#set document(title: "11.2 Parabolas", 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")) == 11.2#h(0.6em)Parabolas #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Before you get started, take this readiness quiz. Graph: #math.equation(block: false, alt: "y equals −3 x squared plus 12 x minus 12 .")[$y = −3 x^(2) + 12 x − 12 .$] #linebreak() If you missed this problem, review . #solutionbox[ #figure(figph[A downward-opening parabola is graphed on a Cartesian coordinate system. Its vertex is located at (2, 0), and it passes through points such as (0, -4) and (4, -4).], alt: "A downward-opening parabola is graphed on a Cartesian coordinate system. Its vertex is located at (2, 0), and it passes through points such as (0, -4) and (4, -4).", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve by completing the square: #math.equation(block: false, alt: "x squared minus 6 x plus 6 equals 0 .")[$x^(2) − 6 x + 6 = 0 .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "x equals 3 plus or minus the square root of 3")[$x = 3 ± sqrt(3)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Write in standard form: #math.equation(block: false, alt: "y equals 3 x squared minus 6 x plus 5 .")[$y = 3 x^(2) − 6 x + 5 .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "y equals 3 x minus 1 squared plus 2")[$y = 3 attach(x − 1, t: 2) + 2$] ] ] === Graph Vertical Parabolas The next conic section we will look at is a #strong[parabola]. We define a parabola as all points in a plane that are the same distance from a fixed point and a fixed line. The fixed point is called the #strong[focus,] and the fixed line is called the #strong[directrix] of the parabola. #figure(figph[This figure shows a double cone. The bottom nappe is intersected by a plane in such a way that the intersection forms a parabola.], alt: "This figure shows a double cone. The bottom nappe is intersected by a plane in such a way that the intersection forms a parabola.", caption: none) #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Parabola] A #strong[parabola] is all points in a plane that are the same distance from a fixed point and a fixed line. The fixed point is called the #strong[focus,] and the fixed line is called the #strong[directrix] of the parabola. #figure(figph[This figure shows a parabola opening upwards. Below the parabola is a horizontal line labeled directrix. A vertical dashed line through the center of the parabola is labeled axis of symmetry. The point where the axis intersects the parabola is labeled vertex. A point on the axis, within the parabola is labeled focus. A line perpendicular to the directrix connects the directrix to a point on the parabola and another line connects this point to the focus. Both these lines are of the same length.], alt: "This figure shows a parabola opening upwards. Below the parabola is a horizontal line labeled directrix. A vertical dashed line through the center of the parabola is labeled axis of symmetry. The point where the axis intersects the parabola is labeled vertex. A point on the axis, within the parabola is labeled focus. A line perpendicular to the directrix connects the directrix to a point on the parabola and another line connects this point to the focus. Both these lines are of the same length.", caption: none) ] Previously, we learned to graph vertical parabolas from the general form or the standard form using properties. Those methods will also work here. We will summarize the properties here. #figure(table( columns: 3, align: left, inset: 6pt, [Vertical Parabolas], [], [], [], [General form #linebreak() #math.equation(block: false, alt: "y equals a x squared plus b x plus c")[$y = a x^(2) + b x + c$]], [Standard form #linebreak() #math.equation(block: false, alt: "y equals a open parenthesis x minus h close parenthesis squared plus k")[$y = a attach(( x − h ), t: 2) + k$]], [#strong[Orientation]], [#math.equation(block: false, alt: "a greater than 0")[$a > 0$] up; #math.equation(block: false, alt: "a less than 0")[$a < 0$] down], [#math.equation(block: false, alt: "a greater than 0")[$a > 0$] up; #math.equation(block: false, alt: "a less than 0")[$a < 0$] down], [#strong[Axis of symmetry]], [#math.equation(block: false, alt: "x equals minus the fraction b over 2 a")[$x = − frac(b, 2 a)$]], [#math.equation(block: false, alt: "x equals h")[$x = h$]], [#strong[Vertex]], [Substitute #math.equation(block: false, alt: "x equals minus the fraction b over 2 a")[$x = − frac(b, 2 a)$] and #linebreak() solve for #emph[y].], [#math.equation(block: false, alt: "open parenthesis h , k close parenthesis")[$( h , k )$]], [#strong[#emph[y]-intercept]], [Let #math.equation(block: false, alt: "x equals 0")[$x = 0$]], [Let #math.equation(block: false, alt: "x equals 0")[$x = 0$]], [#strong[#emph[x]-intercepts]], [Let #math.equation(block: false, alt: "y equals 0")[$y = 0$]], [Let #math.equation(block: false, alt: "y equals 0")[$y = 0$]], )) The graphs show what the parabolas look like when they open up or down. Their position in relation to the #emph[x]- or #emph[y]-axis is merely an example. #figure(figph[This figure shows two parabolas with axis x equals h and vertex h, k. The one on the left opens up and A is greater than 0. The one on the right opens down. Here A is less than 0.], alt: "This figure shows two parabolas with axis x equals h and vertex h, k. The one on the left opens up and A is greater than 0. The one on the right opens down. Here A is less than 0.", caption: none) To graph a parabola from these forms, we used the following steps. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Graph vertical parabolas #math.equation(block: false, alt: "open parenthesis y equals a x squared plus b x plus c or f open parenthesis x close parenthesis equals a open parenthesis x minus h close parenthesis squared plus k close parenthesis")[$( y = a x^(2) + b x + c #h(0.5em) "or" #h(0.5em) f ( x ) = a attach(( x − h ), t: 2) + k )$] using properties.] + Determine whether the parabola opens upward or downward. + Find the axis of symmetry. + Find the vertex. + Find the #emph[y]-intercept. Find the point symmetric to the #emph[y]-intercept across the axis of symmetry. + Find the #emph[x]-intercepts. + Graph the parabola. ] The next example reviews the method of graphing a parabola from the general form of its equation. #examplebox("Example 1")[][ Graph #math.equation(block: false, alt: "y equals − x squared plus 6 x minus 8")[$y = "−" x^(2) + 6 x − 8$] by using properties. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The general form of a quadratic equation and a particular example with y = -x^2 + 6x - 8, showing the relationship between them.], alt: "The general form of a quadratic equation and a particular example with y = -x^2 + 6x - 8, showing the relationship between them.", caption: none)]), [Since #emph[a] is #math.equation(block: false, alt: "−1 ,")[$−1 ,$] the parabola opens downward.], [], [#figure(figph[A red, inverted U-shaped arrow with both ends pointing downwards, suggesting a flow, connection, or dual downward direction.], alt: "A red, inverted U-shaped arrow with both ends pointing downwards, suggesting a flow, connection, or dual downward direction.", caption: none)], [], [To find the axis of symmetry, find #math.equation(block: false, alt: "x equals minus the fraction b over 2 a .")[$x = − frac(b, 2 a) .$]], [#figure(figph[Formula for the symmetry axis of a parabola: x = -b/2a.], alt: "Formula for the symmetry axis of a parabola: x = -b/2a.", caption: none)], [], [#figure(figph[The image shows the mathematical equation: x = -6 / 2(-1).], alt: "The image shows the mathematical equation: x = -6 / 2(-1).", caption: none)], [], [#figure(figph[The equation 'x = 3' is clearly displayed on a white background, representing a fundamental and concise mathematical expression.], alt: "The equation 'x = 3' is clearly displayed on a white background, representing a fundamental and concise mathematical expression.", caption: none)], [], [The axis of symmetry is #math.equation(block: false, alt: "x equals 3 .")[$x = 3 .$]], [], [#figure(figph[Coordinate graph with a vertical dashed line at x=3.], alt: "Coordinate graph with a vertical dashed line at x=3.", caption: none)], [The vertex is on the line #math.equation(block: false, alt: "x equals 3 .")[$x = 3 .$]], [#figure(figph[The image displays the quadratic equation y = -x^2 + 6x - 8.], alt: "The image displays the quadratic equation y = -x^2 + 6x - 8.", caption: none)], [Let #math.equation(block: false, alt: "x equals 3 .")[$x = 3 .$]], [#figure(figph[Equation showing how to calculate y using the formula y equals negative three squared plus six times three minus eight. The numbers three and eight are highlighted.], alt: "Equation showing how to calculate y using the formula y equals negative three squared plus six times three minus eight. The numbers three and eight are highlighted.", caption: none)], [], [#figure(figph[The equation y equals negative nine plus eighteen minus eight.], alt: "The equation y equals negative nine plus eighteen minus eight.", caption: none)], [], [#figure(figph[The equation 'y = 1' written on the image.], alt: "The equation 'y = 1' written on the image.", caption: none)], [], [The vertex is #math.equation(block: false, alt: "open parenthesis 3 , 1 close parenthesis .")[$( 3 , 1 ) .$]], [], [#figure(figph[A graph shows a dashed vertical line at x=3 and a point at (3,1).], alt: "A graph shows a dashed vertical line at x=3 and a point at (3,1).", caption: none)], [The #emph[y]-intercept occurs when #math.equation(block: false, alt: "x equals 0 .")[$x = 0 .$]], [#figure(figph[The equation of a quadratic function is shown as y = -x^2 + 6x - 8.], alt: "The equation of a quadratic function is shown as y = -x^2 + 6x - 8.", caption: none)], [Substitute #math.equation(block: false, alt: "x equals 0 .")[$x = 0 .