#set document(title: "1.3 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")) == 1.3#h(0.6em)Graphs of Functions === Reading Function Values from a Graph The Dow-Jones Industrial Average (DJIA) gives the average of the stock prices of 30 major companies. The graph below shows the DJIA as a function of time during the stock market correction of October 1987. The DJIA is thus #math.equation(block: false, alt: "f open parenthesis t close parenthesis")[$f ( t )$], recorded at noon on day #math.equation(block: false, alt: "t")[$t$] of October. #figure(figph[DJIA in October 1987], alt: "DJIA in October 1987", caption: none) The values of the input variable, time, are displayed on the horizontal axis, and the values of the output variable, DJIA, are displayed on the vertical axis. There is no formula that gives the DJIA for a particular day; but it is still a function, defined by its graph. The value of #math.equation(block: false, alt: "f open parenthesis t close parenthesis")[$f ( t )$] is specified by the vertical coordinate of the point with the given #math.equation(block: false, alt: "t")[$t$]-coordinate. #examplebox("Example 1")[][ + The coordinates of point #math.equation(block: false, alt: "P")[$P$] on the DJIA graph are #math.equation(block: false, alt: "open parenthesis 15 , 2412 close parenthesis")[$( 15 , 2412 )$]. What do the coordinates tell you about the function #math.equation(block: false, alt: "f")[$f$]? + If the DJIA was 1726 at noon on October 20, what can you say about the graph of #math.equation(block: false, alt: "f")[$f$]? #solutionbox[ + The coordinates of point #math.equation(block: false, alt: "P")[$P$] tell us that #math.equation(block: false, alt: "f open parenthesis 15 close parenthesis equals 2412")[$f ( 15 ) = 2412$], so the DJIA was 2412 at noon on October 15. + We can say that #math.equation(block: false, alt: "f open parenthesis 20 close parenthesis equals 1726")[$f ( 20 ) = 1726$], so the point #math.equation(block: false, alt: "open parenthesis 20 , 1726 close parenthesis")[$( 20 , 1726 )$] lies on the graph of #math.equation(block: false, alt: "f")[$f$]. This point is labeled #math.equation(block: false, alt: "Q")[$Q$] in the figure above. ] ] Thus, the coordinates of each point on the graph of the function represent a pair of corresponding values of the two variables. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Graph of a Function] The point #math.equation(block: false, alt: "open parenthesis a , b close parenthesis")[$( a , b )$] lies on the graph of the function #math.equation(block: false, alt: "f")[$f$] if and only if #math.equation(block: false, alt: "f open parenthesis a close parenthesis equals b")[$f ( a ) = b$]. ] Write an equation that says that the point #math.equation(block: false, alt: "open parenthesis 8 , 17 close parenthesis")[$( 8 , 17 )$] lies on the graph of #math.equation(block: false, alt: "g")[$g$]. \_\_\_\_\_ #solutionbox[ #math.equation(block: true, alt: "g open parenthesis 8 close parenthesis equals 17")[$g ( 8 ) = 17$] ] Which equation says that the point #math.equation(block: false, alt: "open parenthesis 8 , 17 close parenthesis")[$( 8 , 17 )$] lies on the graph of #math.equation(block: false, alt: "g")[$g$]? + #math.equation(block: false, alt: "8 plus 17 equals g")[$8 + 17 = g$] + #math.equation(block: false, alt: "g plus 8 equals 17")[$g + 8 = 17$] + #math.equation(block: false, alt: "g open parenthesis 8 close parenthesis equals 17")[$g ( 8 ) = 17$] + #math.equation(block: false, alt: "8 equals the fraction 17 over g")[$8 = display(frac(17, g))$] The water level in Lake Huron alters unpredictably over time. The graph below gives the average water level, #math.equation(block: false, alt: "L open parenthesis t close parenthesis")[$L ( t )$], in meters in the year #math.equation(block: false, alt: "t")[$t$] over a 20-year period. (Source: The Canadian Hydrographic Service) #figure(figph[graph of Lake Huron water level], alt: "graph of Lake Huron water level", caption: none) + The coordinates of point #math.equation(block: false, alt: "H")[$H$] on the graph are #math.equation(block: false, alt: "open parenthesis 1997 , 176.98 close parenthesis")[$( 1997 , 176.98 )$]. What do the coordinates tell you about the function #math.equation(block: false, alt: "L")[$L$]? #linebreak() \_\_\_\_\_ + The average water level in #math.equation(block: false, alt: "2004")[$2004$] was #math.equation(block: false, alt: "176.11")[$176.11$] meters. Write this fact in function notation. What can you say about the graph of #math.equation(block: false, alt: "L")[$L$]? #linebreak() \_\_\_\_\_ #solutionbox[ + #math.equation(block: false, alt: "L open parenthesis 1997 close parenthesis equals 176.98")[$L ( 1997 ) = 176.98$]; the average water level was #math.equation(block: false, alt: "176.98")[$176.98$] meters in #math.equation(block: false, alt: "1997")[$1997$]. + #math.equation(block: false, alt: "L open parenthesis 2004 close parenthesis equals 176.11")[$L ( 2004 ) = 176.11$]. The point #math.equation(block: false, alt: "open parenthesis 2004 , 176.11 close parenthesis")[$( 2004 , 176.11 )$] lies on the graph of #math.equation(block: false, alt: "L")[$L$]. ] The water level in Lake Huron alters unpredictably over time. The graph below gives the average water level, #math.equation(block: false, alt: "L open parenthesis t close parenthesis")[$L ( t )$], in meters in the year #math.equation(block: false, alt: "t")[$t$] over a 20-year period. (Source: The Canadian Hydrographic Service) #figure(figph[graph of Lake Huron water level], alt: "graph of Lake Huron water level", caption: none) + The coordinates of point #math.equation(block: false, alt: "H")[$H$] on the graph are #math.equation(block: false, alt: "open parenthesis 1997 , 176.98 close parenthesis")[$( 1997 , 176.98 )$]. What do the coordinates tell you about the function #math.equation(block: false, alt: "L")[$L$]? + The average water level in #math.equation(block: false, alt: "2004")[$2004$] was #math.equation(block: false, alt: "176.11")[$176.11$] meters. Write this fact in function notation. What can you say about the graph of #math.equation(block: false, alt: "L")[$L$]? #solutionbox[ + #math.equation(block: false, alt: "L open parenthesis 1997 close parenthesis equals 176.98")[$L ( 1997 ) = 176.98$]; the average water level was #math.equation(block: false, alt: "176.98")[$176.98$] meters in #math.equation(block: false, alt: "1997")[$1997$]. + #math.equation(block: false, alt: "L open parenthesis 2004 close parenthesis equals 176.11")[$L ( 2004 ) = 176.11$]. The point #math.equation(block: false, alt: "open parenthesis 2004 , 176.11 close parenthesis")[$( 2004 , 176.11 )$] lies on the graph of #math.equation(block: false, alt: "L")[$L$]. ] Here is another way of describing how a graph depicts a function. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Functions and Coordinates] Each point on the graph of the function #math.equation(block: false, alt: "f")[$f$] has coordinates #math.equation(block: false, alt: "open parenthesis x , f open parenthesis x close parenthesis close parenthesis")[$( x , f ( x ) )$] for some value of #math.equation(block: false, alt: "x")[$x$]. ] #examplebox("Example 2")[][ The figure shows the graph of a function #math.equation(block: false, alt: "g")[$g$]. #figure(figph[graph of a function], alt: "graph of a function", caption: none) + Find #math.equation(block: false, alt: "g open parenthesis minus 2 close parenthesis")[$g ( − 2 )$] and #math.equation(block: false, alt: "g open parenthesis 5 close parenthesis")[$g ( 5 )$]. + For what value(s) of #math.equation(block: false, alt: "t")[$t$] is #math.equation(block: false, alt: "g open parenthesis t close parenthesis equals minus 2")[$g ( t ) = − 2$]? + What is the largest, or maximum, value of #math.equation(block: false, alt: "g open parenthesis t close parenthesis")[$g ( t )$]? For what value of #math.equation(block: false, alt: "t")[$t$] does the function take on its maximum value? + On what intervals is #math.equation(block: false, alt: "g")[$g$] increasing? #solutionbox[ + To find #math.equation(block: false, alt: "g open parenthesis minus 2 close parenthesis")[$g ( − 2 )$], we look for the point with #math.equation(block: false, alt: "t")[$t$]-coordinate #math.equation(block: false, alt: "minus 2")[$− 2$]. The point #math.equation(block: false, alt: "open parenthesis minus 2 , 0 close parenthesis")[$( − 2 , 0 )$] lies on the graph of #math.equation(block: false, alt: "g")[$g$], so #math.equation(block: false, alt: "g open parenthesis minus 2 close parenthesis equals 0")[$g ( − 2 ) = 0$]. Similarly, the point #math.equation(block: false, alt: "open parenthesis 5 , 1 close parenthesis")[$( 5 , 1 )$] lies on the graph, so #math.equation(block: false, alt: "g open parenthesis 5 close parenthesis equals 1")[$g ( 5 ) = 1$]. + We look for points on the graph with #math.equation(block: false, alt: "y")[$y$]-coordinate #math.equation(block: false, alt: "minus 2")[$− 2$]. Because the points #math.equation(block: false, alt: "open parenthesis minus 5 , minus 2 close parenthesis")[$( − 5 , − 2 )$], #math.equation(block: false, alt: "open parenthesis minus 3 , minus 2 close parenthesis")[$( − 3 , − 2 )$], and #math.equation(block: false, alt: "open parenthesis 3 , minus 2 close parenthesis")[$( 3 , − 2 )$] lie on the graph, we know that #math.equation(block: false, alt: "g open parenthesis minus 5 close parenthesis equals minus 2")[$g ( − 5 ) = − 2$], #math.equation(block: false, alt: "g open parenthesis minus 3 close parenthesis equals minus 2")[$g ( − 3 ) = − 2$], and #math.equation(block: false, alt: "g open parenthesis 3 close parenthesis equals minus 2")[$g ( 3 ) = − 2$]. Thus, the #math.equation(block: false, alt: "t")[$t$]-values we want are #math.equation(block: false, alt: "minus 5")[$− 5$], #math.equation(block: false, alt: "minus 3")[$− 3$], and #math.equation(block: false, alt: "3")[$3$]. + The highest point on the graph is #math.equation(block: false, alt: "open parenthesis 1 , 4 close parenthesis")[$( 1 , 4 )$], so the largest #math.equation(block: false, alt: "y")[$y$]-value is #math.equation(block: false, alt: "4")[$4$]. Thus, the maximum value of #math.equation(block: false, alt: "g open parenthesis t close parenthesis")[$g ( t )$] is #math.equation(block: false, alt: "4")[$4$], and it occurs when #math.equation(block: false, alt: "t equals 1")[$t = 1$]. + A graph is increasing if the #math.equation(block: false, alt: "y")[$y$]-values get larger as we read from left to right. The graph of #math.equation(block: false, alt: "g")[$g$] is increasing for #math.equation(block: false, alt: "t")[$t$]-values between #math.equation(block: false, alt: "minus 4")[$− 4$] and #math.equation(block: false, alt: "1")[$1$], and between #math.equation(block: false, alt: "3")[$3$] and #math.equation(block: false, alt: "5")[$5$]. Thus, #math.equation(block: false, alt: "g")[$g$] is increasing on the intervals #math.equation(block: false, alt: "open parenthesis minus 4 , 1 close parenthesis")[$( − 4 , 1 )$] and #math.equation(block: false, alt: "open parenthesis 3 , 5 close parenthesis")[$( 3 , 5 )$]. ] ] Refer to the graph of the function #math.equation(block: false, alt: "g")[$g$] shown in Example. #figure(figph[graph of a function], alt: "graph of a function", caption: none) + #math.equation(block: false, alt: "g open parenthesis 0 close parenthesis equals")[$g ( 0 ) =$]\_\_\_\_\_ + List the value(s) of #math.equation(block: false, alt: "t")[$t$] for which #math.equation(block: false, alt: "g open parenthesis t close parenthesis equals 0")[$g ( t ) = 0$]. Separate different values with commas. #linebreak() \_\_\_\_\_ + What is the smallest, or minimum, value of #math.equation(block: false, alt: "g open parenthesis t close parenthesis")[$g ( t )$]? #linebreak() Minimum: \_\_\_\_\_ #linebreak() For what value of #math.equation(block: false, alt: "t")[$t$] does the function take on its minimum value? #linebreak() #math.equation(block: false, alt: "t equals")[$t =$]\_\_\_\_\_ + Select all the intervals listed below where #math.equation(block: false, alt: "g")[$g$] is decreasing. #linebreak() From #math.equation(block: false, alt: "minus 5")[$− 5$] to #math.equation(block: false, alt: "minus 4")[$− 4$]? \_\_\_\_\_ #linebreak() From #math.equation(block: false, alt: "minus 5")[$− 5$] to #math.equation(block: false, alt: "minus 2")[$− 2$]? \_\_\_\_\_ #linebreak() From 2 to 4? \_\_\_\_\_ #linebreak() From 1 to 3? \_\_\_\_\_ #solutionbox[ + #math.equation(block: false, alt: "3")[$3$] + #math.equation(block: false, alt: "minus 2 , 2 , 4")[$− 2 , 2 , 4$] + #math.equation(block: false, alt: "minus 3")[$− 3$]; #math.equation(block: false, alt: "t equals minus 4")[$t = − 4$] + #math.equation(block: false, alt: "open parenthesis minus 5 , minus 4 close parenthesis")[$( − 5 , − 4 )$] and #math.equation(block: false, alt: "open parenthesis 1 , 3 close parenthesis")[$( 1 , 3 )$] ] Refer to the graph of the function #math.equation(block: false, alt: "g")[$g$] shown in Example. + Find #math.equation(block: false, alt: "g open parenthesis 0 close parenthesis")[$" " g ( 0 )$]. + For what value(s) of #math.equation(block: false, alt: "t")[$t$] is #math.equation(block: false, alt: "g open parenthesis t close parenthesis equals 0")[$" " g ( t ) = 0$]? + What is the smallest, or minimum, value of #math.equation(block: false, alt: "g open parenthesis t close parenthesis")[$g ( t )$]? For what value of #math.equation(block: false, alt: "t")[$t$] does the function take on its minimum value? + On what intervals is #math.equation(block: false, alt: "g")[$g$] is decreasing? #solutionbox[ + #math.equation(block: false, alt: "3")[$3$] + #math.equation(block: false, alt: "minus 2 , 2 , 4")[$− 2 , 2 , 4$] + #math.equation(block: false, alt: "minus 3")[$− 3$]; #math.equation(block: false, alt: "t equals minus 4")[$t = − 4$] + #math.equation(block: false, alt: "open parenthesis minus 5 , minus 4 close parenthesis")[$( − 5 , − 4 )$] and #math.equation(block: false, alt: "open parenthesis 1 , 3 close parenthesis")[$( 1 , 3 )$] ] Which of the following is true? \_\_\_\_\_ #solutionbox[ The maximum value of #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$y = f ( x )$] may occur at two different #math.equation(block: false, alt: "x")[$x$]-values. ] Which of the following statements is true? + It is not possible for the function #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$y = f ( x )$] to take on the same #math.equation(block: false, alt: "y")[$y$]-value at two different #math.equation(block: false, alt: "x")[$x$]-values. + The maximum value of #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$y = f ( x )$] may occur at two different #math.equation(block: false, alt: "x")[$x$]-values. + The maximum value of the function #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$y = f ( x )$] is the largest #math.equation(block: false, alt: "x")[$x$]-value that appears on the graph. + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 0")[$f ( x ) = 0$] at the #math.equation(block: false, alt: "y")[$y$]-intercept of #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]. #notebox("Technology", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Finding Coordinates with a Graphing Utility] We can use the TRACE feature to find the coordinates of points on a graph. For example, graph the equation #math.equation(block: false, alt: "y equals minus 2.6 x minus 5.4")[$y = − 2.6 x − 5.4$] in the window #figure(table( columns: 3, align: left, inset: 6pt, table.header([Xmin#math.equation(block: false, alt: "equals minus 5")[$= − 5$]], [], [Xmax#math.equation(block: false, alt: "equals 4.4")[$= 4.4$]]), [Ymin#math.equation(block: false, alt: "equals minus 20")[$= − 20$]], [], [Ymax#math.equation(block: false, alt: "equals 15")[$= 15$]], )) #figure(figph[graphing calculator trace], alt: "graphing calculator trace", caption: none) Press TRACE, and a “bug” begins flashing on the display. The coordinates of the bug appear at the bottom of the display, as shown in the figure. Use the left and right arrows to move the bug along the graph. You can check that the coordinates of the point #math.equation(block: false, alt: "open parenthesis 2 , minus 10.6 close parenthesis")[$( 2 , − 10.6 )$] do satisfy the equation #math.equation(block: false, alt: "y equals minus 2.6 x minus 5.4")[$y = − 2.6 x − 5.4$]. The points identified by the Trace bug depend on the window settings. If we want to find the #math.equation(block: false, alt: "y")[$y$]-coordinate for a particular #math.equation(block: false, alt: "x")[$x$]-value, we enter the #math.equation(block: false, alt: "x")[$x$]-coordinate of the desired point and press ENTER. ] === Constructing the Graph of a Function Although some functions are defined by their graphs, we can also construct graphs for functions described by tables or equations. We make these graphs the same way we graph equations in two variables: by plotting points whose coordinates satisfy the equation. #examplebox("Example 3")[][ Graph the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the square root of x plus 4")[$f ( x ) = sqrt(x + 4)$] #solutionbox[ We choose several convenient values for #math.equation(block: false, alt: "x")[$x$] and evaluate the function to find the corresponding #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]-values. For this function we cannot choose #math.equation(block: false, alt: "x")[$x$]-values less than #math.equation(block: false, alt: "minus 4")[$− 4$], because the square root of a negative number is not a real number. #math.equation(block: true, alt: "f open parenthesis minus 4 close parenthesis equals the square root of minus 4 plus 4 equals the square root of 0 equals 0")[$f ( − 4 ) = sqrt(− 4 + 4) = sqrt(0) = 0$] #math.equation(block: true, alt: "f open parenthesis minus 3 close parenthesis equals the square root of minus 3 plus 4 equals the square root of 1 equals 1")[$f ( − 3 ) = sqrt(− 3 + 4) = sqrt(1) = 1$] #math.equation(block: true, alt: "f open parenthesis 0 close parenthesis equals the square root of 0 plus 4 equals the square root of 4 equals 2")[$f ( 0 ) = sqrt(0 + 4) = sqrt(4) = 2$] #math.equation(block: true, alt: "f open parenthesis 2 close parenthesis equals the square root of 2 plus 4 equals the square root of 6 approximately equals 2.45")[$f ( 2 ) = sqrt(2 + 4) = sqrt(6) ≈ 2.45$] #math.equation(block: true, alt: "f open parenthesis 5 close parenthesis equals the square root of 5 plus 4 equals the square root of 9 equals 3")[$f ( 5 ) = sqrt(5 + 4) = sqrt(9) = 3$] The results are shown in the table. #figure(table( columns: 2, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]]), [#math.equation(block: false, alt: "minus 4")[$− 4$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "the square root of 6")[$sqrt(6)$]], [#math.equation(block: false, alt: "5")[$5$]], [#math.equation(block: false, alt: "3")[$3$]], )) #figure(figph[square root graph], alt: "square root graph", caption: none) Points on the graph have coordinates #math.equation(block: false, alt: "open parenthesis x , f open parenthesis x close parenthesis close parenthesis")[$( x , f ( x ) )$], so the vertical coordinate of each point is given by the value of #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]. We plot the points and connect them with a smooth curve, as shown in the figure. Notice that no points on the graph have #math.equation(block: false, alt: "x")[$x$]-coordinates less than #math.equation(block: false, alt: "minus 4")[$− 4$]. ] ] How do we find the value of #math.equation(block: false, alt: "f open parenthesis 3 close parenthesis")[$f ( 3 )$] from a graph of #math.equation(block: false, alt: "f")[$f$]? \_\_\_\_\_ #solutionbox[ Find 3 on the #math.equation(block: false, alt: "x")[$x$]-axis, move vertically to the point, then horizontally to the #math.equation(block: false, alt: "y")[$y$]-axis. ] How do we find the value of #math.equation(block: false, alt: "f open parenthesis 3 close parenthesis")[$f ( 3 )$] from a graph of #math.equation(block: false, alt: "f")[$f$]? + Find 3 on the #math.equation(block: false, alt: "x")[$x$]-axis, move vertically to the corresponding point on the graph, then horizontally to the #math.equation(block: false, alt: "y")[$y$]-axis. + Find 3 on the #math.equation(block: false, alt: "y")[$y$]-axis, move horizontally to the corresponding point on the graph, then vertically to the #math.equation(block: false, alt: "x")[$x$]-axis. + Substitute 3 for #math.equation(block: false, alt: "x")[$x$] into the formula for the function. + Substitute 3 for #math.equation(block: false, alt: "y")[$y$] into the formula for the function. #notebox("Technology", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Using Technology to Graph a Function] We can also use a graphing utility to obtain a table and graph for the function in Example. We graph a function just as we graphed an equation. For this function, we enter #math.equation(block: false, alt: "Y sub 1 equals the square root of open parenthesis X plus 4 close parenthesis")[$Y_(1) = sqrt(#h(0.167em)) ( X + 4 )$] and press ZOOM #math.equation(block: false, alt: "6")[$6$] for the standard window. The graph is shown below. #figure(figph[graphing calculator square root graph], alt: "graphing calculator square root graph", caption: none) ] Let #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals func")[$f ( x ) = "func"$] Complete the table of values and sketch a graph of the function. #figure(table( columns: 8, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "minus the fraction 1 over 2")[$− frac(1, 2)$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]]), [#math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]], [\_\_\_\_\_], [\_\_\_\_\_], [\_\_\_\_\_], [\_\_\_\_\_], [\_\_\_\_\_], [\_\_\_\_\_], [\_\_\_\_\_], )) #solutionbox[ #figure(table( columns: 8, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "minus the fraction 1 over 2")[$− frac(1, 2)$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]]), [#math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]], [#math.equation(block: false, alt: "minus 10")[$− 10$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "the fraction minus 17 over 8")[$frac(− 17, 8)$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "the fraction minus 15 over 8")[$frac(− 15, 8)$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "6")[$6$]], )) The graph is shown below. ] #figure(figph[cubic], alt: "cubic", caption: none) ~Let #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x cubed minus 2")[$f ( x ) = x^(3) − 2$] Complete the table of values and sketch a graph of the function. #figure(table( columns: 8, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "minus the fraction 1 over 2")[$− frac(1, 2)$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]]), [#math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]], [$#hide($00$)$], [$#hide($00$)$], [$#hide($00$)$], [$#hide($00$)$], [$#hide($00$)$], [$#hide($00$)$], [$#hide($00$)$], )) #solutionbox[ We evaluate the function at each value of #math.equation(block: false, alt: "x")[$x$]. For example, #math.equation(block: true, alt: "f open parenthesis minus 2 close parenthesis equals open parenthesis minus 2 close parenthesis cubed minus 2 equals minus 8 minus 2 equals minus 10")[$f ( − 2 ) = ( − 2 )^(3) − 2 = − 8 − 2 = − 10$] You can verify the completed table and the graph with your technology tools. #figure(table( columns: 8, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "minus the fraction 1 over 2")[$− frac(1, 2)$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]]), [#math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]], [#math.equation(block: false, alt: "minus 10")[$− 10$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "the fraction minus 17 over 8")[$frac(− 17, 8)$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "the fraction minus 15 over 8")[$frac(− 15, 8)$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "6")[$6$]], )) #figure(figph[cubic], alt: "cubic", caption: none) ] How is graphing a function different from graphing an equation? \_\_\_\_\_ How is graphing a function different from graphing an equation? === The Vertical Line Test In a function, two different outputs cannot be related to the same input. This restriction means that two different ordered pairs cannot have the same first coordinate. What does it mean for the graph of the function? Consider the graph shown in figure (a) below. Every vertical line intersects the graph in at most one point, so there is only one point on the graph for each #math.equation(block: false, alt: "x")[$x$]-value. This graph represents a function. #figure(figph[vertical line test], alt: "vertical line test", caption: none) In figure (b), however, the line #math.equation(block: false, alt: "x equals 2")[$x = 2$] intersects the graph at two points, #math.equation(block: false, alt: "open parenthesis 2 , 1 close parenthesis")[$( 2 , 1 )$] and #math.equation(block: false, alt: "open parenthesis 2 , 4 close parenthesis")[$( 2 , 4 )$]. Two different #math.equation(block: false, alt: "y")[$y$]-values, #math.equation(block: false, alt: "1")[$1$] and #math.equation(block: false, alt: "4")[$4$], are related to the same #math.equation(block: false, alt: "x")[$x$]-value, #math.equation(block: false, alt: "2")[$2$]. This graph cannot be the graph of a function. We summarize these observations as follows. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[The Vertical Line Test] A graph represents a function if and only if every vertical line intersects the graph in at most one point. ] #examplebox("Example 4")[][ Use the vertical line test to decide which of the graphs in the figure represent functions. #figure(figph[vertical line test on three graphs], alt: "vertical line test on three graphs", caption: none) #solutionbox[ - Graph (a) represents a function, because it passes the vertical line test. - Graph (b) is not the graph of a function, because the vertical line at (for example) #math.equation(block: false, alt: "x equals 1")[$x = 1$] intersects the graph at two points. - For graph (c), notice the break in the curve at #math.equation(block: false, alt: "x equals 2")[$x = 2$]: The solid dot at #math.equation(block: false, alt: "open parenthesis 2 , 1 close parenthesis")[$( 2 , 1 )$] is the only point on the graph with #math.equation(block: false, alt: "x equals 2")[$x = 2$]; the open circle at #math.equation(block: false, alt: "open parenthesis 2 , 3 close parenthesis")[$( 2 , 3 )$] indicates that #math.equation(block: false, alt: "open parenthesis 2 , 3 close parenthesis")[$( 2 , 3 )$] is not a point on the graph. Thus, graph (c) is a function, with #math.equation(block: false, alt: "f open parenthesis 2 close parenthesis equals 1")[$f ( 2 ) = 1$]. ] ] What does the vertical line test tell us? \_\_\_\_\_ #solutionbox[ The vertical line test tells us if the graph is a function. ] What does the vertical line test tell us? + If the graph is a vertical line. + If the graph is increasing. + If the graph is decreasing. + If the graph is a function. Use the vertical line test to determine which of the graphs below represent functions. #figure(figph[vertical line test], alt: "vertical line test", caption: none) \_\_\_\_\_ #solutionbox[ Only (b) is a function. ] Use the vertical line test to determine which of the graphs below represent functions. #figure(figph[vertical line test], alt: "vertical line test", caption: none) #solutionbox[ Only (b) is a function. ] === Graphical Solution of Equations and Inequalities The graph of an equation in two variables is just a picture of its solutions. When we read the coordinates of a point on the graph, we are reading a pair of #math.equation(block: false, alt: "x")[$x$]- and #math.equation(block: false, alt: "y")[$y$]-values that make the equation true. For example, the point #math.equation(block: false, alt: "open parenthesis 2 , 7 close parenthesis")[$( 2 , 7 )$] lies on the graph of #math.equation(block: false, alt: "y equals 2 x plus 3")[$" " y = 2 x + 3 " "$] shown at right, so we know that the ordered pair #math.equation(block: false, alt: "open parenthesis 2 , 7 close parenthesis")[$( 2 , 7 )$] is a solution of the equation #math.equation(block: false, alt: "y equals 2 x plus 3")[$y = 2 x + 3$]. You can verify algebraically that #math.equation(block: false, alt: "x equals 2")[$x = 2$] and #math.equation(block: false, alt: "y equals 7")[$y = 7$] satisfy the equation: #math.equation(block: true, alt: "Does 7 equals 2 open parenthesis 2 close parenthesis plus 3 ? Yes")[$"Does " " " 7 = 2 ( 2 ) + 3 "?" " " " " " " " " " " " " " " " " "Yes"$] #figure(figph[graph of line with point], alt: "graph of line with point", caption: none) We can also say that #math.equation(block: false, alt: "x equals 2")[$x = 2$] is a solution of the one-variable equation #math.equation(block: false, alt: "2 x plus 3 equals 7")[$2 x + 3 = 7$]. In fact, we can use the graph of #math.equation(block: false, alt: "y equals 2 x plus 3")[$y = 2 x + 3$] to solve the equation #math.equation(block: false, alt: "2 x plus 3 equals k")[$2 x + 3 = k$] for any value of #math.equation(block: false, alt: "k")[$k$]. Thus, we can use graphs to find solutions to equations in one variable. #examplebox("Example 5")[][ Use the graph of #math.equation(block: false, alt: "y equals 285 minus 15 x")[$" " y = 285 − 15 x " "$] to solve the equation #math.equation(block: false, alt: "150 equals 285 minus 15 x")[$" " 150 = 285 − 15 x$]. #solutionbox[ We begin by locating the point #math.equation(block: false, alt: "P")[$P$] on the graph for which #math.equation(block: false, alt: "y equals 150")[$y = 150$], as shown in the figure. Next we find the #math.equation(block: false, alt: "x")[$x$]-coordinate of point #math.equation(block: false, alt: "P")[$P$] by drawing an imaginary line from #math.equation(block: false, alt: "P")[$P$] straight down to the #math.equation(block: false, alt: "x")[$x$]-axis. The #math.equation(block: false, alt: "x")[$x$]-coordinate of #math.equation(block: false, alt: "P")[$P$] is #math.equation(block: false, alt: "x equals 9")[$x = 9$]. Thus, #math.equation(block: false, alt: "P")[$P$] is the point #math.equation(block: false, alt: "open parenthesis 9 , 150 close parenthesis")[$( 9 , 150 )$], and #math.equation(block: false, alt: "x equals 9")[$x = 9$] when #math.equation(block: false, alt: "y equals 150")[$y = 150$]. The solution of the equation #math.equation(block: false, alt: "150 equals 285 minus 15 x")[$150 = 285 − 15 x$] is #math.equation(block: false, alt: "x equals 9")[$x = 9$]. #figure(figph[graph of line with point], alt: "graph of line with point", caption: none) You can verify the solution algebraically by substituting #math.equation(block: false, alt: "x equals 9")[$x = 9$] into the equation: Does #math.equation(block: false, alt: "150 equals 285 minus 15 open parenthesis 9 close parenthesis")[$150 = 285 − 15 ( 9 )$]? #math.equation(block: true, alt: "285 minus 15 open parenthesis 9 close parenthesis equals 285 minus 135 equals 150. Yes")[$285 − 15 ( 9 ) = 285 − 135 = 150. " " " " " " " " " " " " "Yes"$] ] ] If #math.equation(block: false, alt: "f open parenthesis 8 close parenthesis equals 2")[$f ( 8 ) = 2$], what point lies on the graph of #math.equation(block: false, alt: "f")[$f$]? \_\_\_\_\_ #solutionbox[ #math.equation(block: true, alt: "open parenthesis 8 , 2 close parenthesis")[$( 8 , 2 )$] ] If #math.equation(block: false, alt: "f open parenthesis 8 close parenthesis equals 2")[$f ( 8 ) = 2$], what point lies on the graph of #math.equation(block: false, alt: "f")[$f$]? #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The relationship between an equation and its graph is an important one. For the previous example, make sure you understand that the following three statements are equivalent: - The point #math.equation(block: false, alt: "open parenthesis 9 , 150 close parenthesis")[$( 9 , 150 )$] lies on the graph of #math.equation(block: false, alt: "y equals 285 minus 15 x")[$" " y = 285 − 15 x$]. - The ordered pair #math.equation(block: false, alt: "open parenthesis 9 , 150 close parenthesis")[$( 9 , 150 )$] is a solution of the equation #math.equation(block: false, alt: "y equals 285 minus 15 x")[$" " y = 285 − 15 x$]. - #math.equation(block: false, alt: "x equals 9")[$x = 9$] is a solution of the equation #math.equation(block: false, alt: "150 equals 285 minus 15 x")[$" " 150 = 285 − 15 x$]. ] #figure(figph[function graph], alt: "function graph", caption: none) + Use the graph of #math.equation(block: false, alt: "y equals 30 minus 8 x")[$y = 30 − 8 x$] shown above to solve the equation #math.equation(block: true, alt: "30 minus 8 x equals 50")[$30 − 8 x = 50$] #linebreak() #math.equation(block: false, alt: "x equals")[$x =$]\_\_\_\_\_ + Verify your solution algebraically. #solutionbox[ + The point on the graph where the #math.equation(block: false, alt: "y")[$y$]-coordinate is 50 is the point #math.equation(block: false, alt: "open parenthesis minus 2.5 , 50 close parenthesis")[$( − 2.5 , 50 )$], so #math.equation(block: false, alt: "x equals minus 2.5")[$x = − 2.5$] is the solution. + We verify that we have an identity when we substitute #math.equation(block: false, alt: "x equals minus 2.5")[$x = − 2.5$] into the equation #math.equation(block: false, alt: "30 minus 8 x equals 50")[$30 − 8 x = 50$]. #linebreak() #math.equation(block: true, alt: "30 minus 8 open parenthesis minus 2.5 close parenthesis, equals ? 50; 30 minus open parenthesis minus 20 close parenthesis, equals 50, Yes!")[$30 − 8 ( − 2.5 ) & limits(=)^("?") 50 \ 30 − ( − 20 ) & = 50 & "Yes!"$] ] For part (b): We verify that we have an identity when we substitute #math.equation(block: false, alt: "x equals minus 2.5")[$x = − 2.5$] into the equation #math.equation(block: false, alt: "30 minus 8 x equals 50")[$30 − 8 x = 50$]. #math.equation(block: true, alt: "30 minus 8 open parenthesis minus 2.5 close parenthesis, equals ? 50; 30 minus open parenthesis minus 20 close parenthesis, equals 50, Yes!")[$30 − 8 ( − 2.5 ) & limits(=)^("?") 50 \ 30 − ( − 20 ) & = 50 & "Yes!"$] #figure(figph[function graph], alt: "function graph", caption: none) + Use the graph of #math.equation(block: false, alt: "y equals 30 minus 8 x")[$" " y = 30 − 8 x " "$] shown above to solve the equation #math.equation(block: true, alt: "30 minus 8 x equals 50")[$30 − 8 x = 50$] + Verify your solution algebraically. #solutionbox[ + The point on the line where the #math.equation(block: false, alt: "y")[$y$]-coordinate is #math.equation(block: false, alt: "50")[$50$] is the point #math.equation(block: false, alt: "open parenthesis minus 2.5 , 50 close parenthesis")[$( − 2.5 , 50 )$], so #math.equation(block: false, alt: "x equals minus 2.5")[$x = − 2.5$] is the solution. + We verify that we have an identity when we substitute #math.equation(block: false, alt: "x equals minus 2.5")[$x = − 2.5$] into the equation #math.equation(block: false, alt: "30 minus 8 x equals 50")[$" " 30 − 8 x = 50$]. #math.equation(block: true, alt: "30 minus 8 open parenthesis minus 2.5 close parenthesis, equals ? 50; 30 minus open parenthesis minus 20 close parenthesis, equals 50, Yes!")[$30 − 8 ( − 2.5 ) & limits(=)^("?") 50 \ 30 − ( − 20 ) & = 50 & "Yes!"