: false, alt: "y less than x minus 3 :")[$y < x − 3 ":"$] ⓐ #math.equation(block: false, alt: "open parenthesis 0 , 1 close parenthesis")[$( 0 , 1 )$] ⓑ #math.equation(block: false, alt: "open parenthesis −2 , −4 close parenthesis")[$( −2 , −4 )$] ⓒ #math.equation(block: false, alt: "open parenthesis 5 , 2 close parenthesis")[$( 5 , 2 )$] ⓓ #math.equation(block: false, alt: "open parenthesis 3 , −1 close parenthesis")[$( 3 , −1 )$] #linebreak() ⓔ #math.equation(block: false, alt: "open parenthesis −1 , −5 close parenthesis")[$( −1 , −5 )$] Determine whether each ordered pair is a solution to the inequality #math.equation(block: false, alt: "x plus y greater than 4 :")[$x + y > 4 ":"$] ⓐ #math.equation(block: false, alt: "open parenthesis 6 , 1 close parenthesis")[$( 6 , 1 )$] ⓑ #math.equation(block: false, alt: "open parenthesis −3 , 6 close parenthesis")[$( −3 , 6 )$] ⓒ #math.equation(block: false, alt: "open parenthesis 3 , 2 close parenthesis")[$( 3 , 2 )$] ⓓ #math.equation(block: false, alt: "open parenthesis −5 , 10 close parenthesis")[$( −5 , 10 )$] ⓔ #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] #solutionbox[ ⓐ yes ⓑ no ⓒ yes ⓓ yes; ⓔ no ] #strong[Recognize the Relation Between the Solutions of an Inequality and its Graph] In the following exercises, write the inequality shown by the shaded region. Write the inequality shown by the graph with the boundary line #math.equation(block: false, alt: "y equals − x plus 2 .")[$y = "−" x + 2 .$] #figure(figph[This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, 2), (1, 1), and (2, 0). The line divides the x y-coordinate plane into two halves. The line and the bottom left half are shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, 2), (1, 1), and (2, 0). The line divides the x y-coordinate plane into two halves. The line and the bottom left half are shaded red to indicate that this is where the solutions of the inequality are.", caption: none) Write the inequality shown by the graph with the boundary line #math.equation(block: false, alt: "y equals the fraction 2 over 3 x minus 3 .")[$y = frac(2, 3) x − 3 .$] #figure(figph[This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 3), (3, negative 1), and (6, 1). The line divides the x y-coordinate plane into two halves. The line and the top left half are shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 3), (3, negative 1), and (6, 1). The line divides the x y-coordinate plane into two halves. The line and the top left half are shaded red to indicate that this is where the solutions of the inequality are.", caption: none) #solutionbox[ #math.equation(block: true, alt: "y greater than or equal to the fraction 2 over 3 x minus 3")[$y ≥ frac(2, 3) x − 3$] ] Write the inequality shown by the shaded region in the graph with the boundary line #math.equation(block: false, alt: "x plus y equals −4 .")[$x + y = −4 .$] #figure(figph[This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 4), (negative 2, negative 2), and (negative 4, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 4), (negative 2, negative 2), and (negative 4, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.", caption: none) Write the inequality shown by the shaded region in the graph with the boundary line #math.equation(block: false, alt: "x minus 2 y equals 6 .")[$x − 2 y = 6 .$] #figure(figph[This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 3), (2, negative 2), and (6, 0). The line divides the x y-coordinate plane into two halves. The line and the bottom right half are shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 3), (2, negative 2), and (6, 0). The line divides the x y-coordinate plane into two halves. The line and the bottom right half are shaded red to indicate that this is where the solutions of the inequality are.", caption: none) #solutionbox[ #math.equation(block: true, alt: "x minus 2 y greater than or equal to 6")[$x − 2 y ≥ 6$] ] #strong[Graph Linear Inequalities in Two Variables] In the following exercises, graph each linear inequality. Graph the linear inequality #math.equation(block: false, alt: "y greater than the fraction 2 over 5 x minus 4 .")[$y > frac(2, 5) x − 4 .$] Graph the linear inequality #math.equation(block: false, alt: "y less than or equal to minus the fraction 1 over 4 x plus 3 .")[$y ≤ − frac(1, 4) x + 3 .$] #solutionbox[ #figure(figph[This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, 3), (4, 2), and (8, 1). The line divides the x y-coordinate plane into two halves. The bottom left half is shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, 3), (4, 2), and (8, 1). The line divides the x y-coordinate plane into two halves. The bottom left half is shaded red to indicate that this is where the solutions of the inequality are.", caption: none) ] Graph the linear inequality #math.equation(block: false, alt: "x minus y less than or equal to 5 .")[$x − y ≤ 5 .$] Graph the linear inequality #math.equation(block: false, alt: "3 x plus 2 y greater than 10 .")[$3 x + 2 y > 10 .$] #solutionbox[ #figure(figph[This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, 5), (2, 2), and (4, negative 1). The line divides the x y-coordinate plane into two halves. The top right half is shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, 5), (2, 2), and (4, negative 1). The line divides the x y-coordinate plane into two halves. The top right half is shaded red to indicate that this is where the solutions of the inequality are.", caption: none) ] Graph the linear inequality #math.equation(block: false, alt: "y less than or equal to −3 x .")[$y ≤ −3 x .$] Graph the linear inequality #math.equation(block: false, alt: "y less than 6 .")[$y < 6 .