$]], [#figure(figph[An algebraic equation showing the calculation of 'y' by substituting the value of 0 into the expression: y = -0^2 + 6\*0 - 8.], alt: "An algebraic equation showing the calculation of 'y' by substituting the value of 0 into the expression: y = -0^2 + 6*0 - 8.", caption: none)], [Simplify.], [#figure(figph[Ecuación: y = -8], alt: "Ecuación: y = -8", caption: none)], [], [The #emph[y]-intercept is #math.equation(block: false, alt: "open parenthesis 0 , −8 close parenthesis .")[$( 0 , −8 ) .$]], [The point #math.equation(block: false, alt: "open parenthesis 0 , −8 close parenthesis")[$( 0 , −8 )$] is three units to the left of the #linebreak() line of symmetry. The point three units to the #linebreak() right of the line of symmetry is #math.equation(block: false, alt: "open parenthesis 6 , −8 close parenthesis .")[$( 6 , −8 ) .$]], [Point symmetric to the #emph[y]-intercept is #math.equation(block: false, alt: "open parenthesis 6 , −8 close parenthesis .")[$( 6 , −8 ) .$]], [], [#figure(figph[Cartesian graph with three points: (3,1), (0,-8) and (6,-8). A dashed vertical line passes through x=3.], alt: "Cartesian graph with three points: (3,1), (0,-8) and (6,-8). A dashed vertical line passes through x=3.", caption: none)], [The #emph[x]-intercept occurs when #math.equation(block: false, alt: "y equals 0 .")[$y = 0 .$]], [#figure(figph[The quadratic equation is y equals negative x squared plus six x minus eight.], alt: "The quadratic equation is y equals negative x squared plus six x minus eight.", caption: none)], [Let #math.equation(block: false, alt: "y equals 0 .")[$y = 0 .$]], [#figure(figph[The quadratic equation negative x squared plus six x minus eight equals zero.], alt: "The quadratic equation negative x squared plus six x minus eight equals zero.", caption: none)], [Factor the GCF.], [#figure(figph[Quadratic equation showing zero equals negative of x squared minus six x plus eight.], alt: "Quadratic equation showing zero equals negative of x squared minus six x plus eight.", caption: none)], [Factor the trinomial.], [#figure(figph[The equation 0 = -(x-4)(x-2), an example of a factored algebraic expression.], alt: "The equation 0 = -(x-4)(x-2), an example of a factored algebraic expression.", caption: none)], [Solve for #emph[x].], [#figure(figph[The mathematical expressions X=4 and X=2 appear in the image.], alt: "The mathematical expressions X=4 and X=2 appear in the image.", caption: none)], [], [The #emph[x]-intercepts are #math.equation(block: false, alt: "open parenthesis 4 , 0 close parenthesis , open parenthesis 2 , 0 close parenthesis .")[$( 4 , 0 ) , ( 2 , 0 ) .$]], [Graph the parabola.], [#figure(figph[This graph displays a downward-opening parabola on a coordinate plane. The vertex is at (3, 1), and the dashed line x=3 is the axis of symmetry.], alt: "This graph displays a downward-opening parabola on a coordinate plane. The vertex is at (3, 1), and the dashed line x=3 is the axis of symmetry.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph #math.equation(block: false, alt: "y equals − x squared plus 5 x minus 6")[$y = "−" x^(2) + 5 x − 6$] by using properties. #solutionbox[ #figure(figph[This graph shows a parabola opening downward, with x intercepts (2, 0) and (3, 0) and y intercept (0, negative 6).], alt: "This graph shows a parabola opening downward, with x intercepts (2, 0) and (3, 0) and y intercept (0, negative 6).", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph #math.equation(block: false, alt: "y equals − x squared plus 8 x minus 12")[$y = "−" x^(2) + 8 x − 12$] by using properties. #solutionbox[ #figure(figph[This graph shows a parabola opening downward, with vertex (4, 4) and x intercepts (2, 0) and (6, 0).], alt: "This graph shows a parabola opening downward, with vertex (4, 4) and x intercepts (2, 0) and (6, 0).", caption: none) ] ] The next example reviews the method of graphing a #strong[parabola] from the standard form of its equation, #math.equation(block: false, alt: "y equals a open parenthesis x minus h close parenthesis squared plus k .")[$y = a attach(( x − h ), t: 2) + k .$] #examplebox("Example 2")[][ Write#math.equation(block: false, alt: "y equals 3 x squared minus 6 x plus 5")[$y = 3 x^(2) − 6 x + 5$] in standard form and then use properties of standard form to graph the equation. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Rewrite the function in #math.equation(block: false, alt: "y equals a open parenthesis x minus h close parenthesis squared plus k")[$y = a attach(( x − h ), t: 2) + k$] form #linebreak() by completing the square.], [#math.equation(block: false, alt: "y equals 3 x squared minus 6 x plus 5")[$y = 3 x^(2) − 6 x + 5$]]), [], [#math.equation(block: false, alt: "y equals 3 open parenthesis x squared minus 2 x close parenthesis plus 5")[$y = 3 ( x^(2) − 2 x #h(1.6em) ) + 5$]], [], [#math.equation(block: false, alt: "y equals 3 open parenthesis x squared minus 2 x plus 1 close parenthesis plus 5 minus 3")[$y = 3 ( x^(2) − 2 x + 1 ) + 5 − 3$]], [], [#math.equation(block: false, alt: "y equals 3 open parenthesis x minus 1 close parenthesis squared plus 2")[$y = 3 attach(( x − 1 ), t: 2) + 2$]], [Identify the constants #emph[a, h, k].], [#math.equation(block: false, alt: "a equals 3")[$a = 3$], #math.equation(block: false, alt: "h equals 1")[$h = 1$], #math.equation(block: false, alt: "k equals 2")[$k = 2$]], [Since #math.equation(block: false, alt: "a equals 3 ,")[$a = 3 ,$] the parabola opens upward.], [], [#figure(figph[Two red arrows point upward, connected by a curve, symbolizing an upward trend or a cyclical motion with an upward force.], alt: "Two red arrows point upward, connected by a curve, symbolizing an upward trend or a cyclical motion with an upward force.", caption: none)], [], [The axis of symmetry is #math.equation(block: false, alt: "x equals h .")[$x = h .$]], [The axis of symmetry is #math.equation(block: false, alt: "x equals 1 .")[$x = 1 .$]], [The vertex is #math.equation(block: false, alt: "open parenthesis h , k close parenthesis .")[$( h , k ) .$]], [The vertex is #math.equation(block: false, alt: "open parenthesis 1 , 2 close parenthesis .")[$( 1 , 2 ) .$]], [Find the #emph[y]-intercept by substituting #math.equation(block: false, alt: "x equals 0 .")[$x = 0 .$]], [#math.equation(block: false, alt: "y equals 3 open parenthesis x minus 1 close parenthesis squared plus 2")[$y = 3 attach(( x − 1 ), t: 2) + 2$] #linebreak() #math.equation(block: false, alt: "y equals 3 times 0 squared minus 6 times 0 plus 5")[$y = 3 · 0^(2) − 6 · 0 + 5$]], [], [#math.equation(block: false, alt: "y equals 5")[$y = 5$]], [], [#emph[y]-intercept #math.equation(block: false, alt: "open parenthesis 0 , 5 close parenthesis")[$( 0 , 5 )$]], [Find the point symmetric to #math.equation(block: false, alt: "open parenthesis 0 , 5 close parenthesis")[$( 0 , 5 )$] across the axis of symmetry.], [#math.equation(block: false, alt: "open parenthesis 2 , 5 close parenthesis")[$( 2 , 5 )$]], [Find the #emph[x]-intercepts.], [#math.equation(block: false, alt: "y, equals, 3 open parenthesis x minus 1 close parenthesis squared plus 2; 0, equals, 3 open parenthesis x minus 1 close parenthesis squared plus 2; minus 2, equals, 3 open parenthesis x minus 1 close parenthesis squared; minus the fraction 2 over 3, equals, open parenthesis x minus 1 close parenthesis squared; plus or minus the square root of minus the fraction 2 over 3, equals, x minus 1")[$y & = & 3 attach(( x − 1 ), t: 2) + 2 \ 0 & = & 3 attach(( x − 1 ), t: 2) + 2 \ − 2 & = & 3 attach(( x − 1 ), t: 2) \ − frac(2, 3) & = & attach(( x − 1 ), t: 2) \ ± sqrt(− frac(2, 3)) & = & x − 1$]], [], [The square root of a negative number #linebreak() tells us the solutions are complex #linebreak() numbers. So there are no #emph[x]-intercepts.], [Graph the parabola.], [#figure(figph[A graph of a parabola opening upwards, with its vertex at (1, 2). The axis of symmetry is a dashed vertical line at x=1. Two points (0, 5) and (2, 5) are marked on the curve.], alt: "A graph of a parabola opening upwards, with its vertex at (1, 2). The axis of symmetry is a dashed vertical line at x=1. Two points (0, 5) and (2, 5) are marked on the curve.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ ⓐ Write #math.equation(block: false, alt: "y equals 2 x squared plus 4 x plus 5")[$y = 2 x^(2) + 4 x + 5$] in standard form and ⓑ use properties of standard form to graph the equation. #solutionbox[ ⓐ #math.equation(block: false, alt: "y equals 2 open parenthesis x plus 1 close parenthesis squared plus 3")[$y = 2 attach(( x + 1 ), t: 2) + 3$] #linebreak() ⓑ #linebreak() #figure(figph[This graph shows a parabola opening upwards, with vertex (negative 1, 3) and y intercept (0, 5). It has the point minus (2, 5) on it.], alt: "This graph shows a parabola opening upwards, with vertex (negative 1, 3) and y intercept (0, 5). It has the point minus (2, 5) on it.