#set document(title: "2.3 Transformations of Graphs", 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")) == 2.3#h(0.6em)Transformations of Graphs Models for real situations are often variations of the basic functions introduced in Some Basic Functions. In this section, we explore how certain changes in the formula for a function affect its graph. In particular, we will compare the graph of #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$" " y = f ( x ) " "$] with the graphs of #math.equation(block: true, alt: "y equals f open parenthesis x close parenthesis plus k , y equals f open parenthesis x plus h close parenthesis , and y equals a f open parenthesis x close parenthesis")[$y = f ( x ) + k , " " " " " " " " y = f ( x + h ) , " " " " " " " and " " " " " " " y = a f ( x )$] for different values of the constants #math.equation(block: false, alt: "k")[$k$], #math.equation(block: false, alt: "h")[$h$], and #math.equation(block: false, alt: "a")[$a$]. Such variations are called #strong[transformations] of the graph. Every transformation in this section applied to one basic graph, y = |x| (gray). The blue graph is y = a·|x − h| + k. Slide k to translate vertically and h to translate horizontally — note the graph moves toward positive x when h increases, the counterintuitive direction the section warns about. The slider a is the scale factor: |a| \> 1 stretches the graph vertically, |a| \< 1 compresses it, and a negative a reflects it about the x-axis. Whatever you do, the corner keeps its shape — translations and stretches move and rescale a graph but never bend it.{"functions":\[{"color":"\#6a6a6a","expression":"abs(x)","expression2":"","id":"basic","inequality":"lt","lineDash":\[6,4\],"lineWidth":1.5,"mode":"cartesian","tMax":10,"tMin":-10,"variable":"x","visible":true},{"color":"\#1f4e79","expression":"a\*abs(x - h) + k","expression2":"","id":"transformed","inequality":"lt","lineDash":\[\],"lineWidth":2.5,"mode":"cartesian","tMax":10,"tMin":-10,"variable":"x","visible":true}\],"grid":{"majorSpacing":1,"showAxes":true,"showLabels":true},"parameters":\[{"id":"p-a","label":"Scale factor a","max":3,"min":-2,"name":"a","step":0.1,"value":2},{"id":"p-h","label":"Horizontal shift h","max":4,"min":-4,"name":"h","step":0.1,"value":1},{"id":"p-k","label":"Vertical shift k","max":3,"min":-3,"name":"k","step":0.1,"value":-1}\],"title":"y = a\\u00b7|x \\u2212 h| + k against the basic graph y = |x|","version":1,"viewport":{"centerX":0,"centerY":0,"scale":46}} === Vertical Translations The figure below shows the graphs of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x squared plus 4")[$" " f ( x ) = x^(2) + 4$], #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals x squared minus 4")[$" " g ( x ) = x^(2) − 4$], and the basic parabola, #math.equation(block: false, alt: "y equals x squared")[$" " y = x^(2)$]. By comparing tables of values, we can see exactly how the graphs of #math.equation(block: false, alt: "f")[$f$] and #math.equation(block: false, alt: "g")[$g$] are related to the basic parabola. #figure(figph[vertically shifted parabolas], alt: "vertically shifted parabolas", caption: none) #figure(table( columns: 6, 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: "0")[$" " 0 " "$]], [#math.equation(block: false, alt: "1")[$" " " " 1 " " " "$]], [#math.equation(block: false, alt: "2")[$" " 2 " "$]]), [#math.equation(block: false, alt: "y equals x squared")[$y = x^(2)$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x squared plus 4")[$f ( x ) = x^(2) + 4$]], [#math.equation(block: false, alt: "8")[$8$]], [#math.equation(block: false, alt: "5")[$5$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "5")[$5$]], [#math.equation(block: false, alt: "8")[$8$]], )) #figure(table( columns: 6, 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: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]]), [#math.equation(block: false, alt: "y equals x squared")[$y = x^(2)$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "g open parenthesis x close parenthesis equals x squared minus 4")[$g ( x ) = x^(2) − 4$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "minus 4")[$− 4$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "0")[$0$]], )) Each #math.equation(block: false, alt: "y")[$y$]-value in the table for #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] is four units greater than the corresponding #math.equation(block: false, alt: "y")[$y$]-value for the basic parabola. Consequently, each point on the graph of #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] is four units higher than the corresponding point on the basic parabola, as shown by the arrows. Similarly, each point on the graph of #math.equation(block: false, alt: "g open parenthesis x close parenthesis")[$g ( x )$] is four units lower than the corresponding point on the basic parabola. The graphs of #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$" " y = f ( x ) " "$] and #math.equation(block: false, alt: "y equals g open parenthesis x close parenthesis")[$" " y = g ( x ) " "$] are said to be #strong[translations] of the graph of #math.equation(block: false, alt: "y equals x squared")[$" " y = x^(2)$]. They are shifted to a different location in the plane but retain the same size and shape as the original graph. In general, we have the following principles. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Vertical Translations] Compared with the graph of #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$" " y = f ( x )$], + The graph of #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis plus k , open parenthesis k greater than 0 close parenthesis")[$" " " " y = f ( x ) + k , " " " " ( k > 0 ) " " " "$] is shifted #emph[upward] #math.equation(block: false, alt: "k")[$k$] units. + The graph of #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis minus k , open parenthesis k greater than 0 close parenthesis")[$" " " " y = f ( x ) − k , " " " " ( k > 0 " " " " )$] is shifted #emph[downward] #math.equation(block: false, alt: "k")[$k$] units. ] #examplebox("Example 1")[][ Graph the following functions. + #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals vertical bar x vertical bar plus 3")[$g ( x ) = | x | + 3$] + #math.equation(block: false, alt: "h open parenthesis x close parenthesis equals the fraction 1 over x minus 2")[$h ( x ) = display(frac(1, x)) − 2$] #solutionbox[ + The table shows that the #math.equation(block: false, alt: "y")[$y$]-values for #math.equation(block: false, alt: "g open parenthesis x close parenthesis")[$g ( x )$] are each three units greater than the corresponding #math.equation(block: false, alt: "y")[$y$]-values for the absolute value function. The graph of #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals vertical bar x vertical bar plus 3")[$" " g ( x ) = | x | + 3 " "$] is a translation of the basic graph of #math.equation(block: false, alt: "y equals vertical bar x vertical bar")[$" " y = | x |$], shifted upward three units, as shown below.#figure(figph[translated absolute value graph], alt: "translated absolute value graph", caption: none) #figure(table( columns: 6, 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: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]]), [#math.equation(block: false, alt: "y equals vertical bar x vertical bar")[$y = | x |$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "g open parenthesis x close parenthesis equals vertical bar x vertical bar plus 3")[$g ( x ) = | x | + 3$]], [#math.equation(block: false, alt: "5")[$5$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "5")[$5$]], )) + The table shows that the #math.equation(block: false, alt: "y")[$y$]-values for #math.equation(block: false, alt: "h open parenthesis x close parenthesis")[$h ( x )$] are each two units smaller than the corresponding #math.equation(block: false, alt: "y")[$y$]-values for #math.equation(block: false, alt: "y equals the fraction 1 over x")[$" " y = display(frac(1, x))$]. The graph of #math.equation(block: false, alt: "h open parenthesis x close parenthesis equals the fraction 1 over x minus 2")[$" " h ( x ) = display(frac(1, x)) − 2 " "$] is a translation of the basic graph of #math.equation(block: false, alt: "y equals the fraction 1 over x")[$" " y = display(frac(1, x))$], shifted downward two units, as shown below. #linebreak() #figure(figph[translated reciprocal graph], alt: "translated reciprocal graph", caption: none) #figure(table( columns: 6, 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: "the fraction 1 over 2")[$display(frac(1, 2))$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]]), [#math.equation(block: false, alt: "y equals the fraction 1 over x")[$y = display(frac(1, x))$]], [#math.equation(block: false, alt: "the fraction minus 1 over 2")[$display(frac(− 1, 2))$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "the fraction 1 over 2")[$display(frac(1, 2))$]], [#math.equation(block: false, alt: "h open parenthesis x close parenthesis equals the fraction 1 over x minus 2")[$h ( x ) = display(frac(1, x)) − 2$]], [#math.equation(block: false, alt: "the fraction minus 5 over 2")[$display(frac(− 5, 2))$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "the fraction minus 3 over 2")[$display(frac(− 3, 2))$]], )) ] ] The graph of #math.equation(block: false, alt: "y equals g open parenthesis x close parenthesis")[$y = g ( x )$] has a vertical asymptote at #math.equation(block: false, alt: "x equals minus 4")[$x = − 4$]. What happens to the asymptote under a vertical translation? \_\_\_\_\_ #solutionbox[ Nothing: a vertical asymptote does not change when a graph undergoes a vertical translation. ] The graph of #math.equation(block: false, alt: "y equals g open parenthesis x close parenthesis")[$" " y = g ( x ) " "$] has a vertical asymptote at #math.equation(block: false, alt: "x equals minus 4")[$x = − 4$]. What happens to the asymptote under a vertical translation? + Nothing. + It is compressed vertically. + It is translated vertically. + It is eliminated. + Graph the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals vertical bar x vertical bar plus 1")[$f ( x ) = | x | + 1$]. + How is the graph of #math.equation(block: false, alt: "f")[$f$] different from the graph of #math.equation(block: false, alt: "y equals vertical bar x vertical bar")[$y = | x |$]? #linebreak() To get the graph of #math.equation(block: false, alt: "f")[$f$], #linebreak() \_\_\_\_\_ #solutionbox[ + A graph is below. + Translate #math.equation(block: false, alt: "y equals vertical bar x vertical bar")[$y = | x |$] one unit up. ] A graph for part (a) is below. #figure(figph[shifted absolute value], alt: "shifted absolute value", caption: none) + Graph the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals vertical bar x vertical bar plus 1")[$" " f ( x ) = | x | + 1$]. + How is the graph of #math.equation(block: false, alt: "f")[$f$] different from the graph of #math.equation(block: false, alt: "y equals vertical bar x vertical bar")[$" " y = | x |$]? #solutionbox[ + #figure(figph[shifted absolute value], alt: "shifted absolute value", caption: none) + The graph of #math.equation(block: false, alt: "y equals vertical bar x vertical bar")[$" " y = | x |$] is translated one unit up. ] #examplebox("Example 2")[][ The function #math.equation(block: false, alt: "E equals f open parenthesis h close parenthesis")[$" " E = f ( h ) " "$] graphed below gives the amount of electrical power, in megawatts, drawn by a community from its local power plant as a function of time during a 24-hour period in 2002. Sketch a graph of #math.equation(block: false, alt: "y equals f open parenthesis h close parenthesis plus 300")[$" " y = f ( h ) + 300 " "$] and interpret its meaning. #figure(figph[electrical power], alt: "electrical power", caption: none) #solutionbox[ The graph of #math.equation(block: false, alt: "y equals f open parenthesis h close parenthesis plus 300")[$" " y = f ( h ) + 300 " "$] is a vertical translation of the graph of #math.equation(block: false, alt: "f")[$f$], as shown below. At each hour of the day, or for each value of #math.equation(block: false, alt: "h")[$h$], the #math.equation(block: false, alt: "y")[$y$]-coordinate is 300 greater than on the graph of #math.equation(block: false, alt: "f")[$f$]. So at each hour, the community is drawing 300 megawatts more power than in 2002. #figure(figph[electrical power], alt: "electrical power", caption: none) ] ] #figure(figph[swamp cooler graphs], alt: "swamp cooler graphs", caption: none) An evaporative cooler, or swamp cooler, is an energy-efficient type of air conditioner used in dry climates. A typical swamp cooler can reduce the temperature inside a house by 15 degrees. Figure (a) shows the graph of \\(T = f (t)\\), the temperature inside Kate's house \\(t\\) hours after she turns on the swamp cooler. Write a formula in terms of \\(f\\) for the function \\(g\\) shown in figure (b), and give a possible explanation of its meaning. #math.equation(block: false, alt: "g open parenthesis t close parenthesis equals")[$g ( t ) =$]\_\_\_\_\_ \_\_\_\_\_ #solutionbox[ #math.equation(block: false, alt: "g open parenthesis t close parenthesis equals f open parenthesis t close parenthesis plus 10")[$g ( t ) = f ( t ) + 10$]. The outside temperature was #math.equation(block: false, alt: "10 degrees")[$10^(∘)$] hotter. ] An evaporative cooler, or swamp cooler, is an energy-efficient type of air conditioner used in dry climates. A typical swamp cooler can reduce the temperature inside a house by 15 degrees. Figure (a) shows the graph of #math.equation(block: false, alt: "T equals f open parenthesis t close parenthesis")[$" " T = f ( t )$], the temperature inside Kate's house #math.equation(block: false, alt: "t")[$t$] hours after she turns on the swamp cooler. Write a formula in terms of #math.equation(block: false, alt: "f open parenthesis t close parenthesis")[$f ( t )$] for the function #math.equation(block: false, alt: "g open parenthesis t close parenthesis")[$g ( t )$] shown in figure (b), and give a possible explanation of its meaning. #figure(figph[swamp cooler graphs], alt: "swamp cooler graphs", caption: none) #solutionbox[ #math.equation(block: false, alt: "g open parenthesis t close parenthesis equals f open parenthesis t close parenthesis plus 10")[$g ( t ) = f ( t ) + 10$]. The outside temperature was #math.equation(block: false, alt: "10 degrees")[$10^(∘)$] hotter. ] === Horizontal Translations Now consider the graphs of #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals open parenthesis x plus 2 close parenthesis squared and g open parenthesis x close parenthesis equals open parenthesis x minus 2 close parenthesis squared")[$f ( x ) = ( x + 2 )^(2) " " " " " " " " "and" " " " " " " " " g ( x ) = ( x − 2 )^(2)$] shown below. Compared with the graph of the basic function #math.equation(block: false, alt: "y equals x squared")[$" " y = x^(2)$], the graph of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals open parenthesis x plus 2 close parenthesis squared")[$" " f ( x ) = ( x + 2 )^(2) " "$] is shifted two units to the #emph[left], as shown by the arrows. #figure(figph[graphs], alt: "graphs", caption: none) You can see why this happens by studying the function values in the table. Locate a particular #math.equation(block: false, alt: "y")[$y$]-value for #math.equation(block: false, alt: "y equals x squared")[$y = x^(2)$], say, #math.equation(block: false, alt: "y equals 4")[$y = 4$]. You must move two units to the left in the table to find the same #math.equation(block: false, alt: "y")[$y$]-value for #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$], as shown by the arrow. In fact, each #math.equation(block: false, alt: "y")[$y$]-value for #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] occurs two units to the left when compared to the same #math.equation(block: false, alt: "y")[$y$]-value for #math.equation(block: false, alt: "y equals x squared")[$" " y = x^(2)$]. #figure(figph[table for left translation], alt: "table for left translation", caption: none) #figure(figph[table for right translation], alt: "table for right translation", caption: none) Similarly, the graph of #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals open parenthesis x minus 2 close parenthesis squared")[$" " g ( x ) = ( x − 2 )^(2) " "$] is shifted two units to the #emph[right] compared to the graph of #math.equation(block: false, alt: "y equals x squared")[$" " y = x^(2)$]. In the table for #math.equation(block: false, alt: "g")[$g$], each #math.equation(block: false, alt: "y")[$y$]-value for #math.equation(block: false, alt: "g open parenthesis x close parenthesis")[$g ( x )$] occurs two units to the right of the same #math.equation(block: false, alt: "y")[$y$]-value for #math.equation(block: false, alt: "y equals x squared")[$y = x^(2)$]. In general, we have the following principle. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Horizontal Translations] Compared with the graph of #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$y = f ( x )$], + The graph of #math.equation(block: false, alt: "y equals f open parenthesis x plus h close parenthesis , open parenthesis h greater than 0 close parenthesis")[$" " " " y = f ( x + h ) , " " " " ( h > 0 ) " " " "$] is shifted #math.equation(block: false, alt: "h")[$h$] units to the #emph[left]. + The graph of #math.equation(block: false, alt: "y equals f open parenthesis x minus h close parenthesis , open parenthesis h greater than 0 close parenthesis")[$" " " " y = f ( x − h ) , " " " " ( h > 0 ) " " " "$] is shifted #math.equation(block: false, alt: "h")[$h$] units to the #emph[right]. ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ At first, the direction of a horizontal translation may seem counterintuitive. Look again at the tables above to help you see how the shift occurs. ] #examplebox("Example 3")[][ Graph the following functions. + #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the square root of x plus 1")[$g ( x ) = sqrt(x + 1)$] + #math.equation(block: false, alt: "h open parenthesis x close parenthesis equals the fraction 1 over open parenthesis x minus 3 close parenthesis squared")[$h ( x ) = display(frac(1, ( x − 3 )^(2)))$] #solutionbox[ + Consider the table of values for the function.#figure(table( columns: 6, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "3")[$3$]]), [#math.equation(block: false, alt: "y equals the square root of x")[$y = sqrt(x)$]], [undefined], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "1.414")[$1.414$]], [#math.equation(block: false, alt: "1.732")[$1.732$]], [#math.equation(block: false, alt: "y equals the square root of x plus 1")[$y = sqrt(x + 1)$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "1.414")[$1.414$]], [#math.equation(block: false, alt: "1.732")[$1.732$]], [#math.equation(block: false, alt: "2")[$2$]], )) #linebreak() The table shows that each #math.equation(block: false, alt: "y")[$y$]-value for #math.equation(block: false, alt: "g open parenthesis x close parenthesis")[$g ( x )$] occurs one unit to the left of the same #math.equation(block: false, alt: "y")[$y$]-value for the graph of #math.equation(block: false, alt: "y equals the square root of x")[$" " y = sqrt(x)$]. Consequently, each point on the graph of #math.equation(block: false, alt: "y equals g open parenthesis x close parenthesis")[$y = g ( x )$] is shifted one unit to the left of #math.equation(block: false, alt: "y equals the square root of x")[$" " y = sqrt(x)$], as shown below.#figure(figph[horizontal shift of square root], alt: "horizontal shift of square root", caption: none) + Consider the table of values for the function.#figure(table( columns: 7, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "4")[$4$]]), [#math.equation(block: false, alt: "y equals the fraction 1 over x")[$y = display(frac(1, x))$]], [#math.equation(block: false, alt: "1")[$1$]], [undefined], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "the fraction 1 over 4")[$display(frac(1, 4))$]], [#math.equation(block: false, alt: "the fraction 1 over 9")[$display(frac(1, 9))$]], [#math.equation(block: false, alt: "the fraction 1 over 16")[$display(frac(1, 16))$]], [#math.equation(block: false, alt: "y equals the fraction 1 over open parenthesis x minus 3 close parenthesis squared")[$y = display(frac(1, ( x − 3 )^(2)))$]], [#math.equation(block: false, alt: "the fraction 1 over 16")[$display(frac(1, 16))$]], [#math.equation(block: false, alt: "the fraction 1 over 9")[$display(frac(1, 9))$]], [#math.equation(block: false, alt: "the fraction 1 over 4")[$display(frac(1, 4))$]], [#math.equation(block: false, alt: "1")[$1$]], [undefined], [#math.equation(block: false, alt: "1")[$1$]], )) #linebreak() The table shows that each #math.equation(block: false, alt: "y")[$y$]-value for #math.equation(block: false, alt: "h open parenthesis x close parenthesis")[$h ( x )$] occurs three units to the right of the same #math.equation(block: false, alt: "y")[$y$]-value for the graph of #math.equation(block: false, alt: "y equals the fraction 1 over x squared")[$" " y = display(frac(1, x^(2)))$]. Consequently, each point on the graph of #math.equation(block: false, alt: "y equals h open parenthesis x close parenthesis")[$" " y = h ( x ) " "$] is shifted three units to the right of #math.equation(block: false, alt: "y equals the fraction 1 over x squared")[$" " y = display(frac(1, x^(2)))$], as shown below.