$] ] In a similar fashion, we can solve inequalities with a graph. Consider again the graph of #math.equation(block: false, alt: "y equals 2 x plus 3")[$" " y = 2 x + 3 " "$], shown at right. We saw that #math.equation(block: false, alt: "x equals 2")[$x = 2$] is the solution of the equation #math.equation(block: false, alt: "2 x plus 3 equals 7")[$2 x + 3 = 7$]. When we use #math.equation(block: false, alt: "x equals 2")[$x = 2$] as the input for the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 2 x plus 3")[$f ( x ) = 2 x + 3$], the output is #math.equation(block: false, alt: "y equals 7")[$y = 7$]. Which input values for #math.equation(block: false, alt: "x")[$x$] produce output values greater than #math.equation(block: false, alt: "7")[$7$]? You can see that #math.equation(block: false, alt: "x")[$x$]-values greater than #math.equation(block: false, alt: "2")[$2$] produce #math.equation(block: false, alt: "y")[$y$]-values greater than #math.equation(block: false, alt: "7")[$7$], because points on the graph with #math.equation(block: false, alt: "x")[$x$]-values greater than #math.equation(block: false, alt: "2")[$2$] have #math.equation(block: false, alt: "y")[$y$]-values greater than #math.equation(block: false, alt: "7")[$7$]. Thus, the solutions of the inequality #math.equation(block: false, alt: "2 x plus 3 greater than 7")[$2 x + 3 > 7$] are #math.equation(block: false, alt: "x greater than 2")[$x > 2$]. You can verify this result by solving the inequality algebraically. #figure(figph[graph of line with inequality labeled], alt: "graph of line with inequality labeled", caption: none) #examplebox("Example 6")[][ Use the graph of #math.equation(block: false, alt: "y equals 285 minus 15 x")[$" " y = 285 − 15 x " "$] to solve the inequality #math.equation(block: true, alt: "285 minus 15 x greater than 150")[$285 − 15 x > 150$] #solutionbox[ We begin by locating the point #math.equation(block: false, alt: "P")[$P$] on the graph for which #math.equation(block: false, alt: "y equals 150")[$y = 150$]. Its #math.equation(block: false, alt: "x")[$x$]-coordinate is #math.equation(block: false, alt: "x equals 9")[$x = 9$]. Now, because #math.equation(block: false, alt: "y equals 285 minus 15 x")[$" " y = 285 − 15 x " "$] for points on the graph, the inequality #math.equation(block: true, alt: "285 minus 15 x greater than 150")[$285 − 15 x > 150$] is equivalent to #math.equation(block: false, alt: "y greater than 150")[$" " y > 150$]. #figure(figph[graph of line with inequality labeled], alt: "graph of line with inequality labeled", caption: none) So we are looking for points on the graph with #math.equation(block: false, alt: "y")[$y$]-coordinate greater than #math.equation(block: false, alt: "150")[$150$]. These points are shown in red on the graph. The #math.equation(block: false, alt: "x")[$x$]-coordinates of these points are the #math.equation(block: false, alt: "x")[$x$]-values that satisfy the inequality. From the graph, we see that the solutions are #math.equation(block: false, alt: "x less than 9")[$" " x < 9$]. ] ] You are using a graph of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 48 minus 3.5 x")[$f ( x ) = 48 − 3.5 x$] to solve the inequality #math.equation(block: false, alt: "48 minus 3.5 x greater than minus 15")[$48 − 3.5 x > − 15$]. You find that #math.equation(block: false, alt: "f open parenthesis 18 close parenthesis equals minus 15")[$f ( 18 ) = − 15$]. Your answer is: \_\_\_\_\_ #solutionbox[ #math.equation(block: true, alt: "x less than 18")[$x < 18$] ] You are using a graph of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 48 minus 3.5 x")[$" " f ( x ) = 48 − 3.5 x " "$] to solve the inequality #math.equation(block: false, alt: "48 minus 3.5 x greater than minus 15")[$" " 48 − 3.5 x > − 15 " "$]. You find that #math.equation(block: false, alt: "f open parenthesis 18 close parenthesis equals minus 15")[$" " f ( 18 ) = − 15$]. What is the solution to the inequality? Here is the graph from Practice 5 (Practice 5). #figure(figph[function graph], alt: "function graph", caption: none) + Use the graph of #math.equation(block: false, alt: "y equals 30 minus 8 x")[$y = 30 − 8 x$] above to solve the inequality #math.equation(block: true, alt: "30 minus 8 x less than or equal to 50")[$30 − 8 x ≤ 50$] #linebreak() Answer: \_\_\_\_\_ #linebreak() Note: Use "#math.equation(block: false, alt: "<=")[$<=$]" for the #math.equation(block: false, alt: "less than or equal to")[$≤$] symbol, and use "#math.equation(block: false, alt: ">=")[$>=$]" for #math.equation(block: false, alt: "greater than or equal to")[$≥$]. + Solve the inequality algebraically. #solutionbox[ + As we found in the previous Checkpoint, the point on #math.equation(block: false, alt: "y equals 30 minus 8 x")[$y = 30 − 8 x$] where #math.equation(block: false, alt: "y equals 50")[$y = 50$] occurs when #math.equation(block: false, alt: "x equals minus 2.5")[$x = − 2.5$]. The #math.equation(block: false, alt: "y")[$y$]-coordinates are smaller as we move to the right on the graph, that is, for the points where #math.equation(block: false, alt: "x greater than or equal to minus 2.5")[$x ≥ − 2.5$]. + #math.equation(block: true, alt: "30 minus 8 x, less than or equal to 50, Subtract 30 from both sides.; minus 8 x, less than or equal to 20, Divide by minus 8 and reverse inequality.; x, greater than or equal to minus 2.5")[$30 − 8 x & ≤ 50 & & "Subtract 30 from both sides." \ − 8 x & ≤ 20 & & "Divide by " − 8 " and reverse inequality." \ x & ≥ − 2.5$] ] Here is the graph from Practice 5 (Practice 5). + Use the graph of #math.equation(block: true, alt: "y equals 30 minus 8 x")[$" " y = 30 − 8 x " "$] to solve the inequality #math.equation(block: true, alt: "30 minus 8 x less than or equal to 50")[$30 − 8 x ≤ 50$] + Solve the inequality algebraically. #figure(figph[function graph], alt: "function graph", caption: none) #solutionbox[ + As we found in the previous Checkpoint, the point on #math.equation(block: false, alt: "y equals 30 minus 8 x")[$y = 30 − 8 x$] where #math.equation(block: false, alt: "y equals 50")[$y = 50$] occurs when #math.equation(block: false, alt: "x equals minus 2.5")[$x = − 2.5$]. The #math.equation(block: false, alt: "y")[$y$]-coordinates are smaller as we move to the right on the graph, that is, for the points where #math.equation(block: false, alt: "x greater than or equal to minus 2.5")[$x ≥ − 2.5$]. + #math.equation(block: true, alt: "30 minus 8 x, less than or equal to 50, Subtract 30 from both sides.; minus 8 x, less than or equal to 20, Divide by minus 8 and reverse inequality.; x, greater than or equal to minus 2.5")[$30 − 8 x & ≤ 50 & & "Subtract 30 from both sides." \ − 8 x & ≤ 20 & & "Divide by " − 8 " and reverse inequality." \ x & ≥ − 2.5$] ] We can also use this graphical technique to solve nonlinear equations and inequalities. #examplebox("Example 7")[][ Use a graph of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals minus 2 x cubed plus x squared plus 16 x")[$" " f ( x ) = − 2 x^(3) + x^(2) + 16 x " "$] to solve the equation #math.equation(block: true, alt: "minus 2 x cubed plus x squared plus 16 x equals 15")[$− 2 x^(3) + x^(2) + 16 x = 15$] #solutionbox[ If we sketch in the horizontal line #math.equation(block: false, alt: "y equals 15")[$y = 15$], we can see that there are three points on the graph of #math.equation(block: false, alt: "f")[$f$] that have #math.equation(block: false, alt: "y")[$y$]-coordinate #math.equation(block: false, alt: "15")[$15$], as shown below. The #math.equation(block: false, alt: "x")[$x$]-coordinates of these points are the solutions of the equation. #figure(figph[graph of cubic], alt: "graph of cubic", caption: none) From the graph, we see that the solutions are #math.equation(block: false, alt: "x equals minus 3")[$x = − 3$], #math.equation(block: false, alt: "x equals 1")[$x = 1$], and approximately #math.equation(block: false, alt: "x equals 2.5")[$x = 2.5$]. We can verify each solution algebraically. For example, if #math.equation(block: false, alt: "x equals minus 3")[$x = − 3$], we have #math.equation(block: true, alt: "f open parenthesis minus 3 close parenthesis, equals minus 2 open parenthesis minus 3 close parenthesis 3 plus open parenthesis minus 3 close parenthesis squared plus 16 open parenthesis minus 3 close parenthesis; equals minus 2 open parenthesis minus 27 close parenthesis plus 9 minus 48; equals 54 plus 9 minus 48 equals 15")[$f ( − 3 ) & = − 2 ( − 3 ) 3 + ( − 3 )^(2) + 16 ( − 3 ) \ & = − 2 ( − 27 ) + 9 − 48 \ & = 54 + 9 − 48 = 15$] so #math.equation(block: false, alt: "minus 3")[$− 3$] is a solution. Similarly, you can check that #math.equation(block: false, alt: "x equals 1")[$x = 1$] and #math.equation(block: false, alt: "x equals 2.5")[$x = 2.5$] are solutions. ] ] #figure(figph[graph of parabola], alt: "graph of parabola", caption: none) Use the graph of #math.equation(block: false, alt: "y equals the fraction 1 over 2 n squared plus 2 n minus 10")[$y = frac(1, 2) n^(2) + 2 n − 10$] shown above to solve #math.equation(block: true, alt: "the fraction 1 over 2 n squared plus 2 n minus 10 equals 6")[$frac(1, 2) n^(2) + 2 n − 10 = 6$] and verify your solutions algebraically. #math.equation(block: false, alt: "n equals")[$n =$]\_\_\_\_\_ Separate different values with a comma. #solutionbox[ #math.equation(block: true, alt: "minus 8 , 4")[$− 8 , 4$] ] #figure(figph[graph of parabola], alt: "graph of parabola", caption: none) Use the graph of #math.equation(block: false, alt: "y equals the fraction 1 over 2 n squared plus 2 n minus 10")[$" " y = frac(1, 2) n^(2) + 2 n − 10 " "$] shown above to solve #math.equation(block: true, alt: "the fraction 1 over 2 n squared plus 2 n minus 10 equals 6")[$frac(1, 2) n^(2) + 2 n − 10 = 6$] and verify your solutions algebraically. #solutionbox[ We look for points on the graph that have #math.equation(block: false, alt: "y")[$y$]-coordinate #math.equation(block: false, alt: "6")[$6$]. The #math.equation(block: false, alt: "n")[$n$]-coordinates of those two points are the solutions, #math.equation(block: false, alt: "n equals minus 8")[$n = − 8$] and #math.equation(block: false, alt: "n equals 4")[$n = 4$]. To verify the solutions we evaluate the function at #math.equation(block: false, alt: "n equals minus 8")[$n = − 8$] and #math.equation(block: false, alt: "n equals 4")[$n = 4$] to find #math.equation(block: true, alt: "the fraction 1 over 2 open parenthesis minus 8 close parenthesis squared plus 2 minus 8 minus 10, equals 32 minus 16 minus 10 equals 6; the fraction 1 over 2 open parenthesis 4 close parenthesis squared plus 24 minus 10, equals 8 plus 8 minus 10 equals 6")[$frac(1, 2) ( − 8 )^(2) + 2 − 8 − 10 & = 32 − 16 − 10 = 6 \ frac(1, 2) ( 4 )^(2) + 2 4 − 10 & = 8 + 8 − 10 = 6$] ] When is a function called decreasing? \_\_\_\_\_ #solutionbox[ A function is called decreasing if its #math.equation(block: false, alt: "y")[$y$]-values decrease when its #math.equation(block: false, alt: "x")[$x$]-values increase. ] When is a function called decreasing? + If its #math.equation(block: false, alt: "x")[$x$]-values increase from left to right. + If its #math.equation(block: false, alt: "y")[$y$]-values decrease when its #math.equation(block: false, alt: "x")[$x$]-values decrease. + If its #math.equation(block: false, alt: "y")[$y$]-values decrease when its #math.equation(block: false, alt: "x")[$x$]-values increase. + If the graph lies below the #math.equation(block: false, alt: "x")[$x$]-axis. #notebox("Technology", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Using the Trace Feature] You can use the Trace feature on a graphing utility to approximate solutions to equations. Graph the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] in Example in the window #figure(table( columns: 3, align: left, inset: 6pt, table.header([Xmin#math.equation(block: false, alt: "equals minus 4")[$= − 4$]], [], [Xmax#math.equation(block: false, alt: "equals 4")[$= 4$]]), [Ymin#math.equation(block: false, alt: "equals minus 20")[$= − 20$]], [], [Ymax#math.equation(block: false, alt: "equals 40")[$= 40$]], )) and trace along the curve to the point #math.equation(block: false, alt: "open parenthesis 2.4680851 , 15.512401 close parenthesis")[$( 2.4680851 , 15.512401 )$]. We are close to a solution, because the #math.equation(block: false, alt: "y")[$y$]-value is close to #math.equation(block: false, alt: "15")[$15$]. Try entering #math.equation(block: false, alt: "x")[$x$]-values close to #math.equation(block: false, alt: "2.4680851")[$2.4680851$], for instance, #math.equation(block: false, alt: "x equals 2.4")[$x = 2.4$] and #math.equation(block: false, alt: "x equals 2.5")[$x = 2.5$], to find a better approximation for the solution. ] #examplebox("Example 8")[][ Use the graph in Example to solve the inequality #math.equation(block: true, alt: "minus 2 x cubed plus x squared plus 16 x greater than or equal to 15")[$− 2 x^(3) + x^(2) + 16 x ≥ 15$] #solutionbox[ We first locate all points on the graph that have #math.equation(block: false, alt: "y")[$y$]-coordinates greater than or equal to #math.equation(block: false, alt: "15")[$15$]. The #math.equation(block: false, alt: "x")[$x$]-coordinates of these points are the solutions of the inequality. The figure below shows those points in red, and their #math.equation(block: false, alt: "x")[$x$]-coordinates as intervals on the #math.equation(block: false, alt: "x")[$x$]-axis. The solutions are #math.equation(block: false, alt: "x less than or equal to minus 3")[$x ≤ − 3$] and #math.equation(block: false, alt: "1 less than or equal to x less than or equal to 2.5")[$1 ≤ x ≤ 2.5$], or in interval notation, #math.equation(block: false, alt: "open