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ ⓐ Write #math.equation(block: false, alt: "y equals −2 x squared plus 8 x minus 7")[$y = −2 x^(2) + 8 x − 7$] in standard form and ⓑ use properties of standard form to graph the equation. #solutionbox[ ⓐ #math.equation(block: false, alt: "y equals −2 open parenthesis x minus 2 close parenthesis squared plus 1")[$y = −2 attach(( x − 2 ), t: 2) + 1$] #linebreak() ⓑ #linebreak() #figure(figph[This graph shows a parabola opening downwards, with vertex (2, 1) and axis of symmetry x equals 2. Its y intercept is (0, negative 7).], alt: "This graph shows a parabola opening downwards, with vertex (2, 1) and axis of symmetry x equals 2. Its y intercept is (0, negative 7).", caption: none) ] ] === Graph Horizontal Parabolas Our work so far has only dealt with parabolas that open up or down. We are now going to look at horizontal parabolas. These parabolas open either to the left or to the right. If we interchange the #emph[x] and #emph[y] in our previous equations for parabolas, we get the equations for the parabolas that open to the left or to the right. #figure(table( columns: 3, align: left, inset: 6pt, [Horizontal Parabolas], [], [], [], [General form #linebreak() #math.equation(block: false, alt: "x equals a y squared plus b y plus c")[$x = a y^(2) + b y + c$]], [Standard form #linebreak() #math.equation(block: false, alt: "x equals a open parenthesis y minus k close parenthesis squared plus h")[$x = a attach(( y − k ), t: 2) + h$]], [#strong[Orientation]], [#math.equation(block: false, alt: "a greater than 0")[$a > 0$] right; #math.equation(block: false, alt: "a less than 0")[$a < 0$] left], [#math.equation(block: false, alt: "a greater than 0")[$a > 0$] right; #math.equation(block: false, alt: "a less than 0")[$a < 0$] left], [#strong[Axis of symmetry]], [#math.equation(block: false, alt: "y equals minus the fraction b over 2 a")[$y = − frac(b, 2 a)$]], [#math.equation(block: false, alt: "y equals k")[$y = k$]], [#strong[Vertex]], [Substitute #math.equation(block: false, alt: "y equals minus the fraction b over 2 a")[$y = − frac(b, 2 a)$] and #linebreak() solve for #emph[x].], [#math.equation(block: false, alt: "open parenthesis h , k close parenthesis")[$( h , k )$]], [#strong[#emph[y]-intercepts]], [Let #math.equation(block: false, alt: "x equals 0")[$x = 0$]], [Let #math.equation(block: false, alt: "x equals 0")[$x = 0$]], [#strong[#emph[x]-intercept]], [Let #math.equation(block: false, alt: "y equals 0")[$y = 0$]], [Let #math.equation(block: false, alt: "y equals 0")[$y = 0$]], )) The graphs show what the parabolas look like when they to the left or to the right. Their position in relation to the #emph[x]- or #emph[y]-axis is merely an example. #figure(figph[This figure shows two parabolas with axis of symmetry y equals k,) and vertex (h, k. The one on the left is labeled a greater than 0 and opens to the right. The other parabola opens to the left.], alt: "This figure shows two parabolas with axis of symmetry y equals k,) and vertex (h, k. The one on the left is labeled a greater than 0 and opens to the right. The other parabola opens to the left.", caption: none) Looking at these parabolas, do their graphs represent a function? Since both graphs would fail the vertical line test, they do not represent a function. To graph a #strong[parabola] that opens to the left or to the right is basically the same as what we did for parabolas that open up or down, with the reversal of the #emph[x] and #emph[y] variables. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Graph horizontal parabolas #math.equation(block: false, alt: "open parenthesis x equals a y squared plus b y plus c or x equals a open parenthesis y minus k close parenthesis squared plus h close parenthesis")[$( x = a y^(2) + b y + c #h(0.5em) "or" #h(0.5em) x = a attach(( y − k ), t: 2) + h )$] using properties.] + Determine whether the parabola opens to the left or to the right. + Find the axis of symmetry. + Find the vertex. + Find the #emph[x]-intercept. Find the point symmetric to the #emph[x]-intercept across the axis of symmetry. + Find the #emph[y]-intercepts. + Graph the parabola. ] #examplebox("Example 3")[][ Graph #math.equation(block: false, alt: "x equals 2 y squared")[$x = 2 y^(2)$] by using properties. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[Two mathematical equations are displayed: x = ay^2 + by + c, shown in red, and x = 2y^2, shown in black. Both equations define x in terms of y, with the first being a general quadratic in y and the second a specific instance.], alt: "Two mathematical equations are displayed: x = ay^2 + by + c, shown in red, and x = 2y^2, shown in black. Both equations define x in terms of y, with the first being a general quadratic in y and the second a specific instance.", caption: none)]), [Since #math.equation(block: false, alt: "a equals 2 ,")[$a = 2 ,$] the parabola opens to the right.], [], [#figure(figph[Two red arrows form a curved, cyclical path, suggesting connection, flow, or a continuous process.], alt: "Two red arrows form a curved, cyclical path, suggesting connection, flow, or a continuous process.", caption: none)], [], [To find the axis of symmetry, find #math.equation(block: false, alt: "y equals minus the fraction b over 2 a .")[$y = − frac(b, 2 a) .$]], [#figure(figph[A mathematical equation is displayed, showing y = -b/2a, typically used to find the x-coordinate of the vertex of a parabola given the quadratic equation y = ax^2 + bx + c.], alt: "A mathematical equation is displayed, showing y = -b/2a, typically used to find the x-coordinate of the vertex of a parabola given the quadratic equation y = ax^2 + bx + c.", caption: none)], [], [#figure(figph[A mathematical equation showing y equals negative zero divided by the product of two and two, written as y = - 0 / 2(2).], alt: "A mathematical equation showing y equals negative zero divided by the product of two and two, written as y = - 0 / 2(2).", caption: none)], [], [#figure(figph[The mathematical equation 'y = 0' is displayed on a white background.], alt: "The mathematical equation 'y = 0' is displayed on a white background.", caption: none)], [], [The axis of symmetry is #math.equation(block: false, alt: "y equals 0 .")[$y = 0 .$]], [The vertex is on the line#math.equation(block: false, alt: "y equals 0 .")[$y = 0 .$]], [#figure(figph[A mathematical equation is displayed on a white background, which reads 'x = 2y²'.], alt: "A mathematical equation is displayed on a white background, which reads 'x = 2y²'.", caption: none)], [Let #math.equation(block: false, alt: "y equals 0 .")[$y = 0 .$]], [#figure(figph[The mathematical equation x = 2 \* 0^2 is shown, where the '0' is highlighted with a red outline, indicating a step in a calculation or problem.], alt: "The mathematical equation x = 2 * 0^2 is shown, where the '0' is highlighted with a red outline, indicating a step in a calculation or problem.", caption: none)], [], [#figure(figph[The mathematical notation X = Ø, indicating that X is equal to the empty set.], alt: "The mathematical notation X = Ø, indicating that X is equal to the empty set.", caption: none)], [], [The vertex is #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis .")[$( 0 , 0 ) .$]], )) Since the vertex is #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis ,")[$( 0 , 0 ) ,$] both the #emph[x]- and #emph[y]-intercepts are the point #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis .")[$( 0 , 0 ) .$] To graph the parabola we need more points. In this case it is easiest to choose values of #emph[y]. #linebreak() #figure(figph[In the equation x equals 2 y squared, when y is 1, x is 2 and when y is 2, x is 8. The points are (2, 1) and (8, 2).], alt: "In the equation x equals 2 y squared, when y is 1, x is 2 and when y is 2, x is 8. The points are (2, 1) and (8, 2).", caption: none) #linebreak() We also plot the points symmetric to #math.equation(block: false, alt: "open parenthesis 2 , 1 close parenthesis")[$( 2 , 1 )$] and #math.equation(block: false, alt: "open parenthesis 8 , 2 close parenthesis")[$( 8 , 2 )$] across the #emph[y]-axis, the points #math.equation(block: false, alt: "open parenthesis 2 , −1 close parenthesis ,")[$( 2 , −1 ) ,$]#math.equation(block: false, alt: "open parenthesis 8 , −2 close parenthesis .")[$( 8 , −2 ) .$] Graph the parabola. #figure(figph[This graph shows right opening parabola with vertex (0, 0). Four points are marked on it: point (2, 1), point (2, negative 1), point (8, 2) and point (8 minus 2).], alt: "This graph shows right opening parabola with vertex (0, 0). Four points are marked on it: point (2, 1), point (2, negative 1), point (8, 2) and point (8 minus 2).", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph #math.equation(block: false, alt: "x equals y squared")[$x = y^(2)$] by using properties. #solutionbox[ #figure(figph[This graph shows right opening parabola with vertex at origin. Two points on it are (4, 2) and (4, negative 2).], alt: "This graph shows right opening parabola with vertex at origin. Two points on it are (4, 2) and (4, negative 2).", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph #math.equation(block: false, alt: "x equals − y squared")[$x = "−" y^(2)$] by using properties. #solutionbox[ #figure(figph[This graph shows left opening parabola with vertex at origin. Two points on it are (negative 4, 2) and (negative 4, negative 2).], alt: "This graph shows left opening parabola with vertex at origin. Two points on it are (negative 4, 2) and (negative 4, negative 2).", caption: none) ] ] In the next example, the vertex is not the origin. #examplebox("Example 4")[][ Graph #math.equation(block: false, alt: "x equals − y squared plus 2 y plus 8")[$x = "−" y^(2) + 2 y + 8$] by using properties. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[Two equations representing parabolas opening horizontally are presented: the general form x = ay^2 + by + c, and the specific equation x = -y^2 + 2y + 8.], alt: "Two equations representing parabolas opening horizontally are presented: the general form x = ay^2 + by + c, and the specific equation x = -y^2 + 2y + 8.", caption: none)]), [Since #math.equation(block: false, alt: "a equals −1 ,")[$a = −1 ,$] the parabola opens to the left.], [], [#figure(figph[A red, curved arrow with arrowheads pointing both upwards and downwards, illustrating a continuous loop, cyclical process, or interconnected relationship.], alt: "A red, curved arrow with arrowheads pointing both upwards and downwards, illustrating a continuous loop, cyclical process, or interconnected relationship.", caption: none)], [], [To find the axis of symmetry, find #math.equation(block: false, alt: "y equals minus the fraction b over 2 a .")[$y = − frac(b, 2 a) .