$] #solutionbox[ #figure(figph[This figure has the graph of a straight horizontal dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, 6), (1, 6), and (2, 6). The line divides the x y-coordinate plane into two halves. The bottom half is shaded red to indicate that this is where the solutions of the inequality are.], alt: "This figure has the graph of a straight horizontal dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, 6), (1, 6), and (2, 6). The line divides the x y-coordinate plane into two halves. The bottom half is shaded red to indicate that this is where the solutions of the inequality are.", caption: none) ] #strong[Solve Applications using Linear Inequalities in Two Variables] Shanthie needs to earn at least \$500 a week during her summer break to pay for college. She works two jobs. One as a swimming instructor that pays \$10 an hour and the other as an intern in a law office for \$25 hour. How many hours does Shanthie need to work at each job to earn at least \$500 per week? ⓐ Let #emph[x] be the number of hours she works teaching swimming and let #emph[y] be the number of hours she works as an intern. Write an inequality that would model this situation. #linebreak() ⓑ Graph the inequality. #linebreak() ⓒ Find three ordered pairs #math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$] that would be solutions to the inequality. Then, explain what that means for Shanthie. Atsushi he needs to exercise enough to burn 600 calories each day. He prefers to either run or bike and burns 20 calories per minute while running and 15 calories a minute while biking. ⓐ If #emph[x] is the number of minutes that Atsushi runs and #emph[y] is the number minutes he bikes, find the inequality that models the situation. #linebreak() ⓑ Graph the inequality. #linebreak() ⓒ List three solutions to the inequality. What options do the solutions provide Atsushi? #solutionbox[ ⓐ #math.equation(block: false, alt: "20 x plus 15 y greater than or equal to 600")[$20 x + 15 y ≥ 600$] #linebreak() ⓑ #linebreak() #figure(figph[The figure has a straight line graphed on the x y-coordinate plane. The x-axis runs from 0 to 50. The y-axis runs from 0 to 50. The line goes through the points (0, 40) and (30, 0). The line divides the coordinate plane into two halves. The top right half and the line are colored red to indicate that this is the solution set.], alt: "The figure has a straight line graphed on the x y-coordinate plane. The x-axis runs from 0 to 50. The y-axis runs from 0 to 50. The line goes through the points (0, 40) and (30, 0). The line divides the coordinate plane into two halves. The top right half and the line are colored red to indicate that this is the solution set.", caption: none) #linebreak() #linebreak() ⓒ Answers will vary. ] ==== Relations and Functions #strong[Find the Domain and Range of a Relation] In the following exercises, for each relation, ⓐ find the domain of the relation ⓑ find the range of the relation. #math.equation(block: true, alt: "{ open parenthesis 5 , −2 close parenthesis , open parenthesis 5 , −4 close parenthesis , open parenthesis 7 , −6 close parenthesis ,")[$\{ ( 5 , −2 ) , ( 5 , −4 ) , ( 7 , −6 ) ,$] #linebreak() #math.equation(block: true, alt: "open parenthesis 8 , −8 close parenthesis , open parenthesis 9 , −10 close parenthesis }")[$( 8 , −8 ) , ( 9 , −10 ) \}$] #math.equation(block: true, alt: "{ open parenthesis −3 , 7 close parenthesis , open parenthesis −2 , 3 close parenthesis , open parenthesis −1 , 9 close parenthesis ,")[$\{ ( −3 , 7 ) , ( −2 , 3 ) , ( −1 , 9 ) ,$] #linebreak() #math.equation(block: true, alt: "open parenthesis 0 , −3 close parenthesis , open parenthesis −1 , 8 close parenthesis }")[$( 0 , −3 ) , ( −1 , 8 ) \}$] #solutionbox[ ⓐ D: {−3, −2, −1, 0} #linebreak() ⓑ R: {7, 3, 9, −3, 8} ] In the following exercise, use the mapping of the relation to ⓐ list the ordered pairs of the relation ⓑ find the domain of the relation ⓒ find the range of the relation. The mapping below shows the average weight of a child according to age. #figure(figph[This figure shows two table that each have one column. The table on the left has the header “Age (yrs)” and lists the numbers 1, 2, 3, 4, 5, 6, and 7. The table on the right has the header “Weight (pounds)” and lists the numbers 20, 35, 30, 45, 40, 25, and 50. There are arrows starting at numbers in the age table and pointing towards numbers in the weight table. The first arrow goes from 1 to 20. The second arrow goes from 2 to 25. The third arrow goes from 3 to 30. The fourth arrow goes from 4 to 35. The fifth arrow goes from 5 to 40. The sixth arrow goes from 6 to 45. The seventh arrow goes from 7 to 50.], alt: "This figure shows two table that each have one column. The table on the left has the header “Age (yrs)” and lists the numbers 1, 2, 3, 4, 5, 6, and 7. The table on the right has the header “Weight (pounds)” and lists the numbers 20, 35, 30, 45, 40, 25, and 50. There are arrows starting at numbers in the age table and pointing towards numbers in the weight table. The first arrow goes from 1 to 20. The second arrow goes from 2 to 25. The third arrow goes from 3 to 30. The fourth arrow goes from 4 to 35. The fifth arrow goes from 5 to 40. The sixth arrow goes from 6 to 45. The seventh arrow goes from 7 to 50.", caption: none) In the following exercise, use the graph of the relation to ⓐ list the ordered pairs of the relation ⓑ find the domain of the relation ⓒ find the range of the relation. #figure(figph[The figure shows the graph of some points on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The points (negative 3, 1), (negative 2, negative 1), (negative 2, negative 3), (0, negative 1), (0, 4), and (4, 3).], alt: "The figure shows the graph of some points on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The points (negative 3, 1), (negative 2, negative 1), (negative 2, negative 3), (0, negative 1), (0, 4), and (4, 3).", caption: none) #solutionbox[ ⓐ (4, 3), (−2, −3), (−2, −1), (−3, 1), (0, −1), (0, 4), #linebreak() ⓑ D: {−3, −2, 0, 4} #linebreak() ⓒ R: {−3, −1, 1, 3, 4} ] #strong[Determine if a Relation is a Function] In the following exercises, use the set of ordered pairs to ⓐ determine whether the relation is a function ⓑ find the domain of the relation ⓒ find the range of the relation. #math.equation(block: true, alt: "{ open parenthesis 9 , −5 close parenthesis , open parenthesis 4 , −3 close parenthesis , open parenthesis 1 , −1 close parenthesis ,")[$\{ ( 9 , −5 ) , ( 4 , −3 ) , ( 1 , −1 ) ,$] #linebreak() #math.equation(block: true, alt: "open parenthesis 0 , 0 close parenthesis , open parenthesis 1 , 1 close parenthesis , open parenthesis 4 , 3 close parenthesis , open parenthesis 9 , 5 close parenthesis }")[$( 0 , 0 ) , ( 1 , 1 ) , ( 4 , 3 ) , ( 9 , 5 ) \}$] #math.equation(block: true, alt: "{ open parenthesis −3 , 27 close parenthesis , open parenthesis −2 , 8 close parenthesis , open parenthesis −1 , 1 close parenthesis ,")[$\{ ( −3 , 27 ) , ( −2 , 8 ) , ( −1 , 1 ) ,$] #linebreak() #math.equation(block: true, alt: "open parenthesis 0 , 0 close parenthesis , open parenthesis 1 , 1 close parenthesis , open parenthesis 2 , 8 close parenthesis , open parenthesis 3 , 27 close parenthesis }")[$( 0 , 0 ) , ( 1 , 1 ) , ( 2 , 8 ) , ( 3 , 27 ) \}$] #solutionbox[ ⓐ yes ⓑ {−3, −2, −1, 0, 1, 2, 3} #linebreak() ⓒ {0, 1, 8, 27} ] In the following exercises, use the mapping to ⓐ determine whether the relation is a function ⓑ find the domain of the function ⓒ find the range of the function. #figure(figph[This figure shows two table that each have one column. The table on the left has the header “x” and lists the numbers negative 3, negative 2, negative 1, 0, 1, 2, and 3. The table on the right has the header “x to the fourth power” and lists the numbers 0, 1, 16, and 81. There are arrows starting at numbers in the x table and pointing towards numbers in the x to the fourth power table. The first arrow goes from negative 3 to 81. The second arrow goes from negative 2 to 16. The third arrow goes from negative 1 to 1. The fourth arrow goes from 0 to 0. The fifth arrow goes from 1 to 1. The sixth arrow goes from 2 to 16. The seventh arrow goes from 3 to 81.], alt: "This figure shows two table that each have one column. The table on the left has the header “x” and lists the numbers negative 3, negative 2, negative 1, 0, 1, 2, and 3. The table on the right has the header “x to the fourth power” and lists the numbers 0, 1, 16, and 81. There are arrows starting at numbers in the x table and pointing towards numbers in the x to the fourth power table. The first arrow goes from negative 3 to 81. The second arrow goes from negative 2 to 16. The third arrow goes from negative 1 to 1. The fourth arrow goes from 0 to 0. The fifth arrow goes from 1 to 1. The sixth arrow goes from 2 to 16. The seventh arrow goes from 3 to 81.", caption: none) #figure(figph[This figure shows two table that each have one column. The table on the left has the header “x” and lists the numbers negative 3, negative 2, negative 1, 0, 1, 2, and 3. The table on the right has the header “x to the fifth power” and lists the numbers 0, 1, 32, 243, negative 1, negative 32, and negative 243. There are arrows starting at numbers in the x table and pointing towards numbers in the x to the fifth power table. The first arrow goes from negative 3 to negative 243. The second arrow goes from negative 2 to negative 32. The third arrow goes from negative 1 to 1. The fourth arrow goes from 0 to 0. The fifth arrow goes from 1 to 1. The sixth arrow goes from 2 to 32. The seventh arrow goes from 3 to 243.], alt: "This figure shows two table that each have one column. The table on the left has the header “x” and lists the numbers negative 3, negative 2, negative 1, 0, 1, 2, and 3. The table on the right has the header “x to the fifth power” and lists the numbers 0, 1, 32, 243, negative 1, negative 32, and negative 243. There are arrows starting at numbers in the x table and pointing towards numbers in the x to the fifth power table. The first arrow goes from negative 3 to negative 243. The second arrow goes from negative 2 to negative 32. The third arrow goes from negative 1 to 1. The fourth arrow goes from 0 to 0. The fifth arrow goes from 1 to 1. The sixth arrow goes from 2 to 32. The seventh arrow goes from 3 to 243.", caption: none) #solutionbox[ ⓐ yes #linebreak() ⓑ {−3, −2, −1, 0, 1, 2, 3} #linebreak() ⓒ {−243, −32, −1, 0, 1, 32, 243} ] In the following exercises, determine whether each equation is a function. #math.equation(block: true, alt: "2 x plus y equals −3")[$2 x + y = −3$] #math.equation(block: true, alt: "y equals x squared")[$y = x^(2)$] #solutionbox[ yes ] #math.equation(block: true, alt: "y equals 3 x minus 5")[$y = 3 x − 5$] #math.equation(block: true, alt: "y equals x cubed")[$y = x^(3)$] #solutionbox[ yes ] #math.equation(block: true, alt: "2 x plus y squared equals 4")[$2 x + y^(2) = 4$] #strong[Find the Value of a Function] In the following exercises, evaluate the function: ⓐ #math.equation(block: false, alt: "f open parenthesis −2 close parenthesis")[$f ( −2 )$] ⓑ #math.equation(block: false, alt: "f open parenthesis 3 close parenthesis")[$f ( 3 )$] ⓒ #math.equation(block: false, alt: "f open parenthesis a close parenthesis .")[$f ( a ) .