#figure(figph[horizontal shift of inverse square], alt: "horizontal shift of inverse square", caption: none) ] ] The #math.equation(block: false, alt: "y")[$y$]-intercept of the graph of #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$y = f ( x )$] is #math.equation(block: false, alt: "open parenthesis 0 , 2 close parenthesis")[$( 0 , 2 )$]. What point lies on the graph of #math.equation(block: false, alt: "y equals f open parenthesis x plus 3 close parenthesis")[$y = f ( x + 3 )$]? \_\_\_\_\_ #solutionbox[ #math.equation(block: false, alt: "open parenthesis minus 3 , 2 close parenthesis")[$( − 3 , 2 )$] lies on the graph of #math.equation(block: false, alt: "y equals f open parenthesis x plus 3 close parenthesis")[$y = f ( x + 3 )$]. ] The #math.equation(block: false, alt: "y")[$y$]-intercept of the graph of #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$" " y = f ( x ) " "$] is #math.equation(block: false, alt: "open parenthesis 0 , 2 close parenthesis")[$( 0 , 2 )$]. What point lies on the graph of #math.equation(block: false, alt: "y equals f open parenthesis x plus 3 close parenthesis")[$" " y = f ( x + 3 )$]? + #math.equation(block: false, alt: "open parenthesis 0 , 5 close parenthesis")[$( 0 , 5 )$] + #math.equation(block: false, alt: "open parenthesis 3 , 2 close parenthesis")[$( 3 , 2 )$] + #math.equation(block: false, alt: "open parenthesis minus 3 , 2 close parenthesis")[$( − 3 , 2 )$] + #math.equation(block: false, alt: "open parenthesis 3 , 5 close parenthesis")[$( 3 , 5 )$] + Graph the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals vertical bar x plus 1 vertical bar")[$f ( x ) = | x + 1 |$]. + How is the graph of #math.equation(block: false, alt: "f")[$f$] different from the graph of #math.equation(block: false, alt: "y equals vertical bar x vertical bar")[$y = | x |$]? #linebreak() \_\_\_\_\_ #solutionbox[ + A graph is shown below. + Translate #math.equation(block: false, alt: "y equals vertical bar x vertical bar")[$y = | x |$] one unit left. ] A graph for part (a) is shown below. #figure(figph[shifted absolute value], alt: "shifted absolute value", caption: none) + Graph the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals vertical bar x plus 1 vertical bar")[$" " f ( x ) = | x + 1 |$]. + How is the graph of #math.equation(block: false, alt: "f")[$f$] different from the graph of #math.equation(block: false, alt: "y equals vertical bar x vertical bar")[$" " y = | x |$]? #solutionbox[ + #figure(figph[shifted absolute value], alt: "shifted absolute value", caption: none) + The graph of #math.equation(block: false, alt: "y equals vertical bar x vertical bar")[$" " y = | x | " "$] is translated one unit to the left. ] #examplebox("Example 4")[][ The function #math.equation(block: false, alt: "N equals f open parenthesis p close parenthesis")[$" " N = f ( p ) " "$] graphed below gives the number of people who have a given eye pressure level #math.equation(block: false, alt: "p")[$p$] from a sample of 100 people with healthy eyes, and the function #math.equation(block: false, alt: "g")[$g$] gives the number of people with pressure level #math.equation(block: false, alt: "p")[$p$] in a sample of 100 glaucoma patients. #figure(figph[shift of a bell-shaped curve], alt: "shift of a bell-shaped curve", caption: none) + Write a formula for #math.equation(block: false, alt: "g")[$g$] as a transformation of #math.equation(block: false, alt: "f")[$f$]. + For what pressure readings could a doctor be fairly certain that a patient has glaucoma? #solutionbox[ + The graph of #math.equation(block: false, alt: "g")[$g$] is translated #math.equation(block: false, alt: "10")[$10$] units to the right of #math.equation(block: false, alt: "f")[$f$], so #math.equation(block: false, alt: "g open parenthesis p close parenthesis equals f open parenthesis p minus 10 close parenthesis")[$" " g ( p ) = f ( p − 10 )$]. + Pressure readings above #math.equation(block: false, alt: "40")[$40$] are a strong indication of glaucoma. Readings between #math.equation(block: false, alt: "10")[$10$] and #math.equation(block: false, alt: "40")[$40$] cannot conclusively distinguish healthy eyes from those with glaucoma. ] ] #figure(figph[shift of caffeine surge curve], alt: "shift of caffeine surge curve", caption: none) The function #math.equation(block: false, alt: "C equals f open parenthesis t close parenthesis")[$C = f ( t )$] shown above gives the caffeine level in Delbert's bloodstream at time #math.equation(block: false, alt: "t")[$t$] hours after he drinks a cup of coffee, and #math.equation(block: false, alt: "g open parenthesis t close parenthesis")[$g ( t )$] gives the caffeine level in Francine's bloodstream. Write a formula for #math.equation(block: false, alt: "g")[$g$] in terms of #math.equation(block: false, alt: "f")[$f$], and explain what it tells you about Delbert and Francine. #math.equation(block: false, alt: "g open parenthesis t close parenthesis equals")[$g ( t ) =$]\_\_\_\_\_ \_\_\_\_\_ #solutionbox[ #math.equation(block: false, alt: "g open parenthesis t close parenthesis equals f open parenthesis t minus 3 close parenthesis")[$g ( t ) = f ( t − 3 )$]. Francine drank her coffee #math.equation(block: false, alt: "3")[$3$] hours after Delbert drank his. ] The function #math.equation(block: false, alt: "C equals f open parenthesis t close parenthesis")[$" " C = f ( t ) " "$] shown below gives the caffeine level in Delbert's bloodstream at time #math.equation(block: false, alt: "t")[$t$] hours after he drinks a cup of coffee, and #math.equation(block: false, alt: "g open parenthesis t close parenthesis")[$g ( t )$] gives the caffeine level in Francine's bloodstream. Write a formula for #math.equation(block: false, alt: "g open parenthesis t close parenthesis")[$g ( t )$] in terms of #math.equation(block: false, alt: "f open parenthesis t close parenthesis")[$f ( t )$], and explain what it tells you about Delbert and Francine. #figure(figph[shift of caffeine surge curve], alt: "shift of caffeine surge curve", caption: none) #solutionbox[ #math.equation(block: false, alt: "g open parenthesis t close parenthesis equals f open parenthesis t minus 3 close parenthesis")[$g ( t ) = f ( t − 3 )$]. Francine drank her coffee #math.equation(block: false, alt: "3")[$3$] hours after Delbert drank his. ] #examplebox("Example 5")[][ Graph #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals open parenthesis x plus 4 close parenthesis cubed plus 2")[$" " " " f ( x ) = ( x + 4 )^(3) + 2$] #solutionbox[ We identify the basic graph from the structure of the formula for #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]. In this case, the basic graph is #math.equation(block: false, alt: "y equals x cubed")[$y = x^(3)$], so we begin by locating a few points on that graph, say, #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis , open parenthesis 1 , 1 close parenthesis ,")[$( 0 , 0 ) , " " ( 1 , 1 ) ,$] and #math.equation(block: false, alt: "open parenthesis minus 1 , minus 1 close parenthesis .")[$( − 1 , − 1 ) .$] We'll perform the translations separately, following the order of operations. First, we sketch a graph of #math.equation(block: false, alt: "y equals open parenthesis x plus 4 close parenthesis cubed")[$y = ( x + 4 )^(3)$] by shifting each point on the basic graph four units to the left. We then move each point up two units to obtain the graph of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals open parenthesis x plus 4 close parenthesis cubed plus 2")[$f ( x ) = ( x + 4 )^(3) + 2$]. All three graphs are shown below. #figure(figph[translations of cubic curve], alt: "translations of cubic curve", caption: none) ] ] + Graph the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals vertical bar x minus 2 vertical bar minus 1")[$f ( x ) = | x − 2 | − 1$]. + How is the graph of #math.equation(block: false, alt: "f")[$f$] different from the graph of #math.equation(block: false, alt: "y equals vertical bar x vertical bar")[$y = | x |$]? #linebreak() \_\_\_\_\_ #solutionbox[ + A graph is shown below. + Translate #math.equation(block: false, alt: "y equals vertical bar x vertical bar")[$y = | x |$] one unit down and two units right. ] A graph for part (a): #figure(figph[shifted absolute value], alt: "shifted absolute value", caption: none) + Graph the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals vertical bar x minus 2 vertical bar minus 1")[$" " f ( x ) = | x − 2 | − 1$]. + How is the graph of #math.equation(block: false, alt: "f")[$f$] different from the graph of #math.equation(block: false, alt: "y equals vertical bar x vertical bar")[$" " y = | x |$]? #solutionbox[ + #figure(figph[shifted absolute value], alt: "shifted absolute value", caption: none) + It is translated one unit down and two units to the right from the graph of #math.equation(block: false, alt: "y equals vertical bar x vertical bar")[$y = | x |$] . ] Horizontal translations are less intuitive than vertical translations. What explanation helps you understand them? \_\_\_\_\_ Horizontal translations are less intuitive than vertical translations. What explanation helps you understand them? === Scale Factors We have seen that #emph[adding] a constant to the expression defining a function results in a translation of its graph. What happens if we #emph[multiply] the expression by a constant? Consider the graphs of the functions #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals 2 x squared , g open parenthesis x close parenthesis equals the fraction 1 over 2 x squared , and h open parenthesis x close parenthesis equals minus x squared")[$f ( x ) = 2 x^(2) , " " " " " " " " g ( x ) = frac(1, 2) x^(2) , " " " " " and " " " " " " " h ( x ) = − x^(2)$] shown below, and compare each to the graph of #math.equation(block: false, alt: "y equals x squared")[$y = x^(2)$]. #figure(figph[basic parabola and vertical stretched parabola], alt: "basic parabola and vertical stretched parabola", caption: none) #figure(table( columns: 3, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y equals x squared")[$y = x^(2)$]], [#math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 2 x squared")[$f ( x ) = 2 x^(2)$]]), [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "8")[$8$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "8")[$8$]], )) Compared to the graph of #math.equation(block: false, alt: "y equals x squared")[$" " y = x^(2)$], the graph of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 2 x squared")[$" " f ( x ) = 2 x^(2) " "$] is expanded, or stretched, vertically by a factor of #math.equation(block: false, alt: "2")[$2$]. The #math.equation(block: false, alt: "y")[$y$]-coordinate of each point on the graph has been doubled, as you can see in the table of values, so each point on the graph of #math.equation(block: false, alt: "f")[$f$] is twice as far from the #math.equation(block: false, alt: "x")[$x$]-axis as its counterpart on the basic graph #math.equation(block: false, alt: "y equals x squared")[$y = x^(2)$]. #figure(figph[vertical compressed parabola and basic parabola], alt: "vertical compressed parabola and basic parabola", caption: none) #figure(table( columns: 3, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y equals x squared")[$y = x^(2)$]], [#math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the fraction 1 over 2 x squared")[$g ( x ) = frac(1, 2) x^(2)$]]), [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "2")[$2$]], )) The graph of #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the fraction 1 over 2 x squared")[$" " g ( x ) = display(frac(1, 2)) x^(2) " "$] is compressed vertically by a factor of #math.equation(block: false, alt: "the fraction 1 over 2")[$display(frac(1, 2))$]; each point is half as far from the #math.equation(block: false, alt: "x")[$x$]-axis as its counterpart on the graph of #math.equation(block: false, alt: "y equals x squared")[$y = x^(2)$]. #figure(figph[reflected parabola and basic parabola], alt: "reflected parabola and basic parabola", caption: none) #figure(table( columns: 3, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y equals x squared")[$y = x^(2)$]], [#math.equation(block: false, alt: "h open parenthesis x close parenthesis equals minus x squared")[$h ( x ) = − x^(2)$]]), [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "minus 4")[$− 4$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "minus 4")[$− 4$]], )) The graph of #math.equation(block: false, alt: "h open parenthesis x close parenthesis equals minus x squared")[$" " h ( x ) = − x^(2) " "$] is flipped, or reflected, about the #math.equation(block: false, alt: "x")[$x$]-axis; the #math.equation(block: false, alt: "y")[$y$]-coordinate of each point on the graph of #math.equation(block: false, alt: "y equals x squared")[$" " y = x^(2) " "$] is replaced by its opposite. In general, we have the following principles. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Scale Factors and Reflections] Compared with the graph of #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$" " y = f ( x )$], the graph of #math.equation(block: false, alt: "y equals a f open parenthesis x close parenthesis")[$" " y = a f ( x )$], where #math.equation(block: false, alt: "a not equal to 0")[$a ≠ 0$], is + stretched vertically by a factor of #math.equation(block: false, alt: "vertical bar a vertical bar")[$| a |$] if #math.equation(block: false, alt: "vertical bar a vertical bar greater than 1")[$| a | > 1$], + compressed vertically by a factor of #math.equation(block: false, alt: "vertical bar a vertical bar")[$| a |$] if #math.equation(block: false, alt: "0 less than vertical bar a vertical bar less than 1")[$0 < | a | < 1$], and + reflected about the #math.equation(block: false, alt: "x")[$x$]-axis if #math.equation(block: false, alt: "a less than 0")[$a < 0$]. ] The constant #math.equation(block: false, alt: "a")[$a$] is called the #strong[scale factor] for the graph. The graph of #math.equation(block: false, alt: "y equals F open parenthesis x close parenthesis")[$y = F ( x )$] is symmetric about the #math.equation(block: false, alt: "y")[$y$]-axis. Which of the following graphs is also symmetric about the #math.equation(block: false, alt: "y")[$y$]-axis? \_\_\_\_\_ #solutionbox[ Both (a) #math.equation(block: false, alt: "y equals minus 3 F open parenthesis x close parenthesis")[$y = − 3 F ( x )$] and (b) #math.equation(block: false, alt: "y equals F open parenthesis x close parenthesis minus 3")[$y = F ( x ) − 3$] ] The graph of #math.equation(block: false, alt: "y equals F open parenthesis x close parenthesis")[$y = F ( x )$] is symmetric about the #math.equation(block: false, alt: "y")[$y$]-axis. Which of the following graphs is also symmetric about the #math.equation(block: false, alt: "y")[$y$]-axis? + #math.equation(block: false, alt: "y equals minus 3 F open parenthesis x close parenthesis")[$y = − 3 F ( x )$] + #math.equation(block: false, alt: "y equals F open parenthesis x close parenthesis minus 3")[$y = F ( x ) − 3$] + #math.equation(block: false, alt: "y equals F open parenthesis x minus 3 close parenthesis")[$y = F ( x − 3 )$] + Both (a) and (b) #examplebox("Example 6")[][ Graph the following functions. + #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals 3 the cube root of x")[$" " g ( x ) = 3 root(3, x)$] + #math.equation(block: false, alt: "h open parenthesis x close parenthesis equals the fraction minus 1 over 2 vertical bar x vertical bar")[$" " h ( x ) = display(frac(− 1, 2)) | x |$] #solutionbox[ + The graph of #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals 3 the cube root of x")[$" " g ( x ) = 3 root(3, x) " "$] is a vertical expansion of the basic graph #math.equation(block: false, alt: "y equals the cube root of x")[$" " y = root(3, x) " "$] by a factor of #math.equation(block: false, alt: "3")[$3$], as shown below . Each point on the basic graph has its #math.equation(block: false, alt: "y")[$y$]-coordinate tripled.#figure(figph[scaled cube root], alt: "scaled cube root", caption: none) + The graph of #math.equation(block: false, alt: "h open parenthesis x close parenthesis equals the fraction minus 1 over 2 vertical bar x vertical bar")[$" " h ( x ) = display(frac(− 1, 2)) | x | " "$] is a vertical compression of the basic graph #math.equation(block: false, alt: "y equals vertical bar x vertical bar")[$" " y = | x | " "$] by a factor of #math.equation(block: false, alt: "the fraction 1 over 2")[$display(frac(1, 2))$], combined with a reflection about the #math.equation(block: false, alt: "x")[$x$]-axis. You may find it helpful to graph the function in two steps, as shown below.#figure(figph[scaled absolute value], alt: "scaled absolute value", caption: none) ] ] + Graph the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 2 vertical bar x vertical bar")[$f ( x ) = 2 | x |$]. + How is the graph of #math.equation(block: false, alt: "f")[$f$] different from the graph of #math.equation(block: false, alt: "y equals vertical bar x vertical bar")[$y = | x |$]? #linebreak() \_\_\_\_\_ #solutionbox[ + A graph is shown below. + Stretch #math.equation(block: false, alt: "y equals vertical bar x vertical bar")[$y = | x |$] vertically by a factor of 2 to obtain the graph of #math.equation(block: false, alt: "f")[$f$]. ] A graph for part (a): #figure(figph[scaled absolute value], alt: "scaled absolute value", caption: none) + Graph the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 2 vertical bar x vertical bar")[$" " f ( x ) = 2 | x |$]. + How is the graph of #math.equation(block: false, alt: "f")[$f$] different from the graph of #math.equation(block: false, alt: "y equals vertical bar x vertical bar")[$" " y = | x |$]? #solutionbox[ + #figure(figph[scaled absolute value], alt: "scaled absolute value", caption: none) + Stretch the graph of #math.equation(block: false, alt: "y equals vertical bar x vertical bar")[$" " y = | x | " "$] vertically by a factor of 2 to obtain the graph of #math.equation(block: false, alt: "f")[$f$]. ] Under which transformations does the graph of a function keep the same shape? \_\_\_\_\_ #solutionbox[ The graph of a function keep the same shape under all of these transformations. ] Under which transformations does the graph of a function keep the same shape? + Vertical translation + Horizontal translation + Reflection + All of these #examplebox("Example 7")[][ The function #math.equation(block: false, alt: "A equals f open parenthesis t close parenthesis")[$" " A = f ( t ) " "$] graphed below gives a person's blood alcohol level #math.equation(block: false, alt: "t")[$t$] hours after drinking a martini. Sketch a graph of #math.equation(block: false, alt: "g open parenthesis t close parenthesis equals 2 f open parenthesis t close parenthesis")[$" " g ( t ) = 2 f ( t ) " "$] and explain what it tells you. #figure(figph[alcohol level after martini], alt: "alcohol level after martini", caption: none) #solutionbox[ To sketch a graph of #math.equation(block: false, alt: "g")[$g$], we stretch the graph of #math.equation(block: false, alt: "f")[$f$] vertically by a factor of #math.equation(block: false, alt: "2")[$2$], as shown below. At each time #math.equation(block: false, alt: "t")[$t$], the person’s blood alcohol level is twice the value given by #math.equation(block: false, alt: "f")[$f$]. The function #math.equation(block: false, alt: "g")[$g$] could represent a person's blood alcohol level #math.equation(block: false, alt: "t")[$t$] hours after drinking two martinis. #figure(figph[scaled alcohol level graph], alt: "scaled alcohol level graph", caption: none) ] ] #figure(figph[daylight hours at two latitudes], alt: "daylight hours at two latitudes", caption: none) If the Earth were not tilted on its axis, there would be 12 daylight hours every day all over the planet. But in fact, the length of a day in a particular location depends on the latitude and the time of year. The graph above shows #math.equation(block: false, alt: "H equals f open parenthesis t close parenthesis")[$H = f ( t )$], the length of a day in Helsinki, Finland, #math.equation(block: false, alt: "t")[$t$] days after January 1, and #math.equation(block: false, alt: "R equals g open parenthesis t close parenthesis")[$R = g ( t )$], the length of a day in Rome. Each is expressed as the number of hours greater or less than 12. Write a formula for #math.equation(block: false, alt: "f")[$f$] in terms of #math.equation(block: false, alt: "g")[$g$]. #math.equation(block: false, alt: "f open parenthesis t close parenthesis equals")[$f ( t ) =$]\_\_\_\_\_ What does this formula tell you? On any given day, the number of daylight hours varies from 12 hours by about... \_\_\_\_\_ #solutionbox[ #math.equation(block: false, alt: "f open parenthesis t close parenthesis approximately equals 2 g open parenthesis t close parenthesis")[$f ( t ) ≈ 2 g ( t )$]. On any given day, the number of daylight hours varies from #math.equation(block: false, alt: "12")[$12$] hours about twice as much in Helsinki as it does in Rome. ] If the Earth were not tilted on its axis, there would be 12 daylight hours every day all over the planet. But in fact, the length of a day in a particular location depends on the latitude and the time of year. #figure(figph[daylight hours at two latitudes], alt: "daylight hours at two latitudes", caption: none) The graph above shows #math.equation(block: false, alt: "H equals f open parenthesis t close parenthesis")[$" " H = f ( t )$], the length of a day in Helsinki, Finland, #math.equation(block: false, alt: "t")[$t$] days after January 1, and #math.equation(block: false, alt: "R equals g open parenthesis t close parenthesis")[$" " R = g ( t )$], the length of a day in Rome. Each is expressed as the number of hours greater or less than 12. Write a formula for #math.equation(block: false, alt: "f")[$f$] in terms of #math.equation(block: false, alt: "g")[$g$]. What does this formula tell you? #solutionbox[ #math.equation(block: false, alt: "f open parenthesis t close parenthesis approximately equals 2 g open parenthesis t close parenthesis")[$f ( t ) ≈ 2 g ( t )$]. On any given day, the number of daylight hours varies from #math.equation(block: false, alt: "12")[$12$] hours about twice as much in Helsinki as it does in Rome. ] In this section we did not consider reflections about the #math.equation(block: false, alt: "y")[$y$]-axis. Can you think of a way to alter the formula for #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$y = f ( x )$] to reflect the graph about the #math.equation(block: false, alt: "y")[$y$]-axis? Which of the eight basic graphs would not be affected by such a reflection? \_\_\_\_\_ In this section we did not consider reflections about the #math.equation(block: false, alt: "y")[$y$]-axis. Can you think of a way to alter the formula for #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$y = f ( x )$] to reflect the graph about the #math.equation(block: false, alt: "y")[$y$]-axis? Which of the eight basic graphs would not be affected by such a reflection? === Section Summary ==== Vocabulary Look up the definitions of new