parenthesis minus ∞ , minus 3 close bracket union open bracket 1 , 2.5 close bracket")[$( − upright(∞) , − 3 ] ∪ [ 1 , 2.5 ]$]. (See Appendix A.2 to review interval notation.) #figure(figph[graph of cubic], alt: "graph of cubic", caption: none) ] ] #figure(figph[graph of parabola], alt: "graph of parabola", caption: none) Use the graph above from Practice 7 above to solve the inequality #math.equation(block: true, alt: "the fraction 1 over 2 n squared plus 2 n minus 10 less than 6")[$frac(1, 2) n^(2) + 2 n − 10 < 6$] Answer: \_\_\_\_\_ Note: Use interval notation or inequalities. #solutionbox[ #math.equation(block: false, alt: "minus 8 less than n less than 4")[$− 8 < n < 4$], or in interval notation, #math.equation(block: false, alt: "open parenthesis minus 8 , 4 close parenthesis")[$( − 8 , 4 )$] ] Use the graph above from Practice 6 above to solve the inequality #math.equation(block: true, alt: "the fraction 1 over 2 n squared plus 2 n minus 10 less than 6")[$frac(1, 2) n^(2) + 2 n − 10 < 6$] #solutionbox[ In Practice 6 we found that #math.equation(block: false, alt: "the fraction 1 over 2 n squared plus 2 n minus 10 equals 6")[$frac(1, 2) n^(2) + 2 n − 10 = 6$] when #math.equation(block: false, alt: "n equals minus 8")[$n = − 8$] and #math.equation(block: false, alt: "n equals 4")[$n = 4$]. Looking at the graph, we see that points with #math.equation(block: false, alt: "y")[$y$]-coordinates less than 6 make up the lower portion of the parabola. They have #math.equation(block: false, alt: "x")[$x$]-coordinates between #math.equation(block: false, alt: "n equals minus 8")[$n = − 8$] and #math.equation(block: false, alt: "n equals 4")[$n = 4$]. Thus, the solutions are #math.equation(block: false, alt: "minus 8 less than n less than 4")[$" " − 8 < n < 4$], or in interval notation, #math.equation(block: false, alt: "open parenthesis minus 8 , 4 close parenthesis")[$" " ( − 8 , 4 )$] ] Explain how to use the graph of #math.equation(block: false, alt: "y equals x plus the square root of x")[$y = x + sqrt(x)$] to solve the equation #math.equation(block: false, alt: "x plus the square root of x equals 56")[$x + sqrt(x) = 56$]. \_\_\_\_\_ Explain how to use the graph of #math.equation(block: false, alt: "y equals x plus the square root of x")[$y = x + sqrt(x)$] to solve the equation #math.equation(block: false, alt: "x plus the square root of x equals 56")[$x + sqrt(x) = 56$]. === Section Summary ==== Vocabulary Look up the definitions of new terms in the Glossary. - Coordinates - Maximum - Minimum - Interval - Vertical line test - Inequality - Algebraic solution - Graphical solution ==== CONCEPTS + The point #math.equation(block: false, alt: "open parenthesis a , b close parenthesis")[$( a , b )$] lies on the graph of the function #math.equation(block: false, alt: "f")[$f$] if and only if #math.equation(block: false, alt: "f open parenthesis a close parenthesis equals b")[$f ( a ) = b$]. + Each point on the graph of the function #math.equation(block: false, alt: "f")[$f$] has coordinates #math.equation(block: false, alt: "open parenthesis x , f open parenthesis x close parenthesis close parenthesis")[$( x , f ( x ) )$] for some value of #math.equation(block: false, alt: "x")[$x$]. + The vertical line test tells us whether a graph represents a function. + We can use a graph to solve equations and inequalities in one variable. ==== STUDY QUESTIONS + How can you find the value of #math.equation(block: false, alt: "f open parenthesis 3 close parenthesis")[$f ( 3 )$] from a graph of #math.equation(block: false, alt: "f")[$f$]? + If #math.equation(block: false, alt: "f open parenthesis 8 close parenthesis equals 2")[$f ( 8 ) = 2$], what point lies on the graph of #math.equation(block: false, alt: "f")[$f$]? + Explain how to construct the graph of a function from its equation. + Explain how to use the vertical line test. + How can you solve the equation #math.equation(block: false, alt: "x plus the square root of x equals 56")[$x + sqrt(x) = 56$] using the graph of #math.equation(block: false, alt: "y equals x plus the square root of x")[$y = x + sqrt(x)$]? ==== SKILLS Practice each skill in the Homework problems listed. + Read function values from a graph: \#1–8, 17–20, 33–36 + Recognize the graph of a function: \#9–10, 31 and 32 + Construct a table of values and a graph of a function: \#11–16 + Solve equations and inequalities graphically: \#21–30, 41–50 === Homework 1.3 In Problems 1–8, use the graphs to answer the questions about the functions. + Find #math.equation(block: false, alt: "h open parenthesis minus 3 close parenthesis")[$h ( − 3 )$], #math.equation(block: false, alt: "h open parenthesis 1 close parenthesis")[$h ( 1 )$], and #math.equation(block: false, alt: "h open parenthesis 3 close parenthesis")[$h ( 3 )$]. + For what value(s) of #math.equation(block: false, alt: "z")[$z$] is #math.equation(block: false, alt: "h open parenthesis z close parenthesis equals 3")[$h ( z ) = 3$]? + Find the intercepts of the graph. List the function values given by the intercepts. + What is the maximum value of #math.equation(block: false, alt: "h open parenthesis z close parenthesis")[$h ( z )$]? + For what value(s) of #math.equation(block: false, alt: "z")[$z$] does #math.equation(block: false, alt: "h")[$h$] take on its maximum value? + On what intervals is the function increasing? Decreasing? #figure(figph[curve], alt: "curve", caption: none) + #math.equation(block: false, alt: "minus 2 , 0 , 5")[$− 2 , 0 , 5$] + #math.equation(block: false, alt: "2")[$2$] + #math.equation(block: false, alt: "h open parenthesis minus 2 close parenthesis equals 0 , h open parenthesis 1 close parenthesis equals 0 , h open parenthesis 0 close parenthesis equals minus 2")[$h ( − 2 ) = 0 , " " h ( 1 ) = 0 , " " h ( 0 ) = − 2$] + #math.equation(block: false, alt: "5")[$5$] + #math.equation(block: false, alt: "3")[$3$] + Increasing: #math.equation(block: false, alt: "open parenthesis minus 3 , 0 close parenthesis")[$( − 3 , 0 )$] and #math.equation(block: false, alt: "open parenthesis 1 , 3 close parenthesis")[$( 1 , 3 )$]; decreasing: #math.equation(block: false, alt: "open parenthesis 0 , 1 close parenthesis")[$( 0 , 1 )$] and #math.equation(block: false, alt: "open parenthesis 3 , 5 close parenthesis")[$( 3 , 5 )$] + Find #math.equation(block: false, alt: "G open parenthesis minus 3 close parenthesis")[$G ( − 3 )$], #math.equation(block: false, alt: "G open parenthesis minus 1 close parenthesis")[$G ( − 1 )$], and #math.equation(block: false, alt: "G open parenthesis 2 close parenthesis")[$G ( 2 )$]. + For what value(s) of #math.equation(block: false, alt: "s")[$s$] is #math.equation(block: false, alt: "G open parenthesis s close parenthesis equals 3")[$G ( s ) = 3$]? + Find the intercepts of the graph. List the function values given by the intercepts. + What is the minimum value of #math.equation(block: false, alt: "G open parenthesis s close parenthesis")[$G ( s )$]? + For what value(s) of #math.equation(block: false, alt: "s")[$s$] does #math.equation(block: false, alt: "G")[$G$] take on its minimum value? + On what intervals is the function increasing? Decreasing? #figure(figph[curve], alt: "curve", caption: none) + Find #math.equation(block: false, alt: "R open parenthesis 1 close parenthesis")[$R ( 1 )$] and #math.equation(block: false, alt: "R open parenthesis 3 close parenthesis")[$R ( 3 )$]. + For what value(s) of #math.equation(block: false, alt: "p")[$p$] is #math.equation(block: false, alt: "R open parenthesis p close parenthesis equals 2")[$R ( p ) = 2$]? + Find the intercepts of the graph. List the function values given by the intercepts. + Find the maximum and minimum values of #math.equation(block: false, alt: "R open parenthesis p close parenthesis")[$R ( p )$]. + For what value(s) of #math.equation(block: false, alt: "p")[$p$] does #math.equation(block: false, alt: "R")[$R$] take on its maximum and minimum values? + On what intervals is the function increasing? Decreasing? #figure(figph[curve], alt: "curve", caption: none) + #math.equation(block: false, alt: "minus 1 , 2")[$− 1 , 2$] + #math.equation(block: false, alt: "3 , minus 1.3")[$3 , − 1.3$] + #math.equation(block: false, alt: "R open parenthesis minus 2 close parenthesis equals 0")[$R ( − 2 ) = 0$], #math.equation(block: false, alt: "R open parenthesis 2 close parenthesis equals 0")[$R ( 2 ) = 0$], #math.equation(block: false, alt: "R open parenthesis 4 close parenthesis equals 0")[$R ( 4 ) = 0$], #math.equation(block: false, alt: "R open parenthesis 0 close parenthesis equals 4")[$R ( 0 ) = 4$] + Max: #math.equation(block: false, alt: "4")[$4$]; min: #math.equation(block: false, alt: "minus 5")[$− 5$] + Max at #math.equation(block: false, alt: "p equals 0")[$p = 0$]; min at #math.equation(block: false, alt: "p equals 5")[$p = 5$] + Increasing: #math.equation(block: false, alt: "open parenthesis minus 3 , 0 close parenthesis")[$( − 3 , 0 )$] and #math.equation(block: false, alt: "open parenthesis 1 , 3 close parenthesis")[$( 1 , 3 )$]; decreasing: #math.equation(block: false, alt: "open parenthesis 0 , 1 close parenthesis")[$( 0 , 1 )$] and #math.equation(block: false, alt: "open parenthesis 3 , 5 close parenthesis")[$( 3 , 5 )$] + Find #math.equation(block: false, alt: "f open parenthesis minus 1 close parenthesis")[$f ( − 1 )$] and #math.equation(block: false, alt: "f open parenthesis 3 close parenthesis")[$f ( 3 )$]. + For what value(s) of #math.equation(block: false, alt: "t")[$t$] is #math.equation(block: false, alt: "f open parenthesis t close parenthesis equals 5")[$f ( t ) = 5$]? + Find the intercepts of the graph. List the function values given by the intercepts. + Find the maximum and minimum values of #math.equation(block: false, alt: "f open parenthesis t close parenthesis")[$f ( t )$]. + For what value(s) of #math.equation(block: false, alt: "t")[$t$] does #math.equation(block: false, alt: "f")[$f$] take on its maximum and minimum values? + On what intervals is the function increasing? Decreasing? #figure(figph[curve], alt: "curve", caption: none) + Find #math.equation(block: false, alt: "S open parenthesis 0 close parenthesis")[$S ( 0 )$], #math.equation(block: false, alt: "S open parenthesis the fraction 1 over 6 close parenthesis")[$S ( display(frac(1, 6)) )$], and #math.equation(block: false, alt: "S open parenthesis minus 1 close parenthesis")[$S ( − 1 )$]. + Estimate the value of #math.equation(block: false, alt: "S open parenthesis the fraction 1 over 3 close parenthesis")[$S ( display(frac(1, 3)) )$] from the graph. + For what value(s) of #math.equation(block: false, alt: "x")[$x$] is #math.equation(block: false, alt: "S open parenthesis x close parenthesis equals minus the fraction 1 over 2")[$S ( x ) = − display(frac(1, 2))$]? + Find the maximum and minimum values of #math.equation(block: false, alt: "S open parenthesis x close parenthesis")[$S ( x )$]. + For what value(s) of #math.equation(block: false, alt: "x")[$x$] does #math.equation(block: false, alt: "S")[$S$] take on its maximum and minimum values? #figure(figph[curve], alt: "curve", caption: none) + #math.equation(block: false, alt: "0 , the fraction 1 over 2 , 0")[$0 , display(frac(1, 2)) , 0$] + #math.equation(block: false, alt: "0.9")[$0.9$] + #math.equation(block: false, alt: "the fraction minus 5 over 6")[$display(frac(− 5, 6))$], #math.equation(block: false, alt: "the fraction minus 1 over 6")[$display(frac(− 1, 6))$], #math.equation(block: false, alt: "the fraction 7 over 6")[$display(frac(7, 6))$], #math.equation(block: false, alt: "the fraction 11 over 6")[$display(frac(11, 6))$] + Max: #math.equation(block: false, alt: "1")[$1$]; min: #math.equation(block: false, alt: "minus 1")[$− 1$] + Max at #math.equation(block: false, alt: "x equals minus 1.5 , 0.5")[$x = − 1.5 , 0.5$]; min at #math.equation(block: false, alt: "x equals minus 0.5 , 1.5")[$x = − 0.5 , 1.5$] + Find #math.equation(block: false, alt: "C open parenthesis 0 close parenthesis")[$C ( 0 )$], #math.equation(block: false, alt: "C open parenthesis minus the fraction 1 over 3 close parenthesis")[$C ( − display(frac(1, 3)) )$], and #math.equation(block: false, alt: "C open parenthesis 1 close parenthesis")[$C ( 1 )$]. + Estimate the value of #math.equation(block: false, alt: "C open parenthesis the fraction 1 over 6 close parenthesis")[$C ( display(frac(1, 6)) )$] from the graph. + For what value(s) of #math.equation(block: false, alt: "x")[$x$] is #math.equation(block: false, alt: "C open parenthesis x close parenthesis equals the fraction 1 over 2")[$C ( x ) = display(frac(1, 2))$]? + Find the maximum and minimum values of #math.equation(block: false, alt: "C open parenthesis x close parenthesis")[$C ( x )$]. + For what value(s) of #math.equation(block: false, alt: "x")[$x$] does #math.equation(block: false, alt: "C")[$C$] take on its maximum and minimum values? #figure(figph[curve], alt: "curve", caption: none) + Find #math.equation(block: false, alt: "F open parenthesis minus 3 close parenthesis")[$F ( − 3 )$], #math.equation(block: false, alt: "F open parenthesis minus 2 close parenthesis")[$F ( − 2 )$], and #math.equation(block: false, alt: "F open parenthesis 2 close parenthesis")[$F ( 2 )$]. + For what value(s) of #math.equation(block: false, alt: "s")[$s$] is #math.equation(block: false, alt: "F open parenthesis s close parenthesis equals minus 1")[$F ( s ) = − 1$]? + Find the maximum and minimum values of #math.equation(block: false, alt: "F open parenthesis s close parenthesis")[$F ( s )$]. + For what value(s) of #math.equation(block: false, alt: "s")[$s$] does #math.equation(block: false, alt: "F")[$F$] take on its maximum and minimum values? #figure(figph[curve], alt: "curve", caption: none) + #math.equation(block: false, alt: "2 , 2 , 1")[$2 , 2 , 1$] + #math.equation(block: false, alt: "minus 6 less than or equal to s less than minus 4")[$− 6 ≤ s < − 4 " "$] or #math.equation(block: false, alt: "0 less than or equal to s less than 2")[$" " 0 ≤ s < 2$] + Max: #math.equation(block: false, alt: "2")[$2$]; min: #math.equation(block: false, alt: "minus 1")[$− 1$] + Max for #math.equation(block: false, alt: "minus 3 less than or equal to s less than minus 1")[$− 3 ≤ s < − 1 " "$] or #math.equation(block: false, alt: "3 less than or equal to s less than 5")[$" " 3 ≤ s < 5$]; min for #math.equation(block: false, alt: "minus 6 less than or equal to s less than minus 4")[$− 6 ≤ s < − 4 " "$] or #math.equation(block: false, alt: "0 less than or equal to s less than 2")[$" " 0 ≤ s < 2$] + Find #math.equation(block: false, alt: "P open parenthesis minus 3 close parenthesis")[$P ( − 3 )$], #math.equation(block: false, alt: "P open parenthesis minus 2 close parenthesis")[$P ( − 2 )$], and #math.equation(block: false, alt: "P open parenthesis 1 close parenthesis")[$P ( 1 )$]. + For what value(s) of #math.equation(block: false, alt: "n")[$n$] is #math.equation(block: false, alt: "P open parenthesis n close parenthesis equals 0")[$P ( n ) = 0$]? + Find the maximum and minimum values of #math.equation(block: false, alt: "P open parenthesis n close parenthesis")[$P ( n )$]. + For what value(s) of #math.equation(block: false, alt: "n")[$n$] does #math.equation(block: false, alt: "P")[$P$] take on its maximum and minimum values? #figure(figph[curve], alt: "curve", caption: none) Which of the graphs in Problems 9 and 10 represent functions? #figure(figph[5 curves], alt: "5 curves", caption: none) (a) and (d) #figure(figph[5 curves], alt: "5 curves", caption: none) In Problems 11–16, + Make a table of values and sketch a graph of the function by plotting points. (Use the suggested #math.equation(block: false, alt: "x")[$x$]-values.) + Use a graphing utility to graph the function. Compare the graphing utility's graph with your sketch. #math.equation(block: true, alt: "g open parenthesis x close parenthesis equals x cubed plus 4")[$g ( x ) = x^(3) + 4$]; #math.equation(block: true, alt: "x equals minus 2 , minus 1 , and so on , 2")[$#hide($00000$) x = − 2 , − 1 , … , 2$] #figure(figph[cubic], alt: "cubic", caption: none) #math.equation(block: true, alt: "h open parenthesis x close parenthesis equals 2 plus the square root of x")[$h ( x ) = 2 + sqrt(x)$]; #math.equation(block: true, alt: "x equals 0 , 1 , and so on , 9")[$#hide($00000$) x = 0 , 1 , … , 9$] #math.equation(block: true, alt: "G open parenthesis x close parenthesis equals the square root of 4 minus x")[$G ( x ) = sqrt(4 − x)$]; #math.equation(block: true, alt: "x equals minus 5 , minus 4 , and so on , 4")[$#hide($00000$) x = − 5 , − 4 , … , 4$] #figure(figph[square root], alt: "square root", caption: none) #math.equation(block: true, alt: "F open parenthesis x close parenthesis equals the square root of x minus 1")[$F ( x ) = sqrt(x − 1)$]; #math.equation(block: true, alt: "x equals 1 , 2 , and so on , 10")[$#hide($00000$) x = 1 , 2 , … , 10$] #math.equation(block: true, alt: "v open parenthesis x close parenthesis equals 1 plus 6 x minus x cubed")[$v ( x ) = 1 + 6 x − x^(3)$]; #math.equation(block: true, alt: "x equals minus 3 , minus 2 , and so on , 3")[$#hide($00000$) x = − 3 , − 2 , … , 3$] #figure(figph[cubic], alt: "cubic", caption: none) #math.equation(block: true, alt: "w open parenthesis x close parenthesis equals x cubed minus 8 x")[$w ( x ) = x^(3) − 8 x$]; #math.equation(block: true, alt: "x equals minus 4 , minus 3 , and so on , 4")[$#hide($00000$) x = − 4 , − 3 , … , 4$] The graph shows the speed of sound in the ocean as a function of depth, #math.equation(block: false, alt: "S equals f open parenthesis d close parenthesis")[$S = f ( d )$]. The speed of sound is affected both by increasing water pressure and by dropping temperature. (Source: Scientific American) #figure(figph[speed of sound vs depth], alt: "speed of sound vs depth", caption: none) + Evaluate #math.equation(block: false, alt: "f open parenthesis 1000 close parenthesis")[$f ( 1000 )$] and explain its meaning. + Solve #math.equation(block: false, alt: "f open parenthesis d close parenthesis equals 1500")[$f ( d ) = 1500$] and explain its meaning. + At what depth is the speed of sound the slowest, and what is the speed? Write your answer with function notation. + Describe the behavior of #math.equation(block: false, alt: "f open parenthesis d close parenthesis")[$f ( d )$] as #math.equation(block: false, alt: "d")[$d$] increases. + #math.equation(block: false, alt: "f open parenthesis 1000 close parenthesis equals 1495")[$f ( 1000 ) = 1495$]: The speed of sound at a depth of #math.equation(block: false, alt: "1000")[$1000$] meters is approximately #math.equation(block: false, alt: "1495")[$1495$] meters per second. + #math.equation(block: false, alt: "d equals 570")[$d = 570$] or #math.equation(block: false, alt: "d equals 1070")[$d = 1070$]: The speed of sound is #math.equation(block: false, alt: "1500")[$1500$] meters per second at both a depth of #math.equation(block: false, alt: "570")[$570$] meters and a depth of #math.equation(block: false, alt: "1070")[$1070$] meters. + The slowest speed occurs at a depth of about #math.equation(block: false, alt: "810")[$810$] meters and the speed is about #math.equation(block: false, alt: "1487")[$1487$] meters per second, so #math.equation(block: false, alt: "f open parenthesis 810 close parenthesis equals 1487")[$f ( 810 ) = 1487$]. + #math.equation(block: false, alt: "f")[$f$] increases from about #math.equation(block: false, alt: "1533")[$1533$] to #math.equation(block: false, alt: "1541")[$1541$] in the first #math.equation(block: false, alt: "110")[$110$] meters of depth, then drops to about #math.equation(block: false, alt: "1487")[$1487$] at #math.equation(block: false, alt: "810")[$810$] meters, then rises again, passing #math.equation(block: false, alt: "1553")[$1553$] at a depth of about #math.equation(block: false, alt: "1600")[$1600$] meters. The graph shows the water level (meters) in Lake Superior as a function of time, #math.equation(block: false, alt: "L equals f open parenthesis t close parenthesis")[$L = f ( t )$]. (Source: The Canadian Hydrographic Service) #figure(figph[water level vs time], alt: "water level vs time", caption: none) + Evaluate #math.equation(block: false, alt: "f open parenthesis 1997 close parenthesis")[$f ( 1997 )$] and explain its meaning. + Solve #math.equation(block: false, alt: "f open parenthesis t close parenthesis equals 183.5")[$f ( t ) = 183.5$] and explain its meaning. + In which two years did Lake Superior reach its highest levels, and what were those levels? Write your answers with function notation. + Over which two-year period did the water level drop the most? The graph shows the federal debt as a percentage of the gross domestic product (GDP), as a function of time, #math.equation(block: false, alt: "D equals f open parenthesis t close parenthesis")[$D = f ( t )$]. (Source: Office of Management and Budget) #figure(figph[federal debt], alt: "federal debt", caption: none) + Evaluate #math.equation(block: false, alt: "f open parenthesis 1985 close parenthesis")[$f ( 1985 )$] and explain its meaning. + Solve #math.equation(block: false, alt: "f open parenthesis t close parenthesis equals 70")[$f ( t ) = 70$] and explain its meaning. + When did the federal debt reach its highest level since 1960, and what was that level? Write your answer with function notation. + What is the longest time interval over which the federal debt was decreasing? + #math.equation(block: false, alt: "f open parenthesis 1985 close parenthesis equals 41")[$f ( 1985 ) = 41$]: The federal debt in #math.equation(block: false, alt: "1985")[$1985$] was about #math.equation(block: false, alt: "41 %")[$41 upright(%)$] of the gross domestic product. + #math.equation(block: false, alt: "t equals 1942")[$t = 1942$] or #math.equation(block: false, alt: "t equals 1955")[$t = 1955$]: The federal debt was #math.equation(block: false, alt: "70 %")[$70 upright(%)$] of the gross domestic product in #math.equation(block: false, alt: "1942")[$1942$] and #math.equation(block: false, alt: "1955")[$1955$]. + In about #math.equation(block: false, alt: "1997")[$1997$], the debt was about #math.equation(block: false, alt: "67 %")[$67 upright(%)$] of the gross domestic product, so #math.equation(block: false, alt: "f open parenthesis 1997 close parenthesis approximately equals 67.3")[$f ( 1997 ) ≈ 67.3$]. + The percentage basically dropped from 1946 to 1973, but there were small rises around 1950, 1954, 1958, and 1968, so the longest time interval was from 1958 to 1967. The graph shows the elevation (feet) of the 2005 Los Angeles Marathon course as a function of the distance (miles) into the race, #math.equation(block: false, alt: "a equals f open parenthesis d close parenthesis")[$a = f ( d )$]. (Source: Los Angeles Times, March 3, 2005) #figure(figph[LA marathon elevation], alt: "LA marathon elevation", caption: none) + Evaluate #math.equation(block: false, alt: "f open parenthesis 5 close parenthesis")[$f ( 5 )$] and explain its meaning. + Solve #math.equation(block: false, alt: "f open parenthesis d close parenthesis equals 200")[$f ( d ) = 200$] and explain its meaning. + Where does the marathon course reach its lowest elevation, and what is that elevation? Write your answer with function notation. + Give three intervals over which the elevation is increasing. The figure shows a graph of #math.equation(block: false, alt: "y equals minus 2 x plus 6")[$y = − 2 x + 6$]. #figure(figph[y=-2x+6], alt: "y=-2x+6", caption: none) + Use the graph to find all values of #math.equation(block: false, alt: "x")[$x$] for which + #math.equation(block: false, alt: "y equals 12")[$y = 12$] + #math.equation(block: false, alt: "y greater than 12")[$y > 12$] + #math.equation(block: false, alt: "y less than 12")[$y < 12$] + Use the graph to solve + #math.equation(block: false, alt: "minus 2 x plus 6 equals 12")[$− 2 x + 6 = 12$] + #math.equation(block: false, alt: "minus 2 x plus 6 greater than 12")[$− 2 x + 6 > 12$] + #math.equation(block: false, alt: "minus 2 x plus 6 less than 12")[$− 2 x + 6 < 12$] + Explain why your answers to parts (a) and (b) are the same. + + #math.equation(block: false, alt: "x equals minus 3")[$x = − 3$] + #math.equation(block: false, alt: "x less than minus 3")[$x < − 3$] + #math.equation(block: false, alt: "x greater than minus 3")[$x > − 3$] + + #math.equation(block: false, alt: "x equals minus 3")[$x = − 3$] + #math.equation(block: false, alt: "x less than minus 3")[$x < − 3$] + #math.equation(block: false, alt: "x greater than minus 3")[$x > − 