$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals 3 x minus 4")[$f ( x ) = 3 x − 4$] #solutionbox[ ⓐ #math.equation(block: false, alt: "f open parenthesis −2 close parenthesis equals −10")[$f ( −2 ) = −10$] ⓑ #math.equation(block: false, alt: "f open parenthesis 3 close parenthesis equals 5")[$f ( 3 ) = 5$] ⓒ #math.equation(block: false, alt: "f open parenthesis a close parenthesis equals 3 a minus 4")[$f ( a ) = 3 a − 4$] ] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals −2 x plus 5")[$f ( x ) = −2 x + 5$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals x squared minus 5 x plus 6")[$f ( x ) = x^(2) − 5 x + 6$] #solutionbox[ ⓐ #math.equation(block: false, alt: "f open parenthesis −2 close parenthesis equals 20")[$f ( −2 ) = 20$] ⓑ #math.equation(block: false, alt: "f open parenthesis 3 close parenthesis equals 0")[$f ( 3 ) = 0$] ⓒ #math.equation(block: false, alt: "f open parenthesis a close parenthesis equals a squared minus 5 a plus 6")[$f ( a ) = a^(2) − 5 a + 6$] ] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals 3 x squared minus 2 x plus 1")[$f ( x ) = 3 x^(2) − 2 x + 1$] In the following exercises, evaluate the function. #math.equation(block: true, alt: "g open parenthesis x close parenthesis equals 3 x squared minus 5 x ;")[$g ( x ) = 3 x^(2) − 5 x ;$] #math.equation(block: true, alt: "g open parenthesis 2 close parenthesis")[$g ( 2 )$] #solutionbox[ 2 ] #math.equation(block: true, alt: "F open parenthesis x close parenthesis equals 2 x squared minus 3 x plus 1 ;")[$F ( x ) = 2 x^(2) − 3 x + 1 ;$] #linebreak() #math.equation(block: true, alt: "F open parenthesis −1 close parenthesis")[$F ( −1 )$] #math.equation(block: true, alt: "h open parenthesis t close parenthesis equals 4 | t minus 1 | plus 2 ;")[$h ( t ) = 4 | t − 1 | + 2 ;$] #math.equation(block: true, alt: "h open parenthesis −3 close parenthesis")[$h ( −3 )$] #solutionbox[ 18 ] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the fraction x plus 2 over x minus 1 ;")[$f ( x ) = frac(x + 2, x − 1) ;$] #math.equation(block: true, alt: "f open parenthesis 3 close parenthesis")[$f ( 3 )$] ==== Graphs of Functions #strong[Use the Vertical line Test] In the following exercises, determine whether each graph is the graph of a function. #figure(figph[The figure has a square function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The parabola goes through the points (negative 2, 5), (negative 1, 2), (0, 1), (1, 2), and (2, 5). The lowest point on the graph is (0, 1).], alt: "The figure has a square function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The parabola goes through the points (negative 2, 5), (negative 1, 2), (0, 1), (1, 2), and (2, 5). The lowest point on the graph is (0, 1).", caption: none) #solutionbox[ yes ] #figure(figph[The figure has an s-shaped function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curve goes through the points (negative 1, negative 1), (0, 0), and (1, 1).], alt: "The figure has an s-shaped function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curve goes through the points (negative 1, negative 1), (0, 0), and (1, 1).", caption: none) #figure(figph[The figure has a circle graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The circle goes through the points (negative 5, 0), (5, 0), (0, negative 5), and (0, 5).], alt: "The figure has a circle graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The circle goes through the points (negative 5, 0), (5, 0), (0, negative 5), and (0, 5).", caption: none) #solutionbox[ no ] #figure(figph[The figure has a parabola opening to the right graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The parabola goes through the points (negative 2, 0), (negative 1, 1), (negative 1, negative 1), (2, 2), and (2, negative 2). The left-most point on the graph is (negative 2, 0).], alt: "The figure has a parabola opening to the right graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The parabola goes through the points (negative 2, 0), (negative 1, 1), (negative 1, negative 1), (2, 2), and (2, negative 2). The left-most point on the graph is (negative 2, 0).", caption: none) #figure(figph[The figure has a cube function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line goes through the points (negative 1, negative 1), (0, 0), and (1, 1).], alt: "The figure has a cube function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line goes through the points (negative 1, negative 1), (0, 0), and (1, 1).", caption: none) #solutionbox[ yes ] #figure(figph[The figure has two curved lines graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line on the left goes through the points (negative 3, 0), (negative 4, 2), and (negative 4, negative 2). The curved line on the right goes through the points (3, 0), (4, 2), and (4, negative 2).], alt: "The figure has two curved lines graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line on the left goes through the points (negative 3, 0), (negative 4, 2), and (negative 4, negative 2). The curved line on the right goes through the points (3, 0), (4, 2), and (4, negative 2).", caption: none) #figure(figph[The figure has a sideways absolute value function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The line bends at the point (0, negative 1) and goes to the right. The line goes through the points (1, 0), (1, negative 2), (2, 1), and (2, negative 3).], alt: "The figure has a sideways absolute value function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The line bends at the point (0, negative 1) and goes to the right. The line goes through the points (1, 0), (1, negative 2), (2, 1), and (2, negative 3).", caption: none) #solutionbox[ no ] #strong[Identify Graphs of Basic Functions] In the following exercises, ⓐ graph each function ⓑ state its domain and range. Write the domain and range in interval notation. #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals 5 x plus 1")[$f ( x ) = 5 x + 1$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals −4 x minus 2")[$f ( x ) = −4 x − 2$] #solutionbox[ ⓐ #linebreak() #figure(figph[The figure has a linear function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The line goes through the points (negative 2, 6), (negative 1, 2), and (0, negative 2).], alt: "The figure has a linear function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The line goes through the points (negative 2, 6), (negative 1, 2), and (0, negative 2).", caption: none) #linebreak() #linebreak() ⓑ D: (-∞,∞), R: (-∞,∞) ] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the fraction 2 over 3 x minus 1")[$f ( x ) = frac(2, 3) x − 1$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals −6")[$f ( x ) = −6$] #solutionbox[ ⓐ #linebreak() #figure(figph[The figure has a constant function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 8 to 4. The line goes through the points (0, negative 6), (1, negative 6), and (2, negative 6).], alt: "The figure has a constant function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 8 to 4. The line goes through the points (0, negative 6), (1, negative 6), and (2, negative 6).", caption: none) #linebreak() #linebreak() ⓑ D: (-∞,∞), R: (6) ] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals 2 x")[$f ( x ) = 2 x$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals 3 x squared")[$f ( x ) = 3 x^(2)$] #solutionbox[ ⓐ #linebreak() #figure(figph[The figure has a square function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The parabola goes through the points (negative 1, 3), (0, 0), and (1, 3). The lowest point on the graph is (0, 0).], alt: "The figure has a square function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The parabola goes through the points (negative 1, 3), (0, 0), and (1, 3). The lowest point on the graph is (0, 0).", caption: none) #linebreak() #linebreak() ⓑ D: (-∞,∞), R: \[0,∞) ] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals minus the fraction 1 over 2 x squared")[$f ( x ) = − frac(1, 2) x^(2)$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals x squared plus 2")[$f ( x ) = x^(2) + 2$] #solutionbox[ ⓐ #linebreak() #figure(figph[The figure has a square function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 4 to 8. The parabola goes through the points (negative 2, 6), (negative 1, 3), (0, 2), (1, 3), and (2, 6). The lowest point on the graph is (0, 2).], alt: "The figure has a square function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 4 to 8. The parabola goes through the points (negative 2, 6), (negative 1, 3), (0, 2), (1, 3), and (2, 6). The lowest point on the graph is (0, 2).", caption: none) #linebreak() #linebreak() ⓑ D: (-∞,∞), R: \[2,∞) ] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals x cubed minus 2")[$f ( x ) = x^(3) − 2$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the square root of x plus 2")[$f ( x ) = sqrt(x + 2)$] #solutionbox[ ⓐ #linebreak() #figure(figph[The figure has a square root function graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 8. The y-axis runs from negative 2 to 10. The half-line starts at the point (negative 2, 0) and goes through the points (negative 1, 1) and (2, 2).], alt: "The figure has a square root function graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 8. The y-axis runs from negative 2 to 10. The half-line starts at the point (negative 2, 0) and goes through the points (negative 1, 1) and (2, 2).", caption: none) #linebreak() #linebreak() ⓑ D: \[#math.equation(block: false, alt: "−2 ,")[$−2 ,$] ∞), R: \[0,∞) ] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals − | x |")[$f ( x ) = "−" | x |$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals | x | plus 1")[$f ( x ) = | x | + 1$] #solutionbox[ ⓐ #linebreak() #figure(figph[The figure has an absolute value function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The vertex is at the point (0, 1). The line goes through the points (negative 1, 2) and (1, 2).], alt: "The figure has an absolute value function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The vertex is at the point (0, 1). The line goes through the points (negative 1, 2) and (1, 2).", caption: none) #linebreak() #linebreak() ⓑ D: (-∞,∞), R: \[1,∞) ] #strong[Read Information from a Graph of a Function] In the following exercises, use the graph of the function to find its domain and range. Write the domain and range in interval notation #figure(figph[The figure has a square root function graphed on the x y-coordinate plane. The x-axis runs from 0 to 10. The y-axis runs from 0 to 10. The half-line starts at the point (1, 0) and goes through the points (2, 1) and (5, 2).], alt: "The figure has a square root function graphed on the x y-coordinate plane. The x-axis runs from 0 to 10. The y-axis runs from 0 to 10. The half-line starts at the point (1, 0) and goes through the points (2, 1) and (5, 2).", caption: none) #figure(figph[The figure has an absolute value function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The vertex is at the point (0, 2). The line goes through the points (negative 1, 3) and (1, 3).], alt: "The figure has an absolute value function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The vertex is at the point (0, 2). The line goes through the points (negative 1, 3) and (1, 3).", caption: none) #solutionbox[ D: (-∞,∞), R: \[2,∞) ] #figure(figph[The figure has a cubic function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line goes through the points (negative 2, negative 4), (0, 0), and (2, 4).], alt: "The figure has a cubic function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line goes through the points (negative 2, negative 4), (0, 0), and (2, 4).", caption: none) In the following exercises, use the graph of the function to find the indicated values. #figure(figph[This figure has a wavy curved line graphed on the x y-coordinate plane. The x-axis runs from negative 2 times pi to 2 times pi. The y-axis runs from negative 6 to 6. The curved line segment goes through the points (negative 2 times pi, 0), (negative 3 divided by 2 times pi, 1), (negative pi, 0), (negative 1 divided by 2 times pi, negative 1), (0, 0), (1 divided by 2 times pi, 1), (pi, 0), (3 divided by 2 times pi, negative 1), and (2 times pi, 0). The points (negative 3 divided by 2 times pi, 1) and (1 divided by 2 