terms in the Glossary. - Transformation - Scale factor - Vertical stretch - Vertical compression - Horizontal translation ==== CONCEPTS + #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Vertical Translations] Compared with the graph of #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$y = f ( x )$], + The graph of #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis plus k open parenthesis k greater than 0 close parenthesis")[$y = f ( x ) + k " " " " ( k > 0 )$] is shifted #emph[upward] #math.equation(block: false, alt: "k")[$k$] units. + The graph of #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis minus k open parenthesis k greater than 0 close parenthesis")[$y = f ( x ) − k " " " " ( k > 0 )$] is shifted #emph[downward] #math.equation(block: false, alt: "k")[$k$] units. ] + #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Horizontal Translations] Compared with the graph of #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$y = f ( x )$], + The graph of #math.equation(block: false, alt: "y equals f open parenthesis x plus h close parenthesis open parenthesis h greater than 0 close parenthesis")[$y = f ( x + h ) " " " " ( h > 0 )$] is shifted #math.equation(block: false, alt: "h")[$h$] units to the #emph[left]. + The graph of #math.equation(block: false, alt: "y equals f open parenthesis x minus h close parenthesis open parenthesis h greater than 0 close parenthesis")[$y = f ( x − h ) " " " " ( h > 0 )$] is shifted #math.equation(block: false, alt: "h")[$h$] units to the #emph[right]. ] + #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Scale Factors and Reflections] Compared with the graph of #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$y = f ( x )$], the graph of #math.equation(block: false, alt: "y equals a f open parenthesis x close parenthesis")[$y = a f ( x )$], where #math.equation(block: false, alt: "a not equal to 0")[$a ≠ 0$], is + stretched vertically by a factor of #math.equation(block: false, alt: "vertical bar a vertical bar")[$| a |$] if #math.equation(block: false, alt: "vertical bar a vertical bar greater than 1")[$| a | > 1$], + compressed vertically by a factor of #math.equation(block: false, alt: "vertical bar a vertical bar")[$| a |$] if #math.equation(block: false, alt: "0 less than vertical bar a vertical bar less than 1")[$0 < | a | < 1$], and + reflected about the #math.equation(block: false, alt: "x")[$x$]-axis if #math.equation(block: false, alt: "a less than 0")[$a < 0$]. ] ==== STUDY QUESTIONS + How does a vertical translation affect the formula for a function? Give an example. + How does a horizontal translation affect the formula for a function? Give an example. + How does a scale factor affect the formula for a function? Give an example. + How is the graph of #math.equation(block: false, alt: "y equals minus f open parenthesis x close parenthesis")[$y = − f ( x )$] different from the graph of #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$y = f ( x )$]? ==== SKILLS Practice each skill in the Homework problems listed. + Write formulas for transformations of functions: \#1–6, 19–22, 35–38 + Recognize and sketch translations of the basic graphs: \#7–18 + Recognize and sketch expansions, compression, and reflections of the basic graphs: \#23–34, 43–50 + Identify transformations from tables of values: \#39–42 + Sketch graphs obtained by two or more transformations of a basic graph: \#51–62 + Write a formula for a transformation of a graph: \#63–76 + Interpret transformations of graphs in context: \#71–76 === Homework 2.3 In Problems 1–6, identify the graph as a translation of a basic function, and write a formula for the graph. #figure(figph[transformation of basic graph], alt: "transformation of basic graph", caption: none) #math.equation(block: true, alt: "y equals the square root of x plus 2")[$y = sqrt(x + 2)$] #figure(figph[transformation of basic graph], alt: "transformation of basic graph", caption: none) #figure(figph[transformation of basic graph], alt: "transformation of basic graph", caption: none) #math.equation(block: true, alt: "y equals x cubed minus 1")[$y = x^(3) − 1$] #figure(figph[transformation of basic graph], alt: "transformation of basic graph", caption: none) #figure(figph[transformation of basic graph], alt: "transformation of basic graph", caption: none) #math.equation(block: true, alt: "y equals the fraction 1 over x minus 4")[$y = display(frac(1, x − 4))$] #figure(figph[transformation of basic graph], alt: "transformation of basic graph", caption: none) For Problems 7–18, + Describe how to transform one of the basic graphs to obtain the graph of the given function. + Using guidepoints, sketch the basic graph and the graph of the given function on the same axes. Label the coordinates of three points on the graph of the given function. #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals vertical bar x vertical bar minus 2")[$f ( x ) = | x | − 2$] + Translate #math.equation(block: false, alt: "y equals vertical bar x vertical bar")[$y = | x |$] by #math.equation(block: false, alt: "2")[$2$] units down. + #figure(figph[translated absolute value], alt: "translated absolute value", caption: none) #math.equation(block: true, alt: "g open parenthesis x close parenthesis equals open parenthesis x plus 1 close parenthesis cubed")[$g ( x ) = ( x + 1 )^(3)$] #math.equation(block: true, alt: "g open parenthesis s close parenthesis equals the cube root of s minus 4")[$g ( s ) = root(3, s − 4)$] + Translate #math.equation(block: false, alt: "y equals the cube root of s")[$y = root(3, s)$] by #math.equation(block: false, alt: "4")[$4$] units right. + #figure(figph[translated cube root], alt: "translated cube root", caption: none) #math.equation(block: true, alt: "f open parenthesis s close parenthesis equals s squared plus 3")[$f ( s ) = s^(2) + 3$] #math.equation(block: true, alt: "F open parenthesis t close parenthesis equals the fraction 1 over t squared plus 1")[$F ( t ) = display(frac(1, t^(2))) + 1$] + Translate #math.equation(block: false, alt: "y equals the fraction 1 over t squared")[$y = display(frac(1, t^(2)))$] by #math.equation(block: false, alt: "1")[$1$] unit up. + #figure(figph[translated inverse-square], alt: "translated inverse-square", caption: none) #math.equation(block: true, alt: "G open parenthesis t close parenthesis equals the square root of t minus 2")[$G ( t ) = sqrt(t − 2)$] #math.equation(block: true, alt: "G open parenthesis r close parenthesis equals open parenthesis r plus 2 close parenthesis cubed")[$G ( r ) = ( r + 2 )^(3)$] + Translate #math.equation(block: false, alt: "y equals r cubed")[$y = r^(3)$] by #math.equation(block: false, alt: "2")[$2$] units left. + #figure(figph[translated cubic], alt: "translated cubic", caption: none) #math.equation(block: true, alt: "F open parenthesis r close parenthesis equals the fraction 1 over r minus 4")[$F ( r ) = display(frac(1, r − 4))$] #math.equation(block: true, alt: "H open parenthesis d close parenthesis equals the square root of d minus 3")[$H ( d ) = sqrt(d) − 3$] + Translate #math.equation(block: false, alt: "y equals the square root of d")[$y = sqrt(d)$] by #math.equation(block: false, alt: "3")[$3$] units down. + #figure(figph[translated root], alt: "translated root", caption: none) #math.equation(block: true, alt: "h open parenthesis d close parenthesis equals the cube root of d plus 5")[$h ( d ) = root(3, d) + 5$] #math.equation(block: true, alt: "h open parenthesis v close parenthesis equals the fraction 1 over v plus 6")[$h ( v ) = display(frac(1, v + 6))$] + Translate #math.equation(block: false, alt: "y equals the fraction 1 over v")[$y = display(frac(1, v))$] by #math.equation(block: false, alt: "6")[$6$] units left. + #figure(figph[translated reciprocal], alt: "translated reciprocal", caption: none) #math.equation(block: true, alt: "H open parenthesis v close parenthesis equals the fraction 1 over v squared minus 2")[$H ( v ) = display(frac(1, v^(2))) − 2$] For Problems 19–22, identify the graph as a stretch, compression, or reflection of a basic function, and write a formula for the graph. #figure(figph[transformed reciprocal], alt: "transformed reciprocal", caption: none) A vertical stretch by a factor of #math.equation(block: false, alt: "3")[$3$]: #math.equation(block: false, alt: "y equals the fraction 3 over x")[$y = display(frac(3, x))$] #figure(figph[transformed cube root], alt: "transformed cube root", caption: none) #figure(figph[transformed cubic], alt: "transformed cubic", caption: none) A vertical compression, the scale factor is #math.equation(block: false, alt: "the fraction 1 over 2")[$display(frac(1, 2))$]: #math.equation(block: false, alt: "y equals the fraction 1 over 2 x cubed")[$y = display(frac(1, 2)) x^(3)$] #figure(figph[transformed root], alt: "transformed root", caption: none) For Problems 23–32, + Identify the scale factor for each function and describe how it affects the graph of the corresponding basic function. + Using guidepoints, sketch the basic graph and the graph of the given function on the same axes. Label the coordinates of three points on the graph of the given function. #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the fraction 1 over 3 vertical bar x vertical bar")[$f ( x ) = display(frac(1, 3)) | x |$] + Scale factor #math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$]; #math.equation(block: false, alt: "y equals vertical bar x vertical bar")[$y = | x |$] is compressed vertically by the scale factor. + #figure(figph[transformed absolute value], alt: "transformed absolute value", caption: none) #math.equation(block: true, alt: "H open parenthesis x close parenthesis equals minus 3 vertical bar x vertical bar")[$H ( x ) = − 3 | x |$] #math.equation(block: true, alt: "h open parenthesis z close parenthesis equals the fraction minus 2 over z squared")[$h ( z ) = display(frac(− 2, z^(2)))$] + Scale factor #math.equation(block: false, alt: "minus 2")[$− 2$]; #math.equation(block: false, alt: "y equals the fraction 1 over z squared")[$y = frac(1, z^(2))$] is reflected over the #math.equation(block: false, alt: "z")[$z$]-axis and stretched vertically by a factor of #math.equation(block: false, alt: "2")[$2$]. + #figure(figph[transformed inverse-square], alt: "transformed inverse-square", caption: none) #math.equation(block: true, alt: "g open parenthesis z close parenthesis equals the fraction 2 over z")[$g ( z ) = display(frac(2, z))$] #math.equation(block: true, alt: "G open parenthesis v close parenthesis equals minus 3 the square root of v")[$G ( v ) = − 3 sqrt(v)$] + Scale factor #math.equation(block: false, alt: "minus 3")[$− 3$]; #math.equation(block: false, alt: "y equals the square root of v")[$y = sqrt(v)$] is reflected over the #math.equation(block: false, alt: "v")[$v$]-axis and stretched vertically by a factor of #math.equation(block: false, alt: "3")[$3$]. + #figure(figph[transformed root], alt: "transformed root", caption: none) #math.equation(block: true, alt: "F open parenthesis v close parenthesis equals minus 4 the cube root of v")[$F ( v ) = − 4 root(3, v)$] #math.equation(block: true, alt: "g open