$]], [#figure(figph[A mathematical equation shows 'y equals negative b over two a'.], alt: "A mathematical equation shows 'y equals negative b over two a'.", caption: none)], [], [#figure(figph[A mathematical equation shows 'y = -2 / (2(-1))' centered on a white background.], alt: "A mathematical equation shows 'y = -2 / (2(-1))' centered on a white background.", caption: none)], [], [#figure(figph[The mathematical equation 'y = 1' is shown, indicating a horizontal line on a coordinate plane where the y-value is constant at 1.], alt: "The mathematical equation 'y = 1' is shown, indicating a horizontal line on a coordinate plane where the y-value is constant at 1.", caption: none)], [], [The axis of symmetry is #math.equation(block: false, alt: "y equals 1 .")[$y = 1 .$]], [The vertex is on the line#math.equation(block: false, alt: "y equals 1 .")[$y = 1 .$]], [#figure(figph[The image displays the quadratic equation x = -y^2 + 2y + 8, written in a clear, legible font against a plain white background.], alt: "The image displays the quadratic equation x = -y^2 + 2y + 8, written in a clear, legible font against a plain white background.", caption: none)], [Let #math.equation(block: false, alt: "y equals 1 .")[$y = 1 .$]], [#figure(figph[A mathematical expression on a white background reads X = -1^2 + 2 \* 1 + 8, with the numbers 1 highlighted in red.], alt: "A mathematical expression on a white background reads X = -1^2 + 2 * 1 + 8, with the numbers 1 highlighted in red.", caption: none)], [], [#figure(figph[The equation 'X=9' is visible against a white background.], alt: "The equation 'X=9' is visible against a white background.", caption: none)], [], [The vertex is #math.equation(block: false, alt: "open parenthesis 9 , 1 close parenthesis .")[$( 9 , 1 ) .$]], [The #emph[x]-intercept occurs when #math.equation(block: false, alt: "y equals 0 .")[$y = 0 .$]], [#figure(figph[The image displays a mathematical equation written in black text on a white background. The equation is 'x = -y^2 + 2y + 8', representing a quadratic function where x is expressed in terms of y.], alt: "The image displays a mathematical equation written in black text on a white background. The equation is 'x = -y^2 + 2y + 8', representing a quadratic function where x is expressed in terms of y.", caption: none)], [], [#figure(figph[A mathematical equation, X = -0^2 + 2\*0 + 8, with the number 0 highlighted in red outlines, is shown against a plain white background.], alt: "A mathematical equation, X = -0^2 + 2*0 + 8, with the number 0 highlighted in red outlines, is shown against a plain white background.", caption: none)], [], [#figure(figph[The image displays a mathematical equation 'X = 8' in a simple black font centered against a plain white background.], alt: "The image displays a mathematical equation 'X = 8' in a simple black font centered against a plain white background.", caption: none)], [], [The #emph[x]-intercept is #math.equation(block: false, alt: "open parenthesis 8 , 0 close parenthesis .")[$( 8 , 0 ) .$]], [The point #math.equation(block: false, alt: "open parenthesis 8 , 0 close parenthesis")[$( 8 , 0 )$] is one unit below the line of #linebreak() symmetry. The symmetric point one unit #linebreak() above the line of symmetry is #math.equation(block: false, alt: "open parenthesis 8 , 2 close parenthesis")[$( 8 , 2 )$]], [Symmetric point is #math.equation(block: false, alt: "open parenthesis 8 , 2 close parenthesis .")[$( 8 , 2 ) .$]], [The #emph[y]-intercept occurs when #math.equation(block: false, alt: "x equals 0 .")[$x = 0 .$]], [#figure(figph[A mathematical equation is displayed, reading x = -y^2 + 2y + 8, shown against a plain white background.], alt: "A mathematical equation is displayed, reading x = -y^2 + 2y + 8, shown against a plain white background.", caption: none)], [Substitute #math.equation(block: false, alt: "x equals 0 .")[$x = 0 .$]], [#figure(figph[A quadratic equation is displayed on a white background: 0 = -y^2 + 2y + 8. The equation is presented clearly in black text, ready for solving or analysis.], alt: "A quadratic equation is displayed on a white background: 0 = -y^2 + 2y + 8. The equation is presented clearly in black text, ready for solving or analysis.", caption: none)], [Solve.], [#figure(figph[A quadratic equation displayed on a white background: y^2 - 2y - 8 = 0.], alt: "A quadratic equation displayed on a white background: y^2 - 2y - 8 = 0.", caption: none)], [], [#figure(figph[A mathematical equation is displayed, showing the expression (y-4)(y+2) = 0. The equation appears to be a factored form of a quadratic equation, set equal to zero to find the roots for 'y'.], alt: "A mathematical equation is displayed, showing the expression (y-4)(y+2) = 0. The equation appears to be a factored form of a quadratic equation, set equal to zero to find the roots for 'y'.", caption: none)], [], [#figure(figph[The image displays the mathematical equations 'y=4' and 'y=-2' in a simple, clear font against a white background.], alt: "The image displays the mathematical equations 'y=4' and 'y=-2' in a simple, clear font against a white background.", caption: none)], [], [The #emph[y]-intercepts are #math.equation(block: false, alt: "open parenthesis 0 , 4 close parenthesis")[$( 0 , 4 )$] and #math.equation(block: false, alt: "open parenthesis 0 , −2 close parenthesis .")[$( 0 , −2 ) .$]], [Connect the points to graph the parabola.], [#figure(figph[A graph displays a parabola opening to the left, with its vertex marked at (9, 1). Additional points (0, 4), (0, -2), (8, 2), and (8, 0) are highlighted on the curve.], alt: "A graph displays a parabola opening to the left, with its vertex marked at (9, 1). Additional points (0, 4), (0, -2), (8, 2), and (8, 0) are highlighted on the curve.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph #math.equation(block: false, alt: "x equals − y squared minus 4 y plus 12")[$x = "−" y^(2) − 4 y + 12$] by using properties. #solutionbox[ #figure(figph[This graph shows left opening parabola with vertex (16, negative 2) and x intercept (12, 0).], alt: "This graph shows left opening parabola with vertex (16, negative 2) and x intercept (12, 0).", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph #math.equation(block: false, alt: "x equals − y squared plus 2 y minus 3")[$x = "−" y^(2) + 2 y − 3$] by using properties. #solutionbox[ #figure(figph[This graph shows left opening parabola with vertex (negative 2, 1) and x intercept minus (3, 0).], alt: "This graph shows left opening parabola with vertex (negative 2, 1) and x intercept minus (3, 0).", caption: none) ] ] In , we see the relationship between the equation in standard form and the properties of the parabola. The How To box lists the steps for graphing a parabola in the standard form #math.equation(block: false, alt: "x equals a open parenthesis y minus k close parenthesis squared plus h .")[$x = a attach(( y − k ), t: 2) + h .$] We will use this procedure in the next example. #examplebox("Example 5")[][ Graph #math.equation(block: false, alt: "x equals 2 open parenthesis y minus 2 close parenthesis squared plus 1")[$x = 2 attach(( y − 2 ), t: 2) + 1$] using properties. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[Representation of the standard equation of a parabola x = a(y-k)^2 + h, with a numerical example x = 2(y-2)^2 + 1.], alt: "Representation of the standard equation of a parabola x = a(y-k)^2 + h, with a numerical example x = 2(y-2)^2 + 1.", caption: none)]), [Identify the constants #emph[a, h, k].], [#math.equation(block: false, alt: "a equals 2 ,")[$a = 2 ,$]#math.equation(block: false, alt: "h equals 1 ,")[$h = 1 ,$]#math.equation(block: false, alt: "k equals 2")[$k = 2$]], [Since #math.equation(block: false, alt: "a equals 2 ,")[$a = 2 ,$] the parabola opens to the right.], [], [#figure(figph[Two red arrows in a loop, indicating a cycle or repetition.], alt: "Two red arrows in a loop, indicating a cycle or repetition.", caption: none)], [], [The axis of symmetry is #math.equation(block: false, alt: "y equals k .")[$y = k .$]], [$#h(7.5em)$The axis of symmetry is #math.equation(block: false, alt: "y equals 2 .")[$y = 2 .$]], [The vertex is #math.equation(block: false, alt: "open parenthesis h , k close parenthesis .")[$( h , k ) .$]], [$#h(7.5em)$The vertex is #math.equation(block: false, alt: "open parenthesis 1 , 2 close parenthesis .")[$( 1 , 2 ) .$]], [Find the #emph[x]-intercept by substituting #math.equation(block: false, alt: "y equals 0 .")[$y = 0 .$]], [#math.equation(block: false, alt: "x, equals, 2 open parenthesis y minus 2 close parenthesis squared plus 1; x, equals, 2 open parenthesis 0 minus 2 close parenthesis squared plus 1; x, equals, 9")[$x & = & 2 attach(( y − 2 ), t: 2) + 1 \ x & = & 2 attach(( 0 − 2 ), t: 2) + 1 \ x & = & 9$]], [], [$#h(7.5em)$The #emph[x]-intercept is #math.equation(block: false, alt: "open parenthesis 9 , 0 close parenthesis .")[$( 9 , 0 ) .$]], [Find the point symmetric to #math.equation(block: false, alt: "open parenthesis 9 , 0 close parenthesis")[$( 9 , 0 )$] across the #linebreak() axis of symmetry.], [#math.equation(block: false, alt: "open parenthesis 9 , 4 close parenthesis")[$#h(7.5em) ( 9 , 4 )$]], [Find the #emph[y]-intercepts. Let #math.equation(block: false, alt: "x equals 0 .")[$x = 0 .$]], [#math.equation(block: false, alt: "x, equals, 2 open parenthesis y minus 2 close parenthesis squared plus 1; 0, equals, 2 open parenthesis y minus 2 close parenthesis squared plus 1; −1, equals, 2 open parenthesis y minus 2 close parenthesis squared")[$x & = & 2 attach(( y − 2 ), t: 2) + 1 \ 0 & = & 2 attach(( y − 2 ), t: 2) + 1 \ −1 & = & 2 attach(( y − 2 ), t: 2)$]], [], [A square cannot be negative, so there is no real #linebreak() solution. So there are no #emph[y]-intercepts.], [Graph the parabola.], [#figure(figph[Graph of a horizontal parabola with vertex at (1, 2). Shows the points (9, 4) and (9, 0) and their symmetry axis at y=2.], alt: "Graph of a horizontal parabola with vertex at (1, 2). Shows the points (9, 4) and (9, 0) and their symmetry axis at y=2.