times pi, 1) are the highest points on the graph. The points (negative 1 divided by 2 times pi, negative 1) and (3 divided by 2 times pi, negative 1) are the lowest points on the graph. The pattern extends infinitely to the left and right.], alt: "This figure has a wavy curved line graphed on the x y-coordinate plane. The x-axis runs from negative 2 times pi to 2 times pi. The y-axis runs from negative 6 to 6. The curved line segment goes through the points (negative 2 times pi, 0), (negative 3 divided by 2 times pi, 1), (negative pi, 0), (negative 1 divided by 2 times pi, negative 1), (0, 0), (1 divided by 2 times pi, 1), (pi, 0), (3 divided by 2 times pi, negative 1), and (2 times pi, 0). The points (negative 3 divided by 2 times pi, 1) and (1 divided by 2 times pi, 1) are the highest points on the graph. The points (negative 1 divided by 2 times pi, negative 1) and (3 divided by 2 times pi, negative 1) are the lowest points on the graph. The pattern extends infinitely to the left and right.", caption: none) ⓐ Find #math.equation(block: false, alt: "f open parenthesis 0 close parenthesis .")[$f ( 0 ) .$] #linebreak() ⓑ Find #math.equation(block: false, alt: "f open parenthesis the fraction 1 over 2 π close parenthesis .")[$f ( frac(1, 2) π ) .$] #linebreak() ⓒ Find #math.equation(block: false, alt: "f open parenthesis minus the fraction 3 over 2 π close parenthesis .")[$f ( − frac(3, 2) π ) .$] #linebreak() ⓓ Find the values for #emph[x] when #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 0 .")[$f ( x ) = 0 .$] #linebreak() ⓔ Find the #math.equation(block: false, alt: "x")[$x$]-intercepts. #linebreak() ⓕ Find the #math.equation(block: false, alt: "y")[$y$]-intercepts. #linebreak() ⓖ Find the domain. Write it in interval notation. #linebreak() ⓗ Find the range. Write it in interval notation. #solutionbox[ ⓐ #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 0")[$f ( x ) = 0$] ⓑ #math.equation(block: false, alt: "f the fraction π over 2 equals 1")[$f frac(π, 2) = 1$] #linebreak() ⓒ #math.equation(block: false, alt: "f minus the fraction 3 π over 2 equals 1")[$f − frac(3 π, 2) = 1$] ⓓ #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 0")[$f ( x ) = 0$] for #math.equation(block: false, alt: "x equals −2 π , − π , 0 , π , 2 π")[$x = −2 π , "−" π , 0 , π , 2 π$] #linebreak() ⓔ #math.equation(block: false, alt: "open parenthesis −2 π , 0 close parenthesis ,")[$( −2 π , 0 ) ,$] #math.equation(block: false, alt: "open parenthesis − π , 0 close parenthesis ,")[$( "−" π , 0 ) ,$] #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis ,")[$( 0 , 0 ) ,$] #math.equation(block: false, alt: "open parenthesis π , 0 close parenthesis ,")[$( π , 0 ) ,$] #math.equation(block: false, alt: "open parenthesis 2 π , 0 close parenthesis")[$( 2 π , 0 )$] ⓕ #math.equation(block: false, alt: "0 , 0")[$0 , 0$] #linebreak() ⓖ #math.equation(block: false, alt: "minus ∞ , ∞")[$− ∞ , ∞$] ⓗ #math.equation(block: false, alt: "[ −1 , 1 ]")[$[ −1 , 1 ]$] ] #figure(figph[The figure has a half-circle graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line segment starts at the point (negative 2, 0). The line goes through the point (0, 2) and ends at the point (2, 0). The point (0, 2) is the highest point on the graph.], alt: "The figure has a half-circle graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line segment starts at the point (negative 2, 0). The line goes through the point (0, 2) and ends at the point (2, 0). The point (0, 2) is the highest point on the graph.", caption: none) ⓐ Find #math.equation(block: false, alt: "f open parenthesis 0 close parenthesis .")[$f ( 0 ) .$] #linebreak() ⓑ Find the values for #emph[x] when #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 0 .")[$f ( x ) = 0 .$] #linebreak() ⓒ Find the #math.equation(block: false, alt: "x")[$x$]-intercepts. #linebreak() ⓓ Find the #math.equation(block: false, alt: "y")[$y$]-intercepts. #linebreak() ⓔ Find the domain. Write it in interval notation. #linebreak() ⓕ Find the range. Write it in interval notation. === Practice Test Plot each point in a rectangular coordinate system. ⓐ #math.equation(block: false, alt: "open parenthesis 2 , 5 close parenthesis")[$( 2 , 5 )$] #linebreak() ⓑ #math.equation(block: false, alt: "open parenthesis −1 , −3 close parenthesis")[$( −1 , −3 )$] #linebreak() ⓒ #math.equation(block: false, alt: "open parenthesis 0 , 2 close parenthesis")[$( 0 , 2 )$] #linebreak() ⓓ #math.equation(block: false, alt: "open parenthesis −4 , the fraction 3 over 2 close parenthesis")[$( −4 , frac(3, 2) )$] #linebreak() ⓔ #math.equation(block: false, alt: "open parenthesis 5 , 0 close parenthesis")[$( 5 , 0 )$] #solutionbox[ #figure(figph[This figure shows points plotted on the x y-coordinate plane. The x and y axes run from negative 10 to 10. The point labeled a is 2 units to the right of the origin and 5 units above the origin and is located in quadrant I. The point labeled b is 1 unit to the left of the origin and 3 units below the origin and is located in quadrant III. The point labeled c is 2 units above the origin and is located on the y-axis. The point labeled d is 4 units to the left of the origin and 1.5 units above the origin and is located in quadrant II. The point labeled e is 5 units to the right of the origin and is located on the x-axis.], alt: "This figure shows points plotted on the x y-coordinate plane. The x and y axes run from negative 10 to 10. The point labeled a is 2 units to the right of the origin and 5 units above the origin and is located in quadrant I. The point labeled b is 1 unit to the left of the origin and 3 units below the origin and is located in quadrant III. The point labeled c is 2 units above the origin and is located on the y-axis. The point labeled d is 4 units to the left of the origin and 1.5 units above the origin and is located in quadrant II. The point labeled e is 5 units to the right of the origin and is located on the x-axis.", caption: none) ] Which of the given ordered pairs are solutions to the equation #math.equation(block: false, alt: "3 x minus y equals 6 ?")