parenthesis s close parenthesis equals the fraction minus 1 over 2 s cubed")[$g ( s ) = display(frac(− 1, 2)) s^(3)$] + Scale factor #math.equation(block: false, alt: "the fraction minus 1 over 2")[$frac(− 1, 2)$]; #math.equation(block: false, alt: "y equals s cubed")[$y = s^(3)$] is reflected over the #math.equation(block: false, alt: "s")[$s$]-axis and compressed vertically by a factor of #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$]. + #figure(figph[transformed cubic], alt: "transformed cubic", caption: none) #math.equation(block: true, alt: "f open parenthesis s close parenthesis equals the fraction 1 over 8 s cubed")[$f ( s ) = display(frac(1, 8)) s^(3)$] #math.equation(block: true, alt: "H open parenthesis x close parenthesis equals the fraction 1 over 3 x")[$H ( x ) = display(frac(1, 3 x))$] + Scale factor #math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$]; #math.equation(block: false, alt: "y equals the fraction 1 over x")[$y = frac(1, x)$] is compressed vertically by the scale factor. + #figure(figph[transformed reciprocal], alt: "transformed reciprocal", caption: none) #math.equation(block: true, alt: "h open parenthesis x close parenthesis equals the fraction minus 1 over 4 x squared")[$h ( x ) = display(frac(− 1, 4 x^(2)))$] In Problems 33 and 34, match each graph with its equation. #figure(figph[six graphs], alt: "six graphs", caption: none) + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 3 the square root of x")[$f ( x ) = 3 sqrt(x)$] + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 2 x cubed")[$f ( x ) = 2 x^(3)$] + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the fraction x over 3")[$f ( x ) = display(frac(x, 3))$] + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the fraction 3 over x")[$f ( x ) = display(frac(3, x))$] + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 2 the cube root of x")[$f ( x ) = 2 root(3, x)$] + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 3 x squared")[$f ( x ) = 3 x^(2)$] + vi + ii + iv + i + v + iii #figure(figph[six graphs], alt: "six graphs", caption: none) + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x cubed minus 2")[$f ( x ) = x^(3) − 2$] + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the cube root of x plus 2")[$f ( x ) = root(3, x) + 2$] + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the fraction 1 over open parenthesis x minus 3 close parenthesis squared")[$f ( x ) = display(frac(1, ( x − 3 )^(2)))$] + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals vertical bar x vertical bar minus 3")[$f ( x ) = | x | − 3$] + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x squared plus 3")[$f ( x ) = x^(2) + 3$] + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the square root of x minus 3")[$f ( x ) = sqrt(x − 3)$] In Problems 35–38, the graph of a function is shown. Describe each transformation of the graph; then give a formula for each in terms of the original function. #figure(figph[graph and 4 transformations], alt: "graph and 4 transformations", caption: none) + Vertical stretch by a factor of #math.equation(block: false, alt: "3")[$3$]: #math.equation(block: false, alt: "y equals 3 f open parenthesis x close parenthesis")[$y = 3 f ( x )$] + Reflection about the #math.equation(block: false, alt: "x")[$x$]-axis: #math.equation(block: false, alt: "y equals minus f open parenthesis x close parenthesis")[$y = − f ( x )$] + Translation #math.equation(block: false, alt: "1")[$1$] unit right: #math.equation(block: false, alt: "y equals f open parenthesis x minus 1 close parenthesis")[$y = f ( x − 1 )$] + Translation #math.equation(block: false, alt: "4")[$4$] units up: #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis plus 4")[$y = f ( x ) + 4$] #figure(figph[graph and 4 transformations], alt: "graph and 4 transformations", caption: none) #figure(figph[graph and 4 transformations], alt: "graph and 4 transformations", caption: none) + Reflection about the #math.equation(block: false, alt: "v")[$v$]-axis and vertical stretch by a factor of #math.equation(block: false, alt: "2")[$2$]: #math.equation(block: false, alt: "T equals minus 2 h open parenthesis v close parenthesis")[$T = − 2 h ( v )$] + Vertical stretch by a factor of #math.equation(block: false, alt: "3")[$3$]: #math.equation(block: false, alt: "T equals 3 h open parenthesis v close parenthesis")[$T = 3 h ( v )$] + Translation #math.equation(block: false, alt: "3")[$3$] units up: #math.equation(block: false, alt: "T equals h open parenthesis v close parenthesis plus 3")[$T = h ( v ) + 3$] + Translation #math.equation(block: false, alt: "3")[$3$] units left: #math.equation(block: false, alt: "T equals h open parenthesis v plus 3 close parenthesis")[$T = h ( v + 3 )$] #figure(figph[graph and 4 transformations], alt: "graph and 4 transformations", caption: none) In Problems 39–42, each table in parts (a)–(d) describes a transformation of #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]. Identify the transformation and write a formula for the new function in terms of #math.equation(block: false, alt: "f")[$f$]. #figure(table( columns: 7, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "1")[$" " " " 1 " " " "$]], [#math.equation(block: false, alt: "2")[$" " " " 2 " " " "$]], [#math.equation(block: false, alt: "3")[$" " " " 3 " " " "$]], [#math.equation(block: false, alt: "4")[$" " " " 4 " " " "$]], [#math.equation(block: false, alt: "5")[$" " " " 5 " " " "$]], [#math.equation(block: false, alt: "6")[$" " " " 6 " " " "$]]), [#math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]], [#math.equation(block: false, alt: "8")[$8$]], [#math.equation(block: false, alt: "6")[$6$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "2")[$2$]], )) + #figure(table( columns: 7, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$" " " " x " " " "$]], [#math.equation(block: false, alt: "1")[$" " " " 1 " " " "$]], [#math.equation(block: false, alt: "2")[$" " " " 2 " " " "$]], [#math.equation(block: false, alt: "3")[$" " " " 3 " " " "$]], [#math.equation(block: false, alt: "4")[$" " " " 4 " " " "$]], [#math.equation(block: false, alt: "5")[$" " " " 5 " " " "$]], [#math.equation(block: false, alt: "6")[$" " " " 6 " " " "$]]), [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "8")[$8$]], [#math.equation(block: false, alt: "6")[$6$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "4")[$4$]], )) + #figure(table( columns: 7, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$" " " " x " " " "$]], [#math.equation(block: false, alt: "1")[$" " " " 1 " " " "$]], [#math.equation(block: false, alt: "2")[$" " " " 2 " " " "$]], [#math.equation(block: false, alt: "3")[$" " " " 3 " " " "$]], [#math.equation(block: false, alt: "4")[$" " " " 4 " " " "$]], [#math.equation(block: false, alt: "5")[$" " " " 5 " " " "$]], [#math.equation(block: false, alt: "6")[$" " " " 6 " " " "$]]), [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "minus 4")[$− 4$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], )) + #figure(table( columns: 7, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$" " " " x " " " "$]], [#math.equation(block: false, alt: "1")[$" " " " 1 " " " "$]], [#math.equation(block: false, alt: "2")[$" " " " 2 " " " "$]], [#math.equation(block: false, alt: "3")[$" " " " 3 " " " "$]], [#math.equation(block: false, alt: "4")[$" " " " 4 " " " "$]], [#math.equation(block: false, alt: "5")[$" " " " 5 " " " "$]], [#math.equation(block: false, alt: "6")[$" " " " 6 " " " "$]]), [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], )) + #figure(table( columns: 7, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$" " " " x " " " "$]], [#math.equation(block: false, alt: "1")[$" " " " 1 " " " "$]], [#math.equation(block: false, alt: "2")[$" " " " 2 " " " "$]], [#math.equation(block: false, alt: "3")[$" " " " 3 " " " "$]], [#math.equation(block: false, alt: "4")[$" " " " 4 " " " "$]], [#math.equation(block: false, alt: "5")[$" " " " 5 " " " "$]], [#math.equation(block: false, alt: "6")[$" " " " 6 " " " "$]]), [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "8")[$8$]], [#math.equation(block: false, alt: "6")[$6$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "0")[$0$]], )) + Translation #math.equation(block: false, alt: "2")[$2$] units up: #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis plus 2")[$y = f ( x ) + 2$] + Translation #math.equation(block: false, alt: "4")[$4$] units down: #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis minus 4")[$y = f ( x ) − 4$] + Vertical compression by a factor of #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$]: #math.equation(block: false, alt: "y equals the fraction 1 over 2 f open parenthesis x close parenthesis")[$y = frac(1, 2) f ( x )$] + Translation #math.equation(block: false, alt: "1")[$1$] unit right: #math.equation(block: false, alt: "y equals f open parenthesis x minus 1 close parenthesis")[$y = f ( x − 1 )$] #figure(table( columns: 7, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "minus 3")[$" " − 3 " "$]], [#math.equation(block: false, alt: "minus 2")[$" " − 2 " "$]], [#math.equation(block: false, alt: "minus 1")[$" " − 1 " "$]], [#math.equation(block: false, alt: "0")[$" " " " 0 " " " "$]], [#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: "13")[$13$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "minus 5")[$− 5$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "3")[$3$]], )) + #figure(table( columns: 7, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "minus 3")[$" " − 3 " "$]], [#math.equation(block: false, alt: "minus 2")[$" " − 2 " "$]], [#math.equation(block: false, alt: "minus 1")[$" " − 1 " "$]], [#math.equation(block: false, alt: "0")[$" " " " 0 " " " "$]], [#math.equation(block: false, alt: "1")[$" " " " 1 " " " "$]], [#math.equation(block: false, alt: "2")[$" " " " 2 " " " "$]]), [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "minus 26")[$− 26$]], [#math.equation(block: false, alt: "minus 6")[$− 6$]], [#math.equation(block: false, alt: "6")[$6$]], [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "6")[$6$]], [#math.equation(block: false, alt: "minus 6")[$− 6$]], )) + #figure(table( columns: 7, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "minus 3")[$" " − 3 " "$]], [#math.equation(block: false, alt: "minus 2")[$" " − 2 " "$]], [#math.equation(block: false, alt: "minus 1")[$" " − 1 " "$]], [#math.equation(block: false, alt: "0")[$" " " " 0 " " " "$]], [#math.equation(block: false, alt: "1")[$" " " " 1 " " " "$]], [#math.equation(block: false, alt: "2")[$" " " " 2 " " " "$]]), [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "18")[$18$]], [#math.equation(block: false, alt: "8")[$8$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "8")[$8$]], )) + #figure(table( columns: 7, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "minus 3")[$" " − 3 " "$]], [#math.equation(block: false, alt: "minus 2")[$" " − 2 " "$]], [#math.equation(block: false, alt: "minus 1")[$" " − 1 " "$]], [#math.equation(block: false, alt: "0")[$" " " " 0 " " " "$]], [#math.equation(block: false, alt: "1")[$" " " " 1 " " " "$]], [#math.equation(block: false, alt: "2")[$" " " " 