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph #math.equation(block: false, alt: "x equals 3 open parenthesis y minus 1 close parenthesis squared plus 2")[$x = 3 attach(( y − 1 ), t: 2) + 2$] using properties. #solutionbox[ #figure(figph[This graph shows a parabola opening right with vertex (2, 1) and x intercept (5, 0).], alt: "This graph shows a parabola opening right with vertex (2, 1) and x intercept (5, 0).", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph #math.equation(block: false, alt: "x equals 2 open parenthesis y minus 3 close parenthesis squared plus 2")[$x = 2 attach(( y − 3 ), t: 2) + 2$] using properties. #solutionbox[ #figure(figph[This graph shows a parabola opening right with vertex (2, 3) and symmetric points (4, 2) and (4, 4).], alt: "This graph shows a parabola opening right with vertex (2, 3) and symmetric points (4, 2) and (4, 4).", caption: none) ] ] In the next example, we notice the a is negative and so the parabola opens to the left. #examplebox("Example 6")[][ Graph #math.equation(block: false, alt: "x equals −4 open parenthesis y plus 1 close parenthesis squared plus 4")[$x = −4 attach(( y + 1 ), t: 2) + 4$] using properties. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[Two mathematical equations are displayed on a white background. The top equation, in red, is x = a(y - k)^2 + h. Below it, in black, is a specific example: x = -4(y + 1)^2 + 4.], alt: "Two mathematical equations are displayed on a white background. The top equation, in red, is x = a(y - k)^2 + h. Below it, in black, is a specific example: x = -4(y + 1)^2 + 4.", caption: none)]), [Identify the constants #emph[a, h, k].], [#math.equation(block: false, alt: "a equals −4 ,")[$a = −4 ,$]#math.equation(block: false, alt: "h equals 4 ,")[$h = 4 ,$]#math.equation(block: false, alt: "k equals −1")[$k = −1$]], [Since #math.equation(block: false, alt: "a equals −4 ,")[$a = −4 ,$] the parabola opens to the left.], [], [#figure(figph[Two red curved arrows, one arcing upwards and the other downwards, both pointing to the left from a common origin on the right, suggesting a continuous loop or a back-and-forth interaction.], alt: "Two red curved arrows, one arcing upwards and the other downwards, both pointing to the left from a common origin on the right, suggesting a continuous loop or a back-and-forth interaction.", caption: none)], [], [The axis of symmetry is #math.equation(block: false, alt: "y equals k .")[$y = k .$]], [$#h(8em)$The axis of symmetry is #math.equation(block: false, alt: "y equals −1 .")[$y = −1 .$]], [The vertex is #math.equation(block: false, alt: "open parenthesis h , k close parenthesis .")[$( h , k ) .$]], [$#h(8em)$The vertex is #math.equation(block: false, alt: "open parenthesis 4 , −1 close parenthesis .")[$( 4 , −1 ) .$]], [Find the #emph[x]-intercept by substituting #math.equation(block: false, alt: "y equals 0 .")[$y = 0 .$]], [#math.equation(block: false, alt: "x, equals, −4 open parenthesis y plus 1 close parenthesis squared plus 4; x, equals, −4 open parenthesis 0 plus 1 close parenthesis squared plus 4; x, equals, 0")[$#h(1.7em) x & = & −4 attach(( y + 1 ), t: 2) + 4 \ x & = & −4 attach(( 0 + 1 ), t: 2) + 4 \ x & = & 0$]], [], [$#h(8em)$The #emph[x]-intercept is #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis .")[$( 0 , 0 ) .$]], [Find the point symmetric to #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] across the #linebreak() axis of symmetry.], [#math.equation(block: false, alt: "open parenthesis 0 , −2 close parenthesis")[$#h(8em) ( 0 , −2 )$]], [Find the #emph[y]-intercepts.], [#math.equation(block: false, alt: "x equals −4 open parenthesis y plus 1 close parenthesis squared plus 4")[$#h(8em) x = −4 attach(( y + 1 ), t: 2) + 4$]], [Let #math.equation(block: false, alt: "x equals 0 .")[$x = 0 .$]], [#math.equation(block: false, alt: "0, equals, −4 open parenthesis y plus 1 close parenthesis squared plus 4; −4, equals, −4 open parenthesis y plus 1 close parenthesis squared; 1, equals, open parenthesis y plus 1 close parenthesis squared; y plus 1, equals, plus or minus 1")[$0 & = & −4 attach(( y + 1 ), t: 2) + 4 \ −4 & = & −4 attach(( y + 1 ), t: 2) \ 1 & = & attach(( y + 1 ), t: 2) \ y + 1 & = & ± 1$]], [], [#math.equation(block: false, alt: "y equals −1 plus 1 y equals −1 minus 1")[$y = −1 + 1 #h(1.5em) y = −1 − 1$]], [], [#math.equation(block: false, alt: "y equals 0 y equals −2")[$y = 0 #h(4em) y = −2$]], [], [The #emph[y]-intercepts are #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] and #math.equation(block: false, alt: "open parenthesis 0 , −2 close parenthesis .")[$( 0 , −2 ) .$]], [Graph the parabola.], [#figure(figph[A parabola opening left on a Cartesian plane with vertex (4, -1), passing through (0, 0) and (0, -2). A dashed line at y=-1 indicates the axis of symmetry.], alt: "A parabola opening left on a Cartesian plane with vertex (4, -1), passing through (0, 0) and (0, -2). A dashed line at y=-1 indicates the axis of symmetry.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph #math.equation(block: false, alt: "x equals −4 open parenthesis y plus 2 close parenthesis squared plus 4")[$x = −4 attach(( y + 2 ), t: 2) + 4$] using properties. #solutionbox[ #figure(figph[This figure shows a parabola opening to the left with vertex (4, negative 2) and y intercepts (0, negative 1) and (0, negative 3).], alt: "This figure shows a parabola opening to the left with vertex (4, negative 2) and y intercepts (0, negative 1) and (0, negative 3).", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph #math.equation(block: false, alt: "x equals −2 open parenthesis y plus 3 close parenthesis squared plus 2")[$x = −2 attach(( y + 3 ), t: 2) + 2$] using properties. #solutionbox[ #figure(figph[This figure shows a parabola opening to the left with vertex (2, negative 3) and y intercepts (0, negative 2) and (0, negative 4).], alt: "This figure shows a parabola opening to the left with vertex (2, negative 3) and y intercepts (0, negative 2) and (0, negative 4).", caption: none) ] ] The next example requires that we first put the equation in standard form and then use the properties. #examplebox("Example 7")[][ Write #math.equation(block: false, alt: "x equals 2 y squared plus 12 y plus 17")[$x = 2 y^(2) + 12 y + 17$] in standard form and then use the properties of the standard form to graph the equation. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical equation is displayed: x = 2y^2 + 12y + 17. It shows a quadratic relationship between x and y, where x is expressed as a function of y.], alt: "A mathematical equation is displayed: x = 2y^2 + 12y + 17. It shows a quadratic relationship between x and y, where x is expressed as a function of y.", caption: none)]), [Rewrite the function in #linebreak() #math.equation(block: false, alt: "x equals a open parenthesis y minus k close parenthesis squared plus h")[$x = a attach(( y − k ), t: 2) + h$] form by completing #linebreak() the square.], [#figure(figph[A mathematical equation is displayed on a white background: x = 2(y^2 + 6y) + 17.], alt: "A mathematical equation is displayed on a white background: x = 2(y^2 + 6y) + 17.", caption: none)], [], [#figure(figph[A mathematical equation is displayed on a white background: x = 2(y^2 + 6y + 9) + 17 - 18. The numbers '2', '9', and '18' are highlighted in red, indicating a step in a solution or a focus point.], alt: "A mathematical equation is displayed on a white background: x = 2(y^2 + 6y + 9) + 17 - 18. The numbers '2', '9', and '18' are highlighted in red, indicating a step in a solution or a focus point.", caption: none)], [], [#figure(figph[A mathematical equation is presented against a white background. The equation reads: x = 2(y + 3)^2 - 1.], alt: "A mathematical equation is presented against a white background. The equation reads: x = 2(y + 3)^2 - 1.", caption: none)], [], [#figure(figph[Two math equations are displayed: the general vertex form of a horizontal parabola, x = a(y-k)^2 + h, in red, and a specific example, x = 2(y+3)^2 - 1, in black.], alt: "Two math equations are displayed: the general vertex form of a horizontal parabola, x = a(y-k)^2 + h, in red, and a specific example, x = 2(y+3)^2 - 1, in black.", caption: none)], [Identify the constants #emph[a, h, k].], [#math.equation(block: false, alt: "a equals 2 , h equals −1 , k equals −3")[$a = 2 , #h(1em) h = −1 , #h(1em) k = −3$]], [Since #math.equation(block: false, alt: "a equals 2 ,")[$a = 2 ,$] the parabola opens to #linebreak() the right.], [], [#figure(figph[Two red arrows create a semi-circular path, one pointing right and up, the other left and down, suggesting a cycle, continuous flow, or return motion within a process or system.], alt: "Two red arrows create a semi-circular path, one pointing right and up, the other left and down, suggesting a cycle, continuous flow, or return motion within a process or system.", caption: none)], [], [The axis of symmetry is #math.equation(block: false, alt: "y equals k .")[$y = k .$]], [$#h(10.4em)$The axis of symmetry is #math.equation(block: false, alt: "y equals −3 .")[$y = −3 .$]], [The vertex is #math.equation(block: false, alt: "open parenthesis h , k close parenthesis .")[$( h , k ) .$]], [$#h(10.4em)$The vertex is #math.equation(block: false, alt: "open parenthesis −1 , −3 close parenthesis .")[$( −1 , −3 ) .$]], [Find the #emph[x]-intercept by substituting #linebreak() #math.equation(block: false, alt: "y equals 0 .")[$y = 0 .$]], [#math.equation(block: false, alt: "x, equals, 2 open parenthesis y plus 3 close parenthesis squared minus 1; x, equals, 2 open parenthesis 0 plus 3 close parenthesis squared minus 1; x, equals, 17")[$#h(1.8em) x & = & 2 attach(( y + 3 ), t: 2) − 1 \ x & = & 2 attach(( 0 + 3 ), t: 2) − 1 \ x & = & 17$]], [], [$#h(10.4em)$The #emph[x]-intercept is #math.equation(block: false, alt: "open parenthesis 17 , 0 close parenthesis .")[$( 17 , 0 ) .