[$3 x − y = 6 ?$] ⓐ #math.equation(block: false, alt: "open parenthesis 3 , 3 close parenthesis")[$( 3 , 3 )$] ⓑ #math.equation(block: false, alt: "open parenthesis 2 , 0 close parenthesis")[$( 2 , 0 )$] ⓒ #math.equation(block: false, alt: "open parenthesis 4 , −6 close parenthesis")[$( 4 , −6 )$] Find the slope of each line shown. ⓐ #linebreak() #figure(figph[The figure has a straight line graphed on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (negative 5, 2) (0, negative 1), and (5, negative 4).], alt: "The figure has a straight line graphed on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (negative 5, 2) (0, negative 1), and (5, negative 4).", caption: none) #linebreak() #linebreak() ⓑ #linebreak() #figure(figph[The figure has a straight vertical line graphed on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (2, 0) (2, negative 1), and (2, 1).], alt: "The figure has a straight vertical line graphed on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (2, 0) (2, negative 1), and (2, 1).", caption: none) #solutionbox[ ⓐ #math.equation(block: false, alt: "minus the fraction 3 over 5")[$− frac(3, 5)$] ⓑ undefined ] Find the slope of the line between the points #math.equation(block: false, alt: "open parenthesis 5 , 2 close parenthesis")[$( 5 , 2 )$] and #math.equation(block: false, alt: "open parenthesis −1 , −4 close parenthesis .")[$( −1 , −4 ) .$] Graph the line with slope #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$] containing the point #math.equation(block: false, alt: "open parenthesis −3 , −4 close parenthesis .")[$( −3 , −4 ) .$] #solutionbox[ #figure(figph[The figure has a straight line graphed on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (negative 3, negative 4) (negative 1, negative 3), and (1, negative 2).], alt: "The figure has a straight line graphed on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (negative 3, negative 4) (negative 1, negative 3), and (1, negative 2).", caption: none) ] Find the intercepts of #math.equation(block: false, alt: "4 x plus 2 y equals −8")[$4 x + 2 y = −8$] and graph. #strong[Graph the line for each of the following equations.] #math.equation(block: true, alt: "y equals the fraction 5 over 3 x minus 1")[$y = frac(5, 3) x − 1$] #solutionbox[ #figure(figph[The figure has a straight line graphed on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (negative 3, negative 6) (0, negative 1), and (3, 4).], alt: "The figure has a straight line graphed on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (negative 3, negative 6) (0, negative 1), and (3, 4).", caption: none) ] #math.equation(block: true, alt: "y equals − x")[$y = "−" x$] #math.equation(block: true, alt: "y equals 2")[$y = 2$] #solutionbox[ #figure(figph[The figure has a straight horizontal line graphed on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (negative 1, 2) (0, 2), and (1, 2).], alt: "The figure has a straight horizontal line graphed on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (negative 1, 2) (0, 2), and (1, 2).", caption: none) ] Find the equation of each line. Write the equation in slope-intercept form. slope #math.equation(block: false, alt: "minus the fraction 3 over 4")[$− frac(3, 4)$] and #math.equation(block: false, alt: "y")[$y$]-intercept #math.equation(block: false, alt: "open parenthesis 0 , −2 close parenthesis")[$( 0 , −2 )$] #math.equation(block: false, alt: "m equals 2 ,")[$m = 2 ,$] point #math.equation(block: false, alt: "open parenthesis −3 , −1 close parenthesis")[$( −3 , −1 )$] #solutionbox[ #math.equation(block: true, alt: "y equals 2 x plus 5")[$y = 2 x + 5$] ] containing #math.equation(block: false, alt: "open parenthesis 10 , 1 close parenthesis")[$( 10 , 1 )$] and #math.equation(block: false, alt: "open parenthesis 6 , −1 close parenthesis")[$( 6 , −1 )$] perpendicular to the line #math.equation(block: false, alt: "y equals the fraction 5 over 4 x plus 2 ,")[$y = frac(5, 4) x + 2 ,$] containing the point #math.equation(block: false, alt: "open parenthesis −10 , 3 close parenthesis")[$( −10 , 3 )$] #solutionbox[ #math.equation(block: true, alt: "y equals minus the fraction 4 over 5 x minus 5")[$y = − frac(4, 5) x − 5$] ] Write the inequality shown by the graph with the boundary line #math.equation(block: false, alt: "y equals − x minus 3 .")[$y = "−" x − 3 .$] #figure(figph[The figure has a straight line graphed on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (negative 3, 0), (0, negative 3), and (1, negative 4). The line divides the coordinate plane into two halves. The bottom left half and the line are colored red to indicate that this is the solution set.], alt: "The figure has a straight line graphed on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (negative 3, 0), (0, negative 3), and (1, negative 4). The line divides the coordinate plane into two halves. The bottom left half and the line are colored red to indicate that this is the solution set.", caption: none) Graph each linear inequality. #math.equation(block: true, alt: "y greater than the fraction 3 over 2 x plus 5")[$y > frac(3, 2) x + 5$] #solutionbox[ #figure(figph[The figure has a straight dashed line graphed on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (negative 2, 2), (0, 5), and (2, 8). The line divides the coordinate plane into two halves. The top left half is colored red to indicate that this is the solution set.], alt: "The figure has a straight dashed line graphed on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (negative 2, 2), (0, 5), and (2, 8). The line divides the coordinate plane into two halves. The top left half is colored red