2 " " " "$]]), [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "minus 5")[$− 5$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "13")[$13$]], [#math.equation(block: false, alt: "27")[$27$]], )) + #figure(table( columns: 7, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "minus 3")[$" " − 3 " "$]], [#math.equation(block: false, alt: "minus 2")[$" " − 2 " "$]], [#math.equation(block: false, alt: "minus 1")[$" " − 1 " "$]], [#math.equation(block: false, alt: "0")[$" " " " 0 " " " "$]], [#math.equation(block: false, alt: "1")[$" " " " 1 " " " "$]], [#math.equation(block: false, alt: "2")[$" " " " 2 " " " "$]]), [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "2.6")[$2.6$]], [#math.equation(block: false, alt: "0.6")[$0.6$]], [#math.equation(block: false, alt: "minus 0.6")[$− 0.6$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "minus 0.6")[$− 0.6$]], [#math.equation(block: false, alt: "0.6")[$0.6$]], )) #figure(table( columns: 7, 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: "0")[$" " " " 0 " " " "$]], [#math.equation(block: false, alt: "1")[$" " " " 1 " " " "$]], [#math.equation(block: false, alt: "2")[$" " " " 2 " " " "$]], [#math.equation(block: false, alt: "3")[$" " " " 3 " " " "$]]), [#math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]], [#math.equation(block: false, alt: "minus 9")[$− 9$]], [#math.equation(block: false, alt: "minus 8")[$− 8$]], [#math.equation(block: false, alt: "minus 7")[$− 7$]], [#math.equation(block: false, alt: "minus 6")[$− 6$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "20")[$20$]], )) + #figure(table( columns: 7, 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: "0")[$" " " " 0 " " " "$]], [#math.equation(block: false, alt: "1")[$" " " " 1 " " " "$]], [#math.equation(block: false, alt: "2")[$" " " " 2 " " " "$]], [#math.equation(block: false, alt: "3")[$" " " " 3 " " " "$]]), [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "minus 34")[$− 34$]], [#math.equation(block: false, alt: "minus 9")[$− 9$]], [#math.equation(block: false, alt: "minus 8")[$− 8$]], [#math.equation(block: false, alt: "minus 7")[$− 7$]], [#math.equation(block: false, alt: "minus 6")[$− 6$]], [#math.equation(block: false, alt: "1")[$1$]], )) + #figure(table( columns: 7, 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: "0")[$" " " " 0 " " " "$]], [#math.equation(block: false, alt: "1")[$" " " " 1 " " " "$]], [#math.equation(block: false, alt: "2")[$" " " " 2 " " " "$]], [#math.equation(block: false, alt: "3")[$" " " " 3 " " " "$]]), [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "minus 4")[$− 4$]], [#math.equation(block: false, alt: "21")[$21$]], [#math.equation(block: false, alt: "22")[$22$]], [#math.equation(block: false, alt: "23")[$23$]], [#math.equation(block: false, alt: "24")[$24$]], [#math.equation(block: false, alt: "31")[$31$]], )) + #figure(table( columns: 7, 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: "0")[$" " " " 0 " " " "$]], [#math.equation(block: false, alt: "1")[$" " " " 1 " " " "$]], [#math.equation(block: false, alt: "2")[$" " " " 2 " " " "$]], [#math.equation(block: false, alt: "3")[$" " " " 3 " " " "$]]), [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "18")[$18$]], [#math.equation(block: false, alt: "16")[$16$]], [#math.equation(block: false, alt: "14")[$14$]], [#math.equation(block: false, alt: "12")[$12$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "minus 40")[$− 40$]], )) + #figure(table( columns: 7, 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: "0")[$" " " " 0 " " " "$]], [#math.equation(block: false, alt: "1")[$" " " " 1 " " " "$]], [#math.equation(block: false, alt: "2")[$" " " " 2 " " " "$]], [#math.equation(block: false, alt: "3")[$" " " " 3 " " " "$]]), [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "8")[$8$]], [#math.equation(block: false, alt: "6")[$6$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "minus 12")[$− 12$]], [#math.equation(block: false, alt: "minus 50")[$− 50$]], )) + Translation #math.equation(block: false, alt: "1")[$1$] unit right: #math.equation(block: false, alt: "y equals f open parenthesis x minus 1 close parenthesis")[$y = f ( x − 1 )$] + Part (a) is translated #math.equation(block: false, alt: "30")[$30$] units up: #math.equation(block: false, alt: "y equals f open parenthesis x minus 1 close parenthesis plus 30")[$y = f ( x − 1 ) + 30$] + #math.equation(block: false, alt: "f")[$f$] is reflected about the #math.equation(block: false, alt: "x")[$x$]-axis and stretched vertically by a factor of #math.equation(block: false, alt: "2")[$2$]: #math.equation(block: false, alt: "y equals minus 2 f open parenthesis x close parenthesis")[$y = − 2 f ( x )$] + Part (c) is translated #math.equation(block: false, alt: "10")[$10$] units down: #math.equation(block: false, alt: "y equals minus 2 f open parenthesis x close parenthesis minus 10")[$y = − 2 f ( x ) − 10$] #figure(table( columns: 7, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "1")[$" " " " 1 " " " "$]], [#math.equation(block: false, alt: "2")[$" " " " 2 " " " "$]], [#math.equation(block: false, alt: "3")[$" " " " 3 " " " "$]], [#math.equation(block: false, alt: "4")[$" " " " 4 " " " "$]], [#math.equation(block: false, alt: "5")[$" " " " 5 " " " "$]], [#math.equation(block: false, alt: "6")[$" " " " 6 " " " "$]]), [#math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]], [#math.equation(block: false, alt: "60")[$60$]], [#math.equation(block: false, alt: "30")[$30$]], [#math.equation(block: false, alt: "20")[$20$]], [#math.equation(block: false, alt: "15")[$15$]], [#math.equation(block: false, alt: "12")[$12$]], [#math.equation(block: false, alt: "10")[$10$]], )) + #figure(table( columns: 7, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "1")[$" " " " 1 " " " "$]], [#math.equation(block: false, alt: "2")[$" " " " 2 " " " "$]], [#math.equation(block: false, alt: "3")[$" " " " 3 " " " "$]], [#math.equation(block: false, alt: "4")[$" " " " 4 " " " "$]], [#math.equation(block: false, alt: "5")[$" " " " 5 " " " "$]], [#math.equation(block: false, alt: "6")[$" " " " 6 " " " "$]]), [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "30")[$30$]], [#math.equation(block: false, alt: "15")[$15$]], [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "7.5")[$7.5$]], [#math.equation(block: false, alt: "6")[$6$]], [#math.equation(block: false, alt: "5")[$5$]], )) + #figure(table( columns: 7, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "1")[$" " " " 1 " " " "$]], [#math.equation(block: false, alt: "2")[$" " " " 2 " " " "$]], [#math.equation(block: false, alt: "3")[$" " " " 3 " " " "$]], [#math.equation(block: false, alt: "4")[$" " " " 4 " " " "$]], [#math.equation(block: false, alt: "5")[$" " " " 5 " " " "$]], [#math.equation(block: false, alt: "6")[$" " " " 6 " " " "$]]), [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "35")[$35$]], [#math.equation(block: false, alt: "20")[$20$]], [#math.equation(block: false, alt: "15")[$15$]], [#math.equation(block: false, alt: "12.5")[$12.5$]], [#math.equation(block: false, alt: "11")[$11$]], [#math.equation(block: false, alt: "10")[$10$]], )) + #figure(table( columns: 7, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "1")[$" " " " 1 " " " "$]], [#math.equation(block: false, alt: "2")[$" " " " 2 " " " "$]], [#math.equation(block: false, alt: "3")[$" " " " 3 " " " "$]], [#math.equation(block: false, alt: "4")[$" " " " 4 " " " "$]], [#math.equation(block: false, alt: "5")[$" " " " 5 " " " "$]], [#math.equation(block: false, alt: "6")[$" " " " 6 " " " "$]]), [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "minus 12")[$− 12$]], [#math.equation(block: false, alt: "minus 6")[$− 6$]], [#math.equation(block: false, alt: "minus 4")[$− 4$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "minus 2.4")[$− 2.4$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], )) + #figure(table( columns: 7, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "1")[$" " " " 1 " " " "$]], [#math.equation(block: false, alt: "2")[$" " " " 2 " " " "$]], [#math.equation(block: false, alt: "3")[$" " " " 3 " " " "$]], [#math.equation(block: false, alt: "4")[$" " " " 4 " " " "$]], [#math.equation(block: false, alt: "5")[$" " " " 5 " " " "$]], [#math.equation(block: false, alt: "6")[$" " " " 6 " " " "$]]), [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "minus 10")[$− 10$]], [#math.equation(block: false, alt: "minus 4")[$− 4$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "minus 0.4")[$− 0.4$]], [#math.equation(block: false, alt: "0")[$0$]], )) For Problems 43–50, write the function in the form #math.equation(block: false, alt: "y equals k f open parenthesis x close parenthesis")[$y = k f ( x )$], where #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] is one of the basic functions. Describe how the graph differs from that of the basic function. #math.equation(block: true, alt: "y equals the fraction 1 over 2 x squared")[$y = display(frac(1, 2 x^(2)))$] #math.equation(block: false, alt: "y equals the fraction 1 over 2 times the fraction 1 over x squared")[$y = display(frac(1, 2)) ⋅ display(frac(1, x^(2)))$] is a vertical compression with factor #math.equation(block: false, alt: "d f r a c 12")[$d f r a c 1 2$] of #math.equation(block: false, alt: "y equals the fraction 1 over x squared")[$y = display(frac(1, x^(2)))$]. #math.equation(block: true, alt: "y equals the square root of 9 x")[$y = sqrt(9 x)$] #math.equation(block: true, alt: "y equals the cube root of 8 x")[$y = root(3, 8 x)$] #math.equation(block: false, alt: "y equals 2 the cube root of x")[$y = 2 root(3, x)$] is a vertical stretch with factor #math.equation(block: false, alt: "2")[$2$] of #math.equation(block: false, alt: "y equals the cube root of x")[$y = root(3, x)$]. #math.equation(block: true, alt: "y equals the fraction 1 over 4 x")[$y = display(frac(1, 4 x))$] #math.equation(block: true, alt: "y equals vertical bar 3 x vertical bar")[$y = | 3 x |$] #math.equation(block: false, alt: "y equals 3 vertical bar x vertical bar")[$y = 3 | x |$] is a vertical stretch with factor #math.equation(block: false, alt: "3")[$3$] of #math.equation(block: false, alt: "y equals vertical bar x vertical bar")[$y = | x |$]. #math.equation(block: true, alt: "y equals open parenthesis the fraction x over 2 close parenthesis squared")[$y = attach(( display(frac(x, 2)) ), t: 2)$] #math.equation(block: true, alt: "y equals open parenthesis the fraction x over 2 close parenthesis cubed")[$y = attach(( display(frac(x, 2)) ), t: 3)$] #math.equation(block: false, alt: "y equals the fraction 1 over 8 x cubed")[$y = display(frac(1, 8)) x^(3)$] is a vertical compression with factor #math.equation(block: false, alt: "the fraction 1 over 8")[$display(frac(1, 8))$] of #math.equation(block: false, alt: "y equals x cubed")[$y = x^(3)$]. #math.equation(block: true, alt: "y equals vertical bar the fraction x over 5 vertical bar")[$y = | display(frac(x, 5)) |$] For Problems 51–62, + The graph of each function can be obtained from one of the basic graphs by two or more transformations. Describe the transformations. + Sketch the basic graph and the graph of the given function by