$]], [Find the point symmetric to #math.equation(block: false, alt: "open parenthesis 17 , 0 close parenthesis")[$( 17 , 0 )$] #linebreak() across the axis of symmetry.], [#math.equation(block: false, alt: "open parenthesis 17 , −6 close parenthesis")[$#h(10.4em) ( 17 , −6 )$]], [Find the #emph[y]-intercepts. #linebreak() #linebreak() Let #math.equation(block: false, alt: "x equals 0 .")[$x = 0 .$]], [#math.equation(block: false, alt: "x, equals, 2 open parenthesis y plus 3 close parenthesis squared minus 1; 0, equals, 2 open parenthesis y plus 3 close parenthesis squared minus 1; 1, equals, 2 open parenthesis y plus 3 close parenthesis squared; the fraction 1 over 2, equals, open parenthesis y plus 3 close parenthesis squared; y plus 3, equals, plus or minus the square root of the fraction 1 over 2; y, equals, −3 plus or minus the fraction the square root of 2 over 2")[$x & = & 2 attach(( y + 3 ), t: 2) − 1 \ 0 & = & 2 attach(( y + 3 ), t: 2) − 1 \ 1 & = & 2 attach(( y + 3 ), t: 2) \ frac(1, 2) & = & attach(( y + 3 ), t: 2) \ y + 3 & = & ± sqrt(frac(1, 2)) \ y & = & −3 ± frac(sqrt(2), 2)$]], [], [#math.equation(block: false, alt: "y equals −3 plus the fraction the square root of 2 over 2 y equals −3 minus the fraction the square root of 2 over 2")[$y = −3 + frac(sqrt(2), 2) #h(1em) y = −3 − frac(sqrt(2), 2)$]], [], [#math.equation(block: false, alt: "y approximately equals minus 2.3 y approximately equals minus 3.7")[$y ≈ − 2.3 #h(2em) y ≈ − 3.7$]], [], [The #emph[y]-intercepts are #math.equation(block: false, alt: "open parenthesis 0 , −3 plus the fraction the square root of 2 over 2 close parenthesis , open parenthesis 0 , −3 minus the fraction the square root of 2 over 2 close parenthesis .")[$( 0 , −3 + frac(sqrt(2), 2) ) , ( 0 , −3 − frac(sqrt(2), 2) ) .$]], [Graph the parabola.], [#figure(figph[A parabola on a coordinate plane with vertex (-1, -3), opening to the right. The parabola passes through points (17, 0) and (17, -6). A dashed line marks y=-3.], alt: "A parabola on a coordinate plane with vertex (-1, -3), opening to the right. The parabola passes through points (17, 0) and (17, -6). A dashed line marks y=-3.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ ⓐ Write #math.equation(block: false, alt: "x equals 3 y squared plus 6 y plus 7")[$x = 3 y^(2) + 6 y + 7$] in standard form and ⓑ use properties of the standard form to graph the equation. #solutionbox[ ⓐ #math.equation(block: false, alt: "x equals 3 open parenthesis y plus 1 close parenthesis squared plus 4")[$x = 3 attach(( y + 1 ), t: 2) + 4$] #linebreak() ⓑ #linebreak() #figure(figph[This graph shows a parabola opening to the right with vertex (4, negative 1) and x intercept (7, 0).], alt: "This graph shows a parabola opening to the right with vertex (4, negative 1) and x intercept (7, 0).", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ ⓐ Write #math.equation(block: false, alt: "x equals −4 y squared minus 16 y minus 12")[$x = −4 y^(2) − 16 y − 12$] in standard form and ⓑ use properties of the standard form to graph the equation. #solutionbox[ ⓐ #math.equation(block: false, alt: "x equals −4 open parenthesis y plus 2 close parenthesis squared plus 4")[$x = −4 attach(( y + 2 ), t: 2) + 4$] #linebreak() ⓑ #linebreak() #figure(figph[This graph shows a parabola opening to the left with vertex (4, negative 2) and x intercept minus (12, 0).], alt: "This graph shows a parabola opening to the left with vertex (4, negative 2) and x intercept minus (12, 0).", caption: none) ] ] === Solve Applications with Parabolas Many architectural designs incorporate parabolas. It is not uncommon for bridges to be constructed using parabolas as we will see in the next example. #examplebox("Example 8")[][ Find the equation of the parabolic arch formed in the foundation of the bridge shown. Write the equation in standard form. #figure(figph[This figure shows a parabolic arch formed in the foundation of a bridge. It is 10 feet high and 20 feet wide at the base.], alt: "This figure shows a parabolic arch formed in the foundation of a bridge. It is 10 feet high and 20 feet wide at the base.", caption: none) #solutionbox[ We will first set up a coordinate system and draw the parabola. The graph will give us the information we need to write the equation of the graph in the standard form#math.equation(block: false, alt: "y equals a open parenthesis x minus h close parenthesis squared plus k .")[$y = a attach(( x − h ), t: 2) + k .$] #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([Let the lower left side of the bridge be the #linebreak() origin of the coordinate grid at the point #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis .")[$( 0 , 0 ) .$] #linebreak() Since the base is 20 feet wide the point #linebreak() #math.equation(block: false, alt: "open parenthesis 20 , 0 close parenthesis")[$( 20 , 0 )$] represents the lower right side. #linebreak() The bridge is 10 feet high at the highest #linebreak() point. The highest point is the vertex of #linebreak() the parabola so the #emph[y]-coordinate of the #linebreak() vertex will be 10. #linebreak() Since the bridge is symmetric, the vertex #linebreak() must fall halfway between the left most #linebreak() point, #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis ,")[$( 0 , 0 ) ,$] and the rightmost point #linebreak() #math.equation(block: false, alt: "open parenthesis 20 , 0 close parenthesis .")[$( 20 , 0 ) .$] From this we know that the #linebreak() #emph[x]-coordinate of the vertex will also be 10.], [#figure(figph[A parabolic curve is shown on a coordinate plane. The curve starts at (0,0), peaks at (10,10), and returns to the x-axis at (20,0). The x-axis is labeled from 0 to 20, and the y-axis from 0 to 10.], alt: "A parabolic curve is shown on a coordinate plane. The curve starts at (0,0), peaks at (10,10), and returns to the x-axis at (20,0). The x-axis is labeled from 0 to 20, and the y-axis from 0 to 10.", caption: none)]), [Identify the vertex, #math.equation(block: false, alt: "open parenthesis h , k close parenthesis .")[$( h , k ) .$]], [#math.equation(block: false, alt: "open parenthesis h , k close parenthesis equals open parenthesis 10 , 10 close parenthesis")[$( h , k ) = ( 10 , 10 )$]], [], [#math.equation(block: false, alt: "h equals 10 , k equals 10")[$h = 10 , " " k = 10$]], [Substitute the values into the standard form. #linebreak() #linebreak() The value of #emph[a] is still unknown. To find #linebreak() the value of #emph[a] use one of the other points #linebreak() on the parabola.], [#math.equation(block: false, alt: "y, equals, a open parenthesis x minus h close parenthesis squared plus k; y, equals, a open parenthesis x minus 10 close parenthesis squared plus 10; open parenthesis x , y close parenthesis, equals, open parenthesis 0 , 0 close parenthesis")[$#h(1.3em) y & = & a attach(( x − h ), t: 2) + k \ y & = & a attach(( x − 10 ), t: 2) + 10 \ ( x , y ) & = & ( 0 , 0 )$]], [Substitute the values of the other point #linebreak() into the equation.], [#math.equation(block: false, alt: "y, equals, a open parenthesis x minus 10 close parenthesis squared plus 10; 0, equals, a open parenthesis 0 minus 10 close parenthesis squared plus 10")[$#h(3.35em) y & = & a attach(( x − 10 ), t: 2) + 10 \ 0 & = & a attach(( 0 − 10 ), t: 2) + 10$]], [Solve for #emph[a].], [#math.equation(block: false, alt: "0, equals, a open parenthesis 0 minus 10 close parenthesis squared plus 10; −10, equals, a open parenthesis −10 close parenthesis squared; −10, equals, 100 a; the fraction −10 over 100, equals, a; a, equals, minus the fraction 1 over 10")[$#h(2em) 0 & = & a attach(( 0 − 10 ), t: 2) + 10 \ −10 & = & a attach(( −10 ), t: 2) \ −10 & = & 100 a \ frac(−10, 100) & = & a \ a & = & − frac(1, 10)$]], [], [#math.equation(block: false, alt: "y equals a open parenthesis x minus 10 close parenthesis squared plus 10")[$y = a attach(( x − 10 ), t: 2) + 10$]], [Substitute the value for #emph[a] into the #linebreak() equation.], [#math.equation(block: false, alt: "y equals minus the fraction 1 over 10 open parenthesis x minus 10 close parenthesis squared plus 10")[$#h(1.8em) y = − frac(1, 10) attach(( x − 10 ), t: 2) + 10$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the equation of the parabolic arch formed in the foundation of the bridge shown. Write the equation in standard form. #figure(figph[This figure shows a parabolic arch formed in the foundation of a bridge. It is 20 feet high and 40 feet wide at the base.], alt: "This figure shows a parabolic arch formed in the foundation of a bridge. It is 20 feet high and 40 feet wide at the base.", caption: none) #solutionbox[ #math.equation(block: true, alt: "y equals minus the fraction 1 over 20 open parenthesis x minus 20 close parenthesis squared plus 20")[$y = − frac(1, 20) attach(( x − 20 ), t: 2) + 20$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the equation of the parabolic arch formed in the foundation of the bridge shown. Write the equation in standard form. #figure(figph[This figure shows a parabolic arch formed in the foundation of a bridge. It is 5 feet high and 10 feet wide at the base.], alt: "This figure shows a parabolic arch formed in the foundation of a bridge. It is 5 feet high and 10 feet wide at the base.", caption: none) #solutionbox[ #math.equation(block: true, alt: "y equals minus the fraction 1 over 5 open parenthesis x minus 5 close parenthesis squared plus 5")[$y = − frac(1, 5) attach(( x − 5 ), t: 2) + 5$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Access these online resources for additional instructions and practice with quadratic functions and parabolas. - #link("https://openstax.org/l/37quadfunc")[Quadratic Functions] - #link("https://openstax.org/l/37conhorizpbola")[Introduction to Conics and Graphing Horizontal Parabolas] ] === Key Concepts - #strong[Parabola:] A #strong[parabola] is all points in a plane that are the same distance from a fixed point and a fixed line. The fixed point is called the #strong[focus,] and the fixed line is called the #strong[directrix] of the parabola. #linebreak() #figure(table( columns: 