to indicate that this is the solution set.", caption: none) ] #math.equation(block: true, alt: "x minus y greater than or equal to −4")[$x − y ≥ −4$] #math.equation(block: true, alt: "y less than or equal to −5 x")[$y ≤ −5 x$] #solutionbox[ #figure(figph[The figure has a straight line graphed on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (negative 1, 5), (0, 0), and (1, negative 5). The line divides the coordinate plane into two halves. The bottom left half and the line are colored red to indicate that this is the solution set.], alt: "The figure has a straight line graphed on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (negative 1, 5), (0, 0), and (1, negative 5). The line divides the coordinate plane into two halves. The bottom left half and the line are colored red to indicate that this is the solution set.", caption: none) ] Hiro works two part time jobs in order to earn enough money to meet her obligations of at least \$450 a week. Her job at the mall pays \$10 an hour and her administrative assistant job on campus pays \$15 an hour. How many hours does Hiro need to work at each job to earn at least \$450? ⓐ Let #emph[x] be the number of hours she works at the mall and let #emph[y] be the number of hours she works as administrative assistant. Write an inequality that would model this situation. #linebreak() ⓑ Graph the inequality . #linebreak() ⓒ Find three ordered pairs#math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$] that would be solutions to the inequality. Then explain what that means for Hiro. Use the set of ordered pairs to ⓐ determine whether the relation is a function, ⓑ find the domain of the relation, and ⓒ find the range of the relation. #math.equation(block: true, alt: "{ open parenthesis −3 , 27 close parenthesis , open parenthesis −2 , 8 close parenthesis , open parenthesis −1 , 1 close parenthesis , open parenthesis 0 , 0 close parenthesis ,")[$\{ ( −3 , 27 ) , ( −2 , 8 ) , ( −1 , 1 ) , ( 0 , 0 ) ,$] #linebreak() #math.equation(block: true, alt: "open parenthesis 1 , 1 close parenthesis , open parenthesis 2 , 8 close parenthesis , open parenthesis 3 , 27 close parenthesis }")[$( 1 , 1 ) , ( 2 , 8 ) , ( 3 , 27 ) \}$] #solutionbox[ ⓐ yes ⓑ #math.equation(block: false, alt: "{ −3 , −2 , −1 , 0 , 1 , 2 , 3 }")[$\{ −3 , −2 , −1 , 0 , 1 , 2 , 3 \}$] ⓒ {0, 1, 8, 27} ] Evaluate the function: ⓐ #math.equation(block: false, alt: "f open parenthesis −1 close parenthesis")[$f ( −1 )$] ⓑ #math.equation(block: false, alt: "f open parenthesis 2 close parenthesis")[$f ( 2 )$] ⓒ #math.equation(block: false, alt: "f open parenthesis c close parenthesis .")[$f ( c ) .$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals 4 x squared minus 2 x minus 3")[$f ( x ) = 4 x^(2) − 2 x − 3$] For #math.equation(block: false, alt: "h open parenthesis y close parenthesis equals 3 | y minus 1 | minus 3 ,")[$h ( y ) = 3 | y − 1 | − 3 ,$] evaluate #math.equation(block: false, alt: "h open parenthesis −4 close parenthesis .")[$h ( −4 ) .$] #solutionbox[ 12 ] Determine whether the graph is the graph of a function. Explain your answer. #figure(figph[The figure has a cube function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line goes through the points (negative 1, 1), (0, 2), and (1, 3).], alt: "The figure has a cube function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line goes through the points (negative 1, 1), (0, 2), and (1, 3).", caption: none) In the following exercises, ⓐ graph each function ⓑ state its domain and range. #linebreak() Write the domain and range in interval notation. #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals x squared plus 1")[$f ( x ) = x^(2) + 1$] #solutionbox[ ⓐ #linebreak() #figure(figph[The figure has a square function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The parabola goes through the points (negative 2, 5), (negative 1, 2), (0, 1), (1, 2), and (2, 5). The lowest point on the graph is (0, 1).], alt: "The figure has a square function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The parabola goes through the points (negative 2, 5), (negative 1, 2), (0, 1), (1, 2), and (2, 5). The lowest point on the graph is (0, 1).", caption: none) #linebreak() #linebreak() ⓑ D: (-∞,∞), R: \[1,∞) ] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the square root of x plus 1")[$f ( x ) = sqrt(x + 1)$] #figure(figph[The figure has a square function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The parabola goes through the points (negative 2, 0), (negative 1, negative 3), (0, negative 4), (1, negative 3), and (2, 0). The lowest point on the graph is (0, negative 4).], alt: "The figure has a square function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The parabola goes through the points (negative 2, 0), (negative 1, negative 3), (0, negative 4), (1, negative 3), and (2, 0). The lowest point on the graph is (0, negative 4).", caption: none) ⓐ Find the #math.equation(block: false, alt: "x")[$x$]-intercepts. #linebreak() ⓑ Find the #math.equation(block: false, alt: "y")[$y$]-intercepts. #linebreak() ⓒ Find #math.equation(block: false, alt: "f open parenthesis −1 close parenthesis .")[$f ( −1 ) .$] #linebreak() ⓓ Find #math.equation(block: false, alt: "f open parenthesis 1 close parenthesis .")[$f ( 1 ) .$] #linebreak() ⓔ Find the domain. Write it in interval notation. #linebreak() ⓕ Find the range. Write it in interval notation. #solutionbox[ ⓐ #math.equation(block: false, alt: "x equals −2 , 2")[$x = −2 , 2$] ⓑ #math.equation(block: false, alt: "y equals −4")[$y = −4$] #linebreak() ⓒ #math.equation(block: false, alt: "f open parenthesis −1 close parenthesis equals −3")[$f ( −1 ) = −3$] ⓓ #math.equation(block: false, alt: "f open parenthesis 1 close parenthesis equals −3")[$f ( 1 ) = −3$] #linebreak() ⓔ D: (-∞,∞) ⓕ R: \[#math.equation(block: false, alt: "−4 ,")[$−4 ,$] ∞) ]