3$] + On the graph of #math.equation(block: false, alt: "y equals minus 2 x plus 6")[$y = − 2 x + 6$], a value of #math.equation(block: false, alt: "y")[$y$] is the same as a value of #math.equation(block: false, alt: "minus 2 x plus 6")[$− 2 x + 6$], so parts (a) and (b) are asking for the same #math.equation(block: false, alt: "x")[$x$]'s. The figure shows a graph of #math.equation(block: false, alt: "y equals the fraction minus x over 3 minus 6")[$y = display(frac(− x, 3)) − 6$]. #figure(figph[y=-x/3 - 6], alt: "y=-x/3 - 6", caption: none) + Use the graph to find all values of #math.equation(block: false, alt: "x")[$x$] for which + #math.equation(block: false, alt: "y equals minus 4")[$y = − 4$] + #math.equation(block: false, alt: "y greater than minus 4")[$y > − 4$] + #math.equation(block: false, alt: "y less than minus 4")[$y < − 4$] + Use the graph to solve + #math.equation(block: false, alt: "the fraction minus x over 3 minus 6 equals minus 4")[$display(frac(− x, 3)) − 6 = − 4$] + #math.equation(block: false, alt: "the fraction minus x over 3 minus 6 greater than minus 4")[$display(frac(− x, 3)) − 6 > − 4$] + #math.equation(block: false, alt: "the fraction minus x over 3 minus 6 less than minus 4")[$display(frac(− x, 3)) − 6 < − 4$] + Explain why your answers to parts (a) and (b) are the same. In Problems 23 and 24, use the graph to solve the equation or inequality, and then solve algebraically. (To review solving linear inequalities algebraically, see Algebra Skills Refresher.) The figure shows the graph of #math.equation(block: false, alt: "y equals 1.4 x minus 0.64")[$y = 1.4 x − 0.64$]. Solve the following: #figure(figph[y = 1.4x - 0.64], alt: "y = 1.4x - 0.64", caption: none) + #math.equation(block: false, alt: "1.4 x minus 0.64 equals 0.2")[$1.4 x − 0.64 = 0.2$] + #math.equation(block: false, alt: "minus 1.2 equals 1.4 x minus 0.64")[$− 1.2 = 1.4 x − 0.64$] + #math.equation(block: false, alt: "1.4 x minus 0.64 greater than 0.2")[$1.4 x − 0.64 > 0.2$] + #math.equation(block: false, alt: "minus 1.2 greater than 1.4 x minus 0.64")[$− 1.2 > 1.4 x − 0.64$] + #math.equation(block: false, alt: "x equals 0.6")[$x = 0.6$] + #math.equation(block: false, alt: "x equals minus 0.4")[$x = − 0.4$] + #math.equation(block: false, alt: "x greater than 0.6")[$x > 0.6$] + #math.equation(block: false, alt: "x less than minus 0.4")[$x < − 0.4$] The figure shows the graph of #math.equation(block: false, alt: "y equals minus 2.4 x plus 2.32")[$y = − 2.4 x + 2.32$]. Solve the following: #figure(figph[y = -2.4x + 2.32], alt: "y = -2.4x + 2.32", caption: none) + #math.equation(block: false, alt: "1.6 equals minus 2.4 x plus 2.32")[$1.6 = − 2.4 x + 2.32$] + #math.equation(block: false, alt: "minus 2.4 x plus 2.32 equals 0.4")[$− 2.4 x + 2.32 = 0.4$] + #math.equation(block: false, alt: "minus 2.4 x plus 2.32 greater than or equal to 1.6")[$− 2.4 x + 2.32 ≥ 1.6$] + #math.equation(block: false, alt: "0.4 greater than or equal to minus 2.4 x plus 2.32")[$0.4 ≥ − 2.4 x + 2.32$] For Problems 25–30, use the graphs to estimate solutions to the equations and inequalities. The figure shows the graph of #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the fraction 12 over 2 plus x squared")[$g ( x ) = display(frac(12, 2 + x^(2)))$]. #figure(figph[bell-like curve], alt: "bell-like curve", caption: none) + Solve #math.equation(block: false, alt: "the fraction 12 over 2 plus x squared equals 4")[$display(frac(12, 2 + x^(2))) = 4$] + Solve #math.equation(block: false, alt: "1 less than or equal to the fraction 12 over 2 plus x squared less than or equal to 2")[$1 ≤ display(frac(12, 2 + x^(2))) ≤ 2$] + #math.equation(block: false, alt: "x equals minus 1")[$x = − 1$] or #math.equation(block: false, alt: "x equals 1")[$x = 1$] + Approximately #math.equation(block: false, alt: "minus 3 less than or equal to x less than or equal to minus 2")[$− 3 ≤ x ≤ − 2$] or #math.equation(block: false, alt: "2 less than or equal to x less than or equal to 3")[$2 ≤ x ≤ 3$] The figure shows the graph of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the fraction 30 the square root of x over 1 plus x")[$f ( x ) = display(frac(30 sqrt(x), 1 + x))$]. #figure(figph[curve], alt: "curve", caption: none) + Solve #math.equation(block: false, alt: "the fraction 30 the square root of x over 1 plus x equals 15")[$display(frac(30 sqrt(x), 1 + x)) = 15$] + Solve #math.equation(block: false, alt: "the fraction 30 the square root of x over 1 plus x less than 12")[$display(frac(30 sqrt(x), 1 + x)) < 12$] The figure shows a graph of #math.equation(block: false, alt: "B equals the fraction 1 over 3 p cubed minus 3 p plus 2")[$B = display(frac(1, 3)) p^(3) − 3 p + 2$]. #figure(figph[cubic curve], alt: "cubic curve", caption: none) + Solve #math.equation(block: false, alt: "the fraction 1 over 3 p cubed minus 3 p plus 2 equals 6")[$display(frac(1, 3)) p^(3) − 3 p + 2 = 6$] + Solve #math.equation(block: false, alt: "the fraction 1 over 3 p cubed minus 3 p plus 2 equals 5")[$display(frac(1, 3)) p^(3) − 3 p + 2 = 5$] + Solve #math.equation(block: false, alt: "the fraction 1 over 3 p cubed minus 3 p plus 2 less than 1")[$display(frac(1, 3)) p^(3) − 3 p + 2 < 1$] + What range of values does #math.equation(block: false, alt: "B")[$B$] have for #math.equation(block: false, alt: "p")[$p$] between #math.equation(block: false, alt: "minus 2.5")[$− 2.5$] and #math.equation(block: false, alt: "0.5")[$0.5$]? + For what values of #math.equation(block: false, alt: "p")[$p$] is #math.equation(block: false, alt: "B")[$B$] increasing? + #math.equation(block: false, alt: "3.5")[$3.5$] + #math.equation(block: false, alt: "minus 2.2 , minus 1.2 , 3.4")[$− 2.2 , − 1.2 , 3.4$] + #math.equation(block: false, alt: "p less than minus 3.1")[$p < − 3.1$] or #math.equation(block: false, alt: "0.3 less than p less than 2.8")[$0.3 < p < 2.8$] + #math.equation(block: false, alt: "0.5 less than B less than 5.5")[$0.5 < B < 5.5$] + #math.equation(block: false, alt: "p less than minus 1.7")[$p < − 1.7$] or #math.equation(block: false, alt: "p greater than 1.7")[$p > 1.7$] The figure shows a graph of #math.equation(block: false, alt: "H equals t cubed minus 4 t squared minus 4 t plus 12")[$H = t^(3) − 4 t^(2) − 4 t + 12$]. #figure(figph[cubic curve], alt: "cubic curve", caption: none) + Solve #math.equation(block: false, alt: "t cubed minus 4 t squared minus 4 t plus 12 equals minus 4")[$t^(3) − 4 t^(2) − 4 t + 12 = − 4$] + Solve #math.equation(block: false, alt: "t cubed minus 4 t squared minus 4 t plus 12 equals 17")[$t^(3) − 4 t^(2) − 4 t + 12 = 17$] + Solve #math.equation(block: false, alt: "t cubed minus 4 t squared minus 4 t plus 12 greater than 6")[$t^(3) − 4 t^(2) − 4 t + 12 > 6$] + Estimate the horizontal and vertical intercepts of the graph. + For what values of #math.equation(block: false, alt: "t")[$t$] is #math.equation(block: false, alt: "H")[$H$] increasing? The figure shows a graph of #math.equation(block: false, alt: "M equals g open parenthesis q close parenthesis")[$M = g ( q )$]. #figure(figph[mystery curve], alt: "mystery curve", caption: none) + Find all values of #math.equation(block: false, alt: "q")[$q$] for which + #math.equation(block: false, alt: "g open parenthesis q close parenthesis equals 0")[$g ( q ) = 0$] + #math.equation(block: false, alt: "g open parenthesis q close parenthesis equals 16")[$g ( q ) = 16$] + #math.equation(block: false, alt: "g open parenthesis q close parenthesis less than 6")[$g ( q ) < 6$] + For what values of #math.equation(block: false, alt: "q")[$q$] is #math.equation(block: false, alt: "g open parenthesis q close parenthesis")[$g ( q )$] increasing? + + #math.equation(block: false, alt: "minus 2 , 2")[$− 2 , 2$] + #math.equation(block: false, alt: "minus 2.8 , 0 , 2.8")[$− 2.8 , 0 , 2.8$] + #math.equation(block: false, alt: "minus 2.5 less than q less than minus 1.25")[$− 2.5 < q < − 1.25$] or #math.equation(block: false, alt: "1.25 less than q less than 2.5")[$1.25 < q < 2.5$] + #math.equation(block: false, alt: "minus 2 less than q less than 0")[$− 2 < q < 0$] or #math.equation(block: false, alt: "q greater than 2")[$q > 2$] The figure shows a graph of #math.equation(block: false, alt: "P equals f open parenthesis t close parenthesis")[$P = f ( t )$]. #figure(figph[mystery curve], alt: "mystery curve", caption: none) + Find all values of #math.equation(block: false, alt: "t")[$t$] for which + #math.equation(block: false, alt: "f open parenthesis t close parenthesis equals 3")[$f ( t ) = 3$] + #math.equation(block: false, alt: "f open parenthesis t close parenthesis greater than 4.5")[$f ( t ) > 4.5$] + #math.equation(block: false, alt: "2 less than or equal to f open parenthesis t close parenthesis less than or equal to 4")[$2 ≤ f ( t ) ≤ 4$] + For what values of #math.equation(block: false, alt: "t")[$t$] is #math.equation(block: false, alt: "f open parenthesis t close parenthesis")[$f ( t )$] decreasing? + Delbert reads the following values from the graph of a function: #math.equation(block: true, alt: "f open parenthesis minus 3 close parenthesis equals 5 , f open parenthesis minus 1 close parenthesis equals 2 , f open parenthesis 1 close parenthesis equals 0 ,")[$f ( − 3 ) = 5 , " " f ( − 1 ) = 2 , " " f ( 1 ) = 0 ,$] #math.equation(block: true, alt: "f open parenthesis minus 1 close parenthesis equals minus 4 , f open parenthesis minus 3 close parenthesis equals minus 2")[$f ( − 1 ) = − 4 , " " f ( − 3 ) = − 2$] Can his readings be correct? Explain why or why not. + Francine reads the following values from the graph of a function: #math.equation(block: true, alt: "g open parenthesis minus 2 close parenthesis equals 6 , g open parenthesis 0 close parenthesis equals 0 , g open parenthesis 2 close parenthesis equals 6 ,")[$g ( − 2 ) = 6 , " " g ( 0 ) = 0 , " " g ( 2 ) = 6 ,$] #math.equation(block: true, alt: "g open parenthesis 4 close parenthesis equals 0 , g open parenthesis 6 close parenthesis equals 6")[$g ( 4 ) = 0 , " " g ( 6 ) = 6$] Can her readings be correct? Explain why or why not. + He has an error: #math.equation(block: false, alt: "f open parenthesis minus 3 close parenthesis")[$f ( − 3 )$] cannot have both the value #math.equation(block: false, alt: "5")[$5$] and also the value #math.equation(block: false, alt: "minus 2")[$− 2$], and #math.equation(block: false, alt: "f open parenthesis minus 1 close parenthesis")[$f ( − 1 )$] cannot have both values #math.equation(block: false, alt: "2")[$2$] and #math.equation(block: false, alt: "minus 4")[$− 4$]. + Her readings are possible for a function: each input has only one output. + Sketch the graph of a function that has the following values: #math.equation(block: true, alt: "F open parenthesis minus 2 close parenthesis equals 3 , F open parenthesis minus 1 close parenthesis equals 3 , F open parenthesis 0 close parenthesis equals 3 ,")[$F ( − 2 ) = 3 , " " F ( − 1 ) = 3 , " " F ( 0 ) = 3 ,$] #math.equation(block: true, alt: "F open parenthesis 1 close parenthesis equals 3 , F open parenthesis 2 close parenthesis equals 3")[$F ( 1 ) = 3 , " " F ( 2 ) = 3$] + Sketch the graph of a function that has the following values: #math.equation(block: true, alt: "G open parenthesis minus 2 close parenthesis equals 1 , G open parenthesis minus 1 close parenthesis equals 0 , G open parenthesis 0 close parenthesis equals minus 1 ,")[$G ( − 2 ) = 1 , " " G ( − 1 ) = 0 , " " G ( 0 ) = − 1 ,$] #math.equation(block: true, alt: "G open parenthesis 1 close parenthesis equals 0 , G open parenthesis 2 close parenthesis equals 1")[$G ( 1 ) = 0 , " " G ( 2 ) = 1$] For Problems 33–36, graph each function in the friendly window #math.equation(block: true, alt: "Xmin, equals minus 9.4, Xmax equals 9.4; Ymin, equals minus 10, Ymax equals 10")[$"Xmin" & = − 9.4 & & "Xmax" = 9.4 \ "Ymin" & = − 10 & & "Ymax" = 10$] Then answer the questions about the graph. (See Using a Graphing Calculator for an explanation of friendly windows.) #math.equation(block: true, alt: "g open parenthesis x close parenthesis equals the square root of 36 minus x squared")[$g ( x ) = sqrt(36 − x^(2))$] + Complete the table. (Round values to tenths.)