3, align: left, inset: 6pt, [Vertical Parabolas], [], [], [], [General form #linebreak() #math.equation(block: false, alt: "y equals a x squared plus b x plus c")[$y = a x^(2) + b x + c$]], [Standard form #linebreak() #math.equation(block: false, alt: "y equals a open parenthesis x minus h close parenthesis squared plus k")[$y = a attach(( x − h ), t: 2) + k$]], [#strong[Orientation]], [#math.equation(block: false, alt: "a greater than 0")[$a > 0$] up; #math.equation(block: false, alt: "a less than 0")[$a < 0$] down], [#math.equation(block: false, alt: "a greater than 0")[$a > 0$] up; #math.equation(block: false, alt: "a less than 0")[$a < 0$] down], [#strong[Axis of symmetry]], [#math.equation(block: false, alt: "x equals minus the fraction b over 2 a")[$x = − frac(b, 2 a)$]], [#math.equation(block: false, alt: "x equals h")[$x = h$]], [#strong[Vertex]], [Substitute #math.equation(block: false, alt: "x equals minus the fraction b over 2 a")[$x = − frac(b, 2 a)$] and #linebreak() solve for #emph[y].], [#math.equation(block: false, alt: "open parenthesis h , k close parenthesis")[$( h , k )$]], [#strong[#emph[y]- intercept]], [Let #math.equation(block: false, alt: "x equals 0")[$x = 0$]], [Let #math.equation(block: false, alt: "x equals 0")[$x = 0$]], [#strong[#emph[x]-intercepts]], [Let #math.equation(block: false, alt: "y equals 0")[$y = 0$]], [Let #math.equation(block: false, alt: "y equals 0")[$y = 0$]], )) #linebreak() #figure(figph[This figure shows two parabolas with axis x equals h and vertex (h, k). The one on the left opens up and a is greater than 0. The one on the right opens down. Here a is less than 0.], alt: "This figure shows two parabolas with axis x equals h and vertex (h, k). The one on the left opens up and a is greater than 0. The one on the right opens down. Here a is less than 0.", caption: none) - #strong[How to graph vertical parabolas #math.equation(block: false, alt: "open parenthesis y equals a x squared plus b x plus c")[$\( y = a x^(2) + b x + c$] or #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals a open parenthesis x minus h close parenthesis squared plus k close parenthesis")[$f \( x \) = a attach(\( x − h \), t: 2) + k \)$] using properties.] + Determine whether the parabola opens upward or downward. + Find the axis of symmetry. + Find the vertex. + Find the #emph[y]-intercept. Find the point symmetric to the #emph[y]-intercept across the axis of symmetry. + Find the #emph[x]-intercepts. + Graph the parabola. #linebreak() #figure(table( columns: 3, align: left, inset: 6pt, [Horizontal Parabolas], [], [], [], [General form #linebreak() #math.equation(block: false, alt: "x equals a y squared plus b y plus c")[$x = a y^(2) + b y + c$]], [Standard form #linebreak() #math.equation(block: false, alt: "x equals a open parenthesis y minus k close parenthesis squared plus h")[$x = a attach(( y − k ), t: 2) + h$]], [#strong[Orientation]], [#math.equation(block: false, alt: "a greater than 0")[$a > 0$] right; #math.equation(block: false, alt: "a less than 0")[$a < 0$] left], [#math.equation(block: false, alt: "a greater than 0")[$a > 0$] right; #math.equation(block: false, alt: "a less than 0")[$a < 0$] left], [#strong[Axis of symmetry]], [#math.equation(block: false, alt: "y equals minus the fraction b over 2 a")[$y = − frac(b, 2 a)$]], [#math.equation(block: false, alt: "y equals k")[$y = k$]], [#strong[Vertex]], [Substitute #math.equation(block: false, alt: "y equals minus the fraction b over 2 a")[$y = − frac(b, 2 a)$] and #linebreak() solve for #emph[x].], [#math.equation(block: false, alt: "open parenthesis h , k close parenthesis")[$( h , k )$]], [#strong[#emph[y]-intercepts]], [Let #math.equation(block: false, alt: "x equals 0")[$x = 0$]], [Let #math.equation(block: false, alt: "x equals 0")[$x = 0$]], [#strong[#emph[x]-intercept]], [Let #math.equation(block: false, alt: "y equals 0")[$y = 0$]], [Let #math.equation(block: false, alt: "y equals 0")[$y = 0$]], )) #linebreak() #figure(figph[This figure shows two parabolas with axis of symmetry y equals k, and vertex (h, k). The one on the left is labeled a greater than 0 and opens to the right. The other parabola opens to the left.], alt: "This figure shows two parabolas with axis of symmetry y equals k, and vertex (h, k). The one on the left is labeled a greater than 0 and opens to the right. The other parabola opens to the left.", caption: none) - #strong[How to graph horizontal parabolas #math.equation(block: false, alt: "open parenthesis x equals a y squared plus b y plus c")[$\( x = a y^(2) + b y + c$] or #math.equation(block: false, alt: "x equals a open parenthesis y minus k close parenthesis squared plus h close parenthesis")[$x = a attach(\( y − k \), t: 2) + h \)$] using properties.] + Determine whether the parabola opens to the left or to the right. + Find the axis of symmetry. + Find the vertex. + Find the #emph[x]-intercept. Find the point symmetric to the #emph[x]-intercept across the axis of symmetry. + Find the #emph[y]-intercepts. + Graph the parabola. ==== Practice Makes Perfect #strong[Graph Vertical Parabolas] In the following exercises, graph each equation by using properties. #math.equation(block: true, alt: "y equals − x squared plus 4 x minus 3")[$y = "−" x^(2) + 4 x − 3$] #solutionbox[ #figure(figph[This graph shows a parabola opening downward with vertex (2, 1) and x intercepts (1, 0) and (3, 0).], alt: "This graph shows a parabola opening downward with vertex (2, 1) and x intercepts (1, 0) and (3, 0).", caption: none) ] #math.equation(block: true, alt: "y equals − x squared plus 8 x minus 15")[$y = "−" x^(2) + 8 x − 15$] #math.equation(block: true, alt: "y equals 6 x squared plus 2 x minus 1")[$y = 6 x^(2) + 2 x − 1$] #solutionbox[ #figure(figph[This graph shows a parabola opening upward. The vertex is (negative 0.167, negative 1.167), the x intercepts are (negative 0.608) and (negative 0.274, 0), and the y-intercept is (0, negative 1).], alt: "This graph shows a parabola opening upward. The vertex is (negative 0.167, negative 1.167), the x intercepts are (negative 0.608) and (negative 0.274, 0), and the y-intercept is (0, negative 1).", caption: none) ] #math.equation(block: true, alt: "y equals 8 x squared minus 10 x plus 3")[$y = 8 x^(2) − 10 x + 3$] In the following exercises, ⓐ write the equation in standard form and ⓑ use properties of the standard form to graph the equation. #math.equation(block: true, alt: "y equals − x squared plus 2 x minus 4")[$y = "−" x^(2) + 2 x − 4$] #solutionbox[ ⓐ #math.equation(block: false, alt: "y equals − open parenthesis x minus 1 close parenthesis squared minus 3")[$y = "−" attach(( x − 1 ), t: 2) − 3$] #linebreak() ⓑ #linebreak() #figure(figph[This graph shows a parabola opening downward with vertex (1, negative 3) and y intercept (0, 4).], alt: "This graph shows a parabola opening downward with vertex (1, negative 3) and y intercept (0, 4).", caption: none) ] #math.equation(block: true, alt: "y equals 2 x squared plus 4 x plus 6")[$y = 2 x^(2) + 4 x + 6$] #math.equation(block: true, alt: "y equals −2 x squared minus 4 x minus 5")[$y = −2 x^(2) − 4 x − 5$] #solutionbox[ ⓐ #math.equation(block: false, alt: "y equals −2 open parenthesis x plus 1 close parenthesis squared minus 3")[$y = −2 attach(( x + 1 ), t: 2) − 3$] #linebreak() ⓑ #linebreak() #figure(figph[This graph shows a parabola opening downward with vertex (negative 1, negative 3) and x intercepts (negative 5, 0).], alt: "This graph shows a parabola opening downward with vertex (negative 1, negative 3) and x intercepts (negative 5, 0).", caption: none) ] #math.equation(block: true, alt: "y equals 3 x squared minus 12 x plus 7")[$y = 3 x^(2) − 12 x + 7$] #strong[Graph Horizontal Parabolas] In the following exercises, graph each equation by using properties. #math.equation(block: true, alt: "x equals −2 y squared")[$x = −2 y^(2)$] #solutionbox[ #figure(figph[This graph shows a parabola opening to the left with vertex (0, 0). Two points on it are (negative 2, 1) and (negative 2, negative 1).], alt: "This graph shows a parabola opening to the left with vertex (0, 0). Two points on it are (negative 2, 1) and (negative 2, negative 1).", caption: none) ] #math.equation(block: true, alt: "x equals 3 y squared")[$x = 3 y^(2)$] #math.equation(block: true, alt: "x equals 4 y squared")[$x = 4 y^(2)$] #solutionbox[ #figure(figph[This graph shows a parabola opening to the right with vertex (0, 0). Two points on it are (4, 1) and (4, negative 1).], alt: "This graph shows a parabola opening to the right with vertex (0, 0). Two points on it are (4, 1) and (4, negative 1).", caption: none) ] #math.equation(block: true, alt: "x equals −4 y squared")[$x = −4 y^(2)$] #math.equation(block: true, alt: "x equals − y squared minus 2 y plus 3")[$x = "−" y^(2) − 2 y + 3$] #solutionbox[ #figure(figph[This graph shows a parabola opening to the left with vertex (4, negative 1) and y intercepts (0, 1) and (0, negative 3).], alt: "This graph shows a parabola opening to the left with vertex (4, negative 1) and y intercepts (0, 1) and (0, negative 3).", caption: none) ] #math.equation(block: true, alt: "x equals − y squared minus 4 y plus 5")[$x = "−" y^(2) − 4 y + 5$] #math.equation(block: true, alt: "x equals y squared plus 6 y plus 8")[$x = y^(2) + 6 y + 8$] #solutionbox[ #figure(figph[This graph shows a parabola opening to the right with vertex (negative 1, negative 3) and y intercepts (0, negative 2) and (0, negative 4).], alt: "This graph shows a parabola opening to the right with vertex (negative 1, negative 3) and y intercepts (0, negative 2) and (0, negative 4).", caption: none) ] #math.equation(block: true, alt: "x equals y squared minus 4 y minus 12")[$x = y^(2) − 4 y − 12$] #math.equation(block: true, alt: "x equals open parenthesis y minus 2 close parenthesis squared plus 3")[$x = attach(( y − 2 ), t: 2) + 3$] #solutionbox[ #figure(figph[This graph shows a parabola opening to the right with