hand on the same axes. Label the coordinates of three points on the graph of the given function. #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals 2 plus open parenthesis x minus 3 close parenthesis squared")[$f ( x ) = 2 + ( x − 3 )^(2)$] + Translation by #math.equation(block: false, alt: "2")[$2$] units up and #math.equation(block: false, alt: "3")[$3$] units right + #figure(figph[transformed parabola], alt: "transformed parabola", caption: none) #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals open parenthesis x plus 4 close parenthesis squared plus 1")[$f ( x ) = ( x + 4 )^(2) + 1$] #math.equation(block: true, alt: "g open parenthesis z close parenthesis equals the fraction 1 over z plus 2 minus 3")[$g ( z ) = display(frac(1, z + 2)) − 3$] + Translation by #math.equation(block: false, alt: "2")[$2$] units left and #math.equation(block: false, alt: "3")[$3$] units down. + #figure(figph[transformed reciprocal], alt: "transformed reciprocal", caption: none) #math.equation(block: true, alt: "g open parenthesis z close parenthesis equals the fraction 1 over z minus 1 plus 1")[$g ( z ) = display(frac(1, z − 1)) + 1$] #math.equation(block: true, alt: "F open parenthesis u close parenthesis equals minus 3 the square root of u plus 4 plus 4")[$F ( u ) = − 3 sqrt(u + 4) + 4$] + Reflection across the #math.equation(block: false, alt: "u")[$u$]-axis, vertical stretch by a factor of #math.equation(block: false, alt: "3")[$3$], translation by #math.equation(block: false, alt: "4")[$4$] units left and #math.equation(block: false, alt: "4")[$4$] units up + #figure(figph[transformed reciprocal], alt: "transformed reciprocal", caption: none) #math.equation(block: true, alt: "F open parenthesis u close parenthesis equals 4 the square root of u minus 3 minus 5")[$F ( u ) = 4 sqrt(u − 3) − 5$] #math.equation(block: true, alt: "G open parenthesis t close parenthesis equals 2 vertical bar t minus 5 vertical bar minus 1")[$G ( t ) = 2 | t − 5 | − 1$] + Vertical stretch by a factor of #math.equation(block: false, alt: "2")[$2$], translation by #math.equation(block: false, alt: "5")[$5$] units right and #math.equation(block: false, alt: "1")[$1$] down + #figure(figph[transformed reciprocal], alt: "transformed reciprocal", caption: none) #math.equation(block: true, alt: "G open parenthesis t close parenthesis equals 2 minus vertical bar t plus 4 vertical bar")[$G ( t ) = 2 − | t + 4 |$] #math.equation(block: true, alt: "H open parenthesis w close parenthesis equals 6 minus the fraction 2 over open parenthesis w minus 1 close parenthesis squared")[$H ( w ) = 6 − display(frac(2, ( w − 1 )^(2)))$] + Reflection across the #math.equation(block: false, alt: "w")[$w$]-axis, vertical stretch by a factor of #math.equation(block: false, alt: "2")[$2$], translation by #math.equation(block: false, alt: "6")[$6$] units up and #math.equation(block: false, alt: "1")[$1$] unit right + #figure(figph[transformed inverse-square], alt: "transformed inverse-square", caption: none) #math.equation(block: true, alt: "H open parenthesis w close parenthesis equals the fraction 3 over open parenthesis w plus 2 close parenthesis squared minus 1")[$H ( w ) = display(frac(3, ( w + 2 )^(2))) − 1$] #math.equation(block: true, alt: "f open parenthesis t close parenthesis equals the cube root of t minus 8 minus 1")[$f ( t ) = root(3, t − 8) − 1$] + Translation by #math.equation(block: false, alt: "8")[$8$] units right and #math.equation(block: false, alt: "1")[$1$] unit down + #figure(figph[transformed cube root], alt: "transformed cube root", caption: none) #math.equation(block: true, alt: "f open parenthesis t close parenthesis equals the cube root of t plus 1 plus 8")[$f ( t ) = root(3, t + 1) + 8$] In Problems 63 and 64, each graph can be obtained by two transformations of the given graph. Describe the transformations and write a formula for the new graph in terms of f. #figure(figph[graph and two transformations], alt: "graph and two transformations", caption: none) + Translation by #math.equation(block: false, alt: "4")[$4$] units up and #math.equation(block: false, alt: "1")[$1$] unit right: #math.equation(block: false, alt: "y equals f open parenthesis x minus 1 close parenthesis plus 4")[$y = f ( x − 1 ) + 4$] + Vertical stretch by a factor of #math.equation(block: false, alt: "2")[$2$] and a translation by #math.equation(block: false, alt: "4")[$4$] units up: #math.equation(block: false, alt: "y equals 2 f open parenthesis x close parenthesis plus 4")[$y = 2 f ( x ) + 4$] #figure(figph[graph and two transformations], alt: "graph and two transformations", caption: none) For Problems 65–70, + Describe the graph as a transformation of a basic function. + Give an equation for the function shown. #figure(figph[transformed basic], alt: "transformed basic", caption: none) + #math.equation(block: false, alt: "y equals vertical bar x vertical bar")[$y = | x |$] translated by #math.equation(block: false, alt: "1")[$1$] unit left and #math.equation(block: false, alt: "2")[$2$] units down + #math.equation(block: false, alt: "y equals vertical bar x plus 1 vertical bar minus 2")[$y = | x + 1 | − 2$] #figure(figph[transformed basic], alt: "transformed basic", caption: none) #figure(figph[transformed basic], alt: "transformed basic", caption: none) + #math.equation(block: false, alt: "y equals the square root of x")[$y = sqrt(x)$] reflected about the #math.equation(block: false, alt: "x")[$x$]-axis and shifted #math.equation(block: false, alt: "3")[$3$] units up + #math.equation(block: false, alt: "y equals minus the square root of x plus 3")[$y = − sqrt(x) + 3$] #figure(figph[transformed basic], alt: "transformed basic", caption: none) #figure(figph[transformed basic], alt: "transformed basic", caption: none) + #math.equation(block: false, alt: "y equals x cubed")[$y = x^(3)$] translated by #math.equation(block: false, alt: "3")[$3$] units right and #math.equation(block: false, alt: "1")[$1$] unit up + #math.equation(block: false, alt: "y equals open parenthesis x minus 3 close parenthesis cubed plus 1")[$y = ( x − 3 )^(3) + 1$] #figure(figph[transformed basic], alt: "transformed basic", caption: none) The graph of #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] shows the number of students in Professor Hilbert's class who scored #math.equation(block: false, alt: "x")[$x$] points on a quiz. Write a formula for each transformation of #math.equation(block: false, alt: "f")[$f$] ((a) and (b) of the figure below); then explain how the quiz results in that class compare to the results in Professor Hilbert's class. #figure(figph[bell-shaped curve], alt: "bell-shaped curve", caption: none) #figure(figph[bell-shaped curve], alt: "bell-shaped curve", caption: none) #figure(figph[bell-shaped curve], alt: "bell-shaped curve", caption: none) + #math.equation(block: false, alt: "y equals f open parenthesis x minus 20 close parenthesis")[$y = f ( x − 20 )$]: Students scored #math.equation(block: false, alt: "20")[$20$] points higher than Professor Hilbert's class. + #math.equation(block: false, alt: "y equals 1.5 f open parenthesis x close parenthesis")[$y = 1.5 f ( x )$]: The class is about #math.equation(block: false, alt: "50 %")[$50 upright(%)$] larger than Hilbert's, but the classes scored the same. The graph of #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] shows the number of men at Tyler College who are #math.equation(block: false, alt: "x")[$x$] inches tall. Write a formula for each transformation of #math.equation(block: false, alt: "f")[$f$] ; then explain how the heights in that population compare to the Tyler College men. #figure(figph[bell-shaped curve centered above x=70], alt: "bell-shaped curve centered above x=70", caption: none) #figure(figph[bell-shaped curve centered above x=70], alt: "bell-shaped curve centered above x=70", caption: none) #figure(figph[bell-shaped curve centered above x=60], alt: "bell-shaped curve centered above x=60", caption: none) The graph of #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] shows the California state income tax rate, in percent, for a single taxpayer whose annual taxable income is #math.equation(block: false, alt: "x")[$x$] dollars. Write a formula for each transformation of #math.equation(block: false, alt: "f")[$f$]; then explain what it tells you about the income tax scheme in that state. #figure(figph[step function], alt: "step function", caption: none) #figure(figph[step function], alt: "step function", caption: none) #figure(figph[step function], alt: "step function", caption: none) + #math.equation(block: false, alt: "y equals f open parenthesis x minus 5000 close parenthesis")[$y = f ( x − 5000 )$]: Taxpayers earn \$#math.equation(block: false, alt: "5000")[$5000$] more than Californians in each tax rate + #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis minus 0.2")[$y = f ( x ) − 0.2$]: Taxpayers pay #math.equation(block: false, alt: "0.2 %")[$0.2 upright(%)$] less tax than Californians on the same income. The graph of #math.equation(block: false, alt: "f open parenthesis w close parenthesis")[$f ( w )$] shows the shipping rate at SendIt for a package that weighs #math.equation(block: false, alt: "w")[$w$] pounds. Write a formula for each transformation of #math.equation(block: false, alt: "f")[$f$] and explain how the shipping rates compare to the rates at SendIt. #figure(figph[step function], alt: "step function", caption: none) #figure(figph[step function], alt: "step function", caption: none) #figure(figph[step function], alt: "step function", caption: none) The graph of #math.equation(block: false, alt: "g open parenthesis t close parenthesis")[$g ( t )$] shows the population of marmots in a national park #math.equation(block: false, alt: "t")[$t$] months after January 1. Write a formula for each transformation of #math.equation(block: false, alt: "g")[$g$] and explain how the population of that species compares to the population of marmots. #figure(figph[periodic], alt: "periodic", caption: none) #figure(figph[periodic], alt: "periodic", caption: none) #figure(figph[periodic], alt: "periodic", caption: none) + #math.equation(block: false, alt: "y equals g open parenthesis t plus 2 close parenthesis")[$y = g ( t + 2 )$]: This population has its maximum and minimum two months before the marmots. + #math.equation(block: false, alt: "y equals g open parenthesis t close parenthesis minus 20")[$y = g ( t ) − 20$]: This population remains #math.equation(block: false, alt: "20")[$20$] fewer than that of the marmots. The graph of #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] is a dose-response curve. It shows the intensity of the response to a drug as a function of the dosage #math.equation(block: false, alt: "x")[$x$] milligrams administered. The intensity is given as a percentage of the maximum response. Write a formula for each transformation of #math.equation(block: false, alt: "f")[$f$] and explain what it tells you about the response to that drug #figure(figph[sigmoid], alt: "sigmoid", caption: none) #figure(figph[sigmoid], alt: "sigmoid", caption: none) #figure(figph[sigmoid], alt: "sigmoid", caption: none)