#figure(table( columns: 5, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "minus 4")[$− 4$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "5")[$5$]]), [#math.equation(block: false, alt: "g open parenthesis x close parenthesis")[$g ( x )$]], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], )) + Find all points on the graph for which #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals 3.6")[$g ( x ) = 3.6$]. + #figure(table( columns: 5, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "minus 4")[$− 4$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "5")[$5$]]), [#math.equation(block: false, alt: "g open parenthesis x close parenthesis")[$g ( x )$]], [#math.equation(block: false, alt: "4.5")[$4.5$]], [#math.equation(block: false, alt: "5.7")[$5.7$]], [#math.equation(block: false, alt: "5.2")[$5.2$]], [#math.equation(block: false, alt: "3.3")[$3.3$]], )) + #math.equation(block: false, alt: "minus 4.8 , 4.8")[$− 4.8 , 4.8$] #math.equation(block: true, alt: "g open parenthesis x close parenthesis equals the square root of x squared minus 6")[$g ( x ) = sqrt(x^(2)) − 6$] + Complete the table. (Round values to tenths.)#figure(table( columns: 5, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "minus 8")[$− 8$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "6")[$6$]]), [#math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], )) + Find all points on the graph for which #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals minus 2")[$f ( x ) = − 2$]. #math.equation(block: true, alt: "F open parenthesis x close parenthesis equals 0.5 x cubed minus 4 x")[$F ( x ) = 0.5 x^(3) − 4 x$] + Estimate the coordinates of the turning points of the graph, that is, where the graph changes from increasing to decreasing or vice versa. + Write an equation of the form #math.equation(block: false, alt: "F open parenthesis a close parenthesis equals b")[$F ( a ) = b$] for each turning point. + #math.equation(block: false, alt: "open parenthesis minus 1.6 , 4.352 close parenthesis , open parenthesis 1.6 , minus 4.352 close parenthesis")[$( − 1.6 , 4.352 ) , ( 1.6 , − 4.352 )$] + #math.equation(block: false, alt: "F open parenthesis minus 1.6 close parenthesis equals 4.352")[$F ( − 1.6 ) = 4.352$]; #math.equation(block: false, alt: "F open parenthesis 1.6 close parenthesis equals minus 4.352")[$F ( 1.6 ) = − 4.352$] #math.equation(block: true, alt: "G open parenthesis x close parenthesis equals 2 plus 4 x minus x cubed")[$G ( x ) = 2 + 4 x − x^(3)$] + Estimate the coordinates of the turning points of the graph, that is, where the graph changes from increasing to decreasing or vice versa. + Write an equation of the form #math.equation(block: false, alt: "G open parenthesis a close parenthesis equals b")[$G ( a ) = b$] for each turning point. For Problems 37–40, graph the function + first using the standard window. + then using the suggested window. Explain how the window alters the appearance of the graph in each case. #math.equation(block: true, alt: "h open parenthesis x close parenthesis equals the fraction 1 over x squared plus 10")[$h ( x ) = display(frac(1, x^(2) + 10))$] #math.equation(block: true, alt: "Xmin, equals minus 5, Xmax equals 5; Ymin, equals 0, Ymax equals 0.5")[$"Xmin" & = − 5 & & "Xmax" = 5 \ "Ymin" & = 0 & & "Ymax" = 0.5$] + #figure(figph[standard GC window], alt: "standard GC window", caption: none) + #figure(figph[curve visible in GC window], alt: "curve visible in GC window", caption: none) #linebreak() The curve cannot be distinguished from the #math.equation(block: false, alt: "x")[$x$]-axis in the standard window because the values of #math.equation(block: false, alt: "y")[$y$] are closer to zero than the resolution of the calculator can display. The second window provides sufficient resolution to see the curve. #math.equation(block: true, alt: "H open parenthesis x close parenthesis equals the square root of 1 minus x squared")[$H ( x ) = sqrt(1 − x^(2))$] #math.equation(block: true, alt: "Xmin, equals minus 2, Xmax equals 2; Ymin, equals minus 2, Ymax equals 2")[$"Xmin" & = − 2 & & "Xmax" = 2 \ "Ymin" & = − 2 & & "Ymax" = 2$] #math.equation(block: true, alt: "P open parenthesis x close parenthesis equals open parenthesis x minus 8 close parenthesis open parenthesis x plus 6 close parenthesis open parenthesis x minus 15 close parenthesis")[$P ( x ) = ( x − 8 ) ( x + 6 ) ( x − 15 )$] #math.equation(block: true, alt: "Xmin, equals minus 10, Xmax equals 20; Ymin, equals minus 250, Ymax equals 750")[$"Xmin" & = − 10 & & "Xmax" = 20 \ "Ymin" & = − 250 & & "Ymax" = 750$] + #figure(figph[standard GC window], alt: "standard GC window", caption: none) + #figure(figph[curve visible in GC window], alt: "curve visible in GC window", caption: none) #linebreak() The curve looks like two vertical lines in the standard window because that window covers too small a region of the plane. The second window allows us to see the turning points of the curve. #math.equation(block: true, alt: "p open parenthesis x close parenthesis equals 200 x cubed")[$p ( x ) = 200 x^(3)$] #math.equation(block: true, alt: "Xmin, equals minus 5, Xmax equals 5; Ymin, equals minus 10 , 000, Ymax equals 10 , 000")[$"Xmin" & = − 5 & & "Xmax" = 5 \ "Ymin" & = − 10 , 000 & & "Ymax" = 10 , 000$] For Problems 41–44, graph the equation with the ZInteger setting. (Press ZOOM #math.equation(block: false, alt: "6")[$6$],then ZOOM #math.equation(block: false, alt: "8")[$8$] ENTER.) Use the graph to answer each question. Use the equation to verify your answers. Graph #math.equation(block: false, alt: "y equals 2 x minus 3")[$y = 2 x − 3$] + For what value of #math.equation(block: false, alt: "x")[$x$] is #math.equation(block: false, alt: "y equals 5")[$y = 5$]? + For what value of #math.equation(block: false, alt: "x")[$x$] is #math.equation(block: false, alt: "y equals minus 13")[$y = − 13$]? + For what values of #math.equation(block: false, alt: "x")[$x$] is #math.equation(block: false, alt: "y greater than minus 1")[$y > − 1$]? + For what values of #math.equation(block: false, alt: "x")[$x$] is #math.equation(block: false, alt: "y less than 25")[$y < 25$]? + #math.equation(block: false, alt: "x equals 4")[$x = 4$] + #math.equation(block: false, alt: "x equals minus 5")[$x = − 5$] + #math.equation(block: false, alt: "x greater than 1")[$x > 1$] + #math.equation(block: false, alt: "x less than 14")[$x < 14$] Graph #math.equation(block: false, alt: "y equals 4 minus 2 x")[$y = 4 − 2 x$] + For what value of #math.equation(block: false, alt: "x")[$x$] is #math.equation(block: false, alt: "y equals 6")[$y = 6$]? + For what value of #math.equation(block: false, alt: "x")[$x$] is #math.equation(block: false, alt: "y equals minus 4")[$y = − 4$]? + For what values of #math.equation(block: false, alt: "x")[$x$] is #math.equation(block: false, alt: "y greater than minus 12")[$y > − 12$]? + For what values of #math.equation(block: false, alt: "x")[$x$] is #math.equation(block: false, alt: "y less than 18")[$y < 18$]? Graph #math.equation(block: false, alt: "y equals 6.5 minus 1.8 x")[$y = 6.5 − 1.8 x$] + For what value of #math.equation(block: false, alt: "x")[$x$] is #math.equation(block: false, alt: "y equals minus 13.3")[$y = − 13.3$]? + For what value of #math.equation(block: false, alt: "x")[$x$] is #math.equation(block: false, alt: "y equals 24.5")[$y = 24.5$]? + For what values of #math.equation(block: false, alt: "x")[$x$] is #math.equation(block: false, alt: "y less than or equal to 15.5")[$y ≤ 15.5$]? + For what values of #math.equation(block: false, alt: "x")[$x$] is #math.equation(block: false, alt: "y greater than or equal to minus 7.9")[$y ≥ − 7.9$]? + #math.equation(block: false, alt: "x equals 11")[$x = 11$] + #math.equation(block: false, alt: "x equals minus 10")[$x = − 10$] + #math.equation(block: false, alt: "x greater than or equal to minus 5")[$x ≥ − 5$] + #math.equation(block: false, alt: "x less than or equal to 8")[$x ≤ 8$] Graph #math.equation(block: false, alt: "y equals 0.2 x plus 1.4")[$y = 0.2 x + 1.4$] + For what value of #math.equation(block: false, alt: "x")[$x$] is #math.equation(block: false, alt: "y equals minus 5.2")[$y = − 5.2$]? + For what value of #math.equation(block: false, alt: "x")[$x$] is #math.equation(block: false, alt: "y equals 2.8")[$y = 2.8$]? + For what values of #math.equation(block: false, alt: "x")[$x$] is #math.equation(block: false, alt: "y less than or equal to minus 3.2")[$y ≤ − 3.2$]? + For what values of #math.equation(block: false, alt: "x")[$x$] is #math.equation(block: false, alt: "y greater than or equal to 4.4")[$y ≥ 4.4$]? For Problems 45–48, graph the equation with the ZInteger setting. Use the graph to solve each equation or inequality. Check your solutions algebraically. Graph #math.equation(block: false, alt: "y equals minus 0.4 x plus 3.7")[$y = − 0.4 x + 3.7$] + Solve #math.equation(block: false, alt: "minus 0.4 x plus 3.7 equals 2.1")[$− 0.4 x + 3.7 = 2.1$] + Solve #math.equation(block: false, alt: "minus 0.4 x plus 3.7 greater than minus 5.1")[$− 0.4 x + 3.7 > − 5.1$] + #math.equation(block: false, alt: "x equals 4")[$x = 4$] + #math.equation(block: false, alt: "x less than 22")[$x < 22$] Graph #math.equation(block: false, alt: "y equals 0.4 open parenthesis x minus 1.5 close parenthesis")[$y = 0.4 ( x − 1.5 )$] + Solve #math.equation(block: false, alt: "0.4 open parenthesis x minus 1.5 close parenthesis equals minus 8.6")[$0.4 ( x − 1.5 ) = − 8.6$] + Solve #math.equation(block: false, alt: "0.4 open parenthesis x minus 1.5 close parenthesis less than 8.6")[$0.4 ( x − 1.5 ) < 8.6$] Graph #math.equation(block: false, alt: "y equals the fraction 2 over 3 x minus 24")[$y = display(frac(2, 3)) x − 24$] + Solve #math.equation(block: false, alt: "the fraction 2 over 3 x minus 24 equals minus 10 the fraction 2 over 3")[$display(frac(2, 3)) x − 24 = − 10 display(frac(2, 3))$] + Solve #math.equation(block: false, alt: "the fraction 2 over 3 x minus 24 less than or equal to minus 19 the fraction 1 over 3")[$display(frac(2, 3)) x − 24 ≤ − 19 display(frac(1, 3))$] + #math.equation(block: false, alt: "x equals 20")[$x = 20$] + #math.equation(block: false, alt: "x less than or equal to 7")[$x ≤ 7$] Graph #math.equation(block: false, alt: "y equals the fraction 80 minus 3 x over 5")[$y = display(frac(80 − 3 x, 5))$]. + Solve #math.equation(block: false, alt: "the fraction 80 minus 3 x over 5 equals 22 the fraction 3 over 5")[$display(frac(80 − 3 x, 5)) = 22 display(frac(3, 5))$]. + Solve #math.equation(block: false, alt: "the fraction 80 minus 3 x over 5 less than or equal to minus 9 the fraction 2 over 5")[$display(frac(80 − 3 x, 5)) ≤ − 9 display(frac(2, 5))$]. Graph #math.equation(block: false, alt: "y equals 0.01 x cubed minus 0.1 x squared minus 2.75 x plus 15")[$y = 0.01 x^(3) − 0.1 x^(2) − 2.75 x + 15$]. + Use your graph to solve #math.equation(block: false, alt: "0.01 x cubed minus 0.1 x squared minus 2.75 x plus 15 equals 0")[$0.01 x^(3) − 0.1 x^(2) − 2.75 x + 15 = 0$]. + Press Y= and enter #math.equation(block: false, alt: "Y sub 2 equals 10")[$Y_(2) = 10$]. Press GRAPH, and you should see the horizontal line #math.equation(block: false, alt: "y equals 10")[$y = 10$] superimposed on your previous graph. How many solutions does the equation #math.equation(block: true, alt: "0.01 x cubed minus 0.1 x squared minus 2.75 x plus 15 equals 10")[$0.01 x^(3) − 0.1 x^(2) − 2.75 x + 15 = 10$] have? Estimate each solution to the nearest whole number. + #math.equation(block: false, alt: "minus 15 , 5 , 20")[$− 15 , 5 , 20$] + #math.equation(block: false, alt: "minus 13 , 2 , 22")[$− 13 , 2 , 22$] Graph #math.equation(block: false, alt: "y equals 2.5 x minus 0.025 x squared minus 0.005 x cubed")[$y = 2.5 x − 0.025 x^(2) − 0.005 x^(3)$]. + Use your graph to solve #math.equation(block: false, alt: "2.5 x minus 0.025 x squared minus 0.005 x cubed equals 0")[$2.5 x − 0.025 x^(2) − 0.005 x^(3) = 0$]. + Press Y= and enter #math.equation(block: false, alt: "Y sub 2 equals minus 5")[$Y_(2) = − 5$]. Press GRAPH, and you should see the horizontal line #math.equation(block: false, alt: "y equals minus 5")[$y = − 5$] superimposed on your previous graph. How many solutions does the equation #math.equation(block: true, alt: "2.5 x minus 0.025 x squared minus 0.005 x cubed equals minus 5")[$2.5 x − 0.025 x^(2) − 0.005 x^(3) = − 5$] have? Estimate each solution to the nearest whole number.