vertex (3, 2) and x intercept (7, 0).], alt: "This graph shows a parabola opening to the right with vertex (3, 2) and x intercept (7, 0).", caption: none) ] #math.equation(block: true, alt: "x equals open parenthesis y minus 1 close parenthesis squared plus 4")[$x = attach(( y − 1 ), t: 2) + 4$] #math.equation(block: true, alt: "x equals − open parenthesis y minus 1 close parenthesis squared plus 2")[$x = "−" attach(( y − 1 ), t: 2) + 2$] #solutionbox[ #figure(figph[This graph shows a parabola opening to the left with vertex (2, 1) and x intercept (1, 0).], alt: "This graph shows a parabola opening to the left with vertex (2, 1) and x intercept (1, 0).", caption: none) ] #math.equation(block: true, alt: "x equals − open parenthesis y minus 4 close parenthesis squared plus 3")[$x = "−" attach(( y − 4 ), t: 2) + 3$] #math.equation(block: true, alt: "x equals open parenthesis y plus 2 close parenthesis squared plus 1")[$x = attach(( y + 2 ), t: 2) + 1$] #solutionbox[ #figure(figph[This graph shows a parabola opening to the right with vertex (1, negative 2) and x intercept (5, 0).], alt: "This graph shows a parabola opening to the right with vertex (1, negative 2) and x intercept (5, 0).", caption: none) ] #math.equation(block: true, alt: "x equals open parenthesis y plus 1 close parenthesis squared plus 2")[$x = attach(( y + 1 ), t: 2) + 2$] #math.equation(block: true, alt: "x equals − open parenthesis y plus 3 close parenthesis squared plus 2")[$x = "−" attach(( y + 3 ), t: 2) + 2$] #solutionbox[ #figure(figph[This graph shows a parabola opening to the left with vertex (2, negative 3). Two points on it are (negative 2, negative 1) and (negative 2, 5).], alt: "This graph shows a parabola opening to the left with vertex (2, negative 3). Two points on it are (negative 2, negative 1) and (negative 2, 5).", caption: none) ] #math.equation(block: true, alt: "x equals − open parenthesis y plus 4 close parenthesis squared plus 3")[$x = "−" attach(( y + 4 ), t: 2) + 3$] #math.equation(block: true, alt: "x equals −3 open parenthesis y minus 2 close parenthesis squared plus 3")[$x = −3 attach(( y − 2 ), t: 2) + 3$] #solutionbox[ #figure(figph[This graph shows a parabola opening to the left with vertex (3, 2) and y intercepts (0, 1) and (0, 3).], alt: "This graph shows a parabola opening to the left with vertex (3, 2) and y intercepts (0, 1) and (0, 3).", caption: none) ] #math.equation(block: true, alt: "x equals −2 open parenthesis y minus 1 close parenthesis squared plus 2")[$x = −2 attach(( y − 1 ), t: 2) + 2$] #math.equation(block: true, alt: "x equals 4 open parenthesis y plus 1 close parenthesis squared minus 4")[$x = 4 attach(( y + 1 ), t: 2) − 4$] #solutionbox[ #figure(figph[This graph shows a parabola opening to the right with vertex (negative 4, negative 1) and y intercepts (0, 0) and (0, negative 2).], alt: "This graph shows a parabola opening to the right with vertex (negative 4, negative 1) and y intercepts (0, 0) and (0, negative 2).", caption: none) ] #math.equation(block: true, alt: "x equals 2 open parenthesis y plus 4 close parenthesis squared minus 2")[$x = 2 attach(( y + 4 ), t: 2) − 2$] In the following exercises, ⓐ write the equation in standard form and ⓑ use properties of the standard form to graph the equation. #math.equation(block: true, alt: "x equals y squared plus 4 y minus 5")[$x = y^(2) + 4 y − 5$] #solutionbox[ ⓐ #math.equation(block: false, alt: "x equals open parenthesis y plus 2 close parenthesis squared minus 9")[$x = attach(( y + 2 ), t: 2) − 9$] #linebreak() ⓑ #linebreak() #figure(figph[This graph shows a parabola opening to the right with vertex (negative 9, negative 2) and y intercepts (0, 1) and (0, negative 5).], alt: "This graph shows a parabola opening to the right with vertex (negative 9, negative 2) and y intercepts (0, 1) and (0, negative 5).", caption: none) ] #math.equation(block: true, alt: "x equals y squared plus 2 y minus 3")[$x = y^(2) + 2 y − 3$] #math.equation(block: true, alt: "x equals −2 y squared minus 12 y minus 16")[$x = −2 y^(2) − 12 y − 16$] #solutionbox[ ⓐ #math.equation(block: false, alt: "x equals −2 open parenthesis y plus 3 close parenthesis squared plus 2")[$x = −2 attach(( y + 3 ), t: 2) + 2$] #linebreak() ⓑ #linebreak() #figure(figph[This graph shows a parabola opening to the left with vertex (2, negative 3) and y intercepts (0, negative 2) and (0, negative 4).], alt: "This graph shows a parabola opening to the left with vertex (2, negative 3) and y intercepts (0, negative 2) and (0, negative 4).", caption: none) ] #math.equation(block: true, alt: "x equals −3 y squared minus 6 y minus 5")[$x = −3 y^(2) − 6 y − 5$] #strong[Mixed Practice] In the following exercises, match each graph to one of the following equations: ⓐ #emph[x]#super[2] + #emph[y]#super[2] = 64 ⓑ #emph[x]#super[2] + #emph[y]#super[2] = 49 #linebreak() ⓒ (#emph[x] + 5)#super[2] + (#emph[y] + 2)#super[2] = 4 ⓓ (#emph[x] − 2)#super[2] + (#emph[y] − 3)#super[2] = 9 ⓔ #emph[y] = −#emph[x]#super[2] + 8#emph[x] − 15 ⓕ #emph[y] = 6#emph[x]#super[2] + 2#emph[x] − 1 #figure(figph[This graph shows circle with center (0, 0) and radius 8 units.], alt: "This graph shows circle with center (0, 0) and radius 8 units.", caption: none) #solutionbox[ ⓐ ] #figure(figph[This graph shows a parabola opening upwards. Its vertex has an x value of slightly less than 0 and a y value of slightly less than negative 1. A point on it is close to (negative 1, 3).], alt: "This graph shows a parabola opening upwards. Its vertex has an x value of slightly less than 0 and a y value of slightly less than negative 1. A point on it is close to (negative 1, 3).", caption: none) #figure(figph[This graph shows circle with center (0, 0) and radius 7 units.], alt: "This graph shows circle with center (0, 0) and radius 7 units.", caption: none) #solutionbox[ ⓑ ] #figure(figph[This graph shows a parabola opening downwards with vertex (4, 1) and x intercepts (3, 0) and (5, 0).], alt: "This graph shows a parabola opening downwards with vertex (4, 1) and x intercepts (3, 0) and (5, 0).", caption: none) #figure(figph[This graph shows circle with center (2, 3) and radius 3 units.], alt: "This graph shows circle with center (2, 3) and radius 3 units.", caption: none) #solutionbox[ ⓓ ] #figure(figph[This graph shows circle with center (negative 5, negative 2) and radius 2 units.], alt: "This graph shows circle with center (negative 5, negative 2) and radius 2 units.", caption: none) #strong[Solve Applications with Parabolas] Write the equation in standard form of the parabolic arch formed in the foundation of the bridge shown. Write the equation in standard form. #figure(figph[This graph shows circle with center (negative 5, negative 2) and radius 2 units.], alt: "This graph shows circle with center (negative 5, negative 2) and radius 2 units.", caption: none) #solutionbox[ #math.equation(block: true, alt: "y equals minus the fraction 1 over 15 open parenthesis x minus 15 close parenthesis squared plus 15")[$y = − frac(1, 15) attach(( x − 15 ), t: 2) + 15$] ] Write the equation in standard form of the parabolic arch formed in the foundation of the bridge shown. Write the equation in standard form. #figure(figph[This figure shows a parabolic arch formed in the foundation of a bridge. It is 50 feet high and 100 feet wide at the base.], alt: "This figure shows a parabolic arch formed in the foundation of a bridge. It is 50 feet high and 100 feet wide at the base.", caption: none) Write the equation in standard form of the parabolic arch formed in the foundation of the bridge shown. Write the equation in standard form. #figure(figph[This figure shows a parabolic arch formed in the foundation of a bridge. It is 90 feet high and 60 feet wide at the base.], alt: "This figure shows a parabolic arch formed in the foundation of a bridge. It is 90 feet high and 60 feet wide at the base.", caption: none) #solutionbox[ #math.equation(block: true, alt: "y equals minus the fraction 1 over 10 open parenthesis x minus 30 close parenthesis squared plus 90")[$y = − frac(1, 10) attach(( x − 30 ), t: 2) + 90$] ] Write the equation in standard form of the parabolic arch formed in the foundation of the bridge shown. Write the equation in standard form. #figure(figph[This figure shows a parabolic arch formed in the foundation of a bridge. It is 45 feet high and 30 feet wide at the base.], alt: "This figure shows a parabolic arch formed in the foundation of a bridge. It is 45 feet high and 30 feet wide at the base.", caption: none) ==== Writing Exercises In your own words, define a parabola. #solutionbox[ Answers will vary. ] Is the parabola #math.equation(block: false, alt: "y equals x squared")[$y = x^(2)$] a function? Is the parabola #math.equation(block: false, alt: "x equals y squared")[$x = y^(2)$] a function? Explain why or why not. Write the equation of a parabola that opens up or down in standard form and the equation of a parabola that opens left or right in standard form. Provide a sketch of the parabola for each one, label the vertex and axis of symmetry. #solutionbox[ Answers will vary. ] Explain in your own words, how you can tell from its equation whether a parabola opens up, down, left or right. ==== Self Check ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. #figure(figph[This table has four columns, 3 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don’t get it. The first column has the following statements: graph vertical parabolas, graph horizontal parabolas, solve applications with parabolas. The remaining columns are blank.], alt: "This table has four columns, 3 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don’t get it. The first column has the following statements: graph vertical parabolas, graph horizontal parabolas, solve applications with parabolas. The remaining columns are blank.", caption: none) ⓑ After reviewing this checklist, what will you do to become confident for all objectives?