#set document(title: "4.2 Exponential Functions", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 4.2#h(0.6em)Exponential Functions === Introduction In Exponential Growth and Decay, we studied functions that describe exponential growth or decay. More formally, we define an #strong[exponential function] as follows. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Exponential Function] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals a b to the power x , where b greater than 0 and b not equal to 1 , a not equal to 0")[$f ( x ) = a b^(x) , " " " " " " " " " where " " " " " " " b > 0 " " " " " " " and " " " " " " " b ≠ 1 ", " " " " " " " a ≠ 0$] ] Some examples of exponential functions are #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals 5 to the power x , P open parenthesis t close parenthesis equals 250 open parenthesis 1.7 close parenthesis to the power t , and g open parenthesis n close parenthesis equals 2.4 open parenthesis 0.3 close parenthesis to the power n")[$f ( x ) = 5^(x) , " " " " " " " " P ( t ) = 250 ( 1.7 )^(t) , " " " " " " " " "and " " " " " " " " " g ( n ) = 2.4 ( 0.3 )^(n)$] The constant #math.equation(block: false, alt: "a")[$a$] is the #math.equation(block: false, alt: "y")[$y$]-intercept of the graph because #math.equation(block: true, alt: "f open parenthesis 0 close parenthesis equals a times b to the power 0 equals a times 1 equals a")[$f ( 0 ) = a ⋅ b^(0) = a ⋅ 1 = a$] For the examples above, we find that the #math.equation(block: false, alt: "y")[$y$]-intercepts are #math.equation(block: true, alt: "f open parenthesis 0 close parenthesis, equals 5 to the power 0 equals 1 ,; P open parenthesis 0 close parenthesis, equals 250 open parenthesis 1.7 close parenthesis to the power 0 equals 250 , and; g open parenthesis 0 close parenthesis, equals 2.4 open parenthesis 0.3 close parenthesis to the power 0 equals 2.4")[$f ( 0 ) & = 5^(0) = 1 "," \ P ( 0 ) & = 250 ( 1.7 )^(0) = 250 ", and" \ g ( 0 ) & = 2.4 ( 0.3 )^(0) = 2.4$] The positive constant #math.equation(block: false, alt: "b")[$b$] is called the #strong[base] of the exponential function. Which of the following is an exponential function? \_\_\_\_\_ #solutionbox[ #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals 3 open parenthesis 4 close parenthesis to the power x")[$f ( x ) = 3 ( 4 )^(x)$] ] Which of the following is an exponential function? + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 3 x to the power 4")[$f ( x ) = 3 x^(4)$] + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 3 open parenthesis 4 close parenthesis to the power x")[$f ( x ) = 3 ( 4 )^(x)$] + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 2 x to the power 3 / 4")[$f ( x ) = 2 x^(3 / 4)$] + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the fraction 4 over x cubed")[$f ( x ) = display(frac(4, x^(3)))$] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ - We do not allow #math.equation(block: false, alt: "b")[$b$] to be negative, because if #math.equation(block: false, alt: "b less than 0")[$b < 0$], then #math.equation(block: false, alt: "b to the power x")[$b^(x)$] is not a real number for some values of #math.equation(block: false, alt: "x")[$x$]. For example, if #math.equation(block: false, alt: "b equals minus 4")[$b = − 4$] and #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals open parenthesis minus 4 close parenthesis to the power x")[$" " f ( x ) = ( − 4 )^(x)$], then #math.equation(block: false, alt: "f open parenthesis 1 / 2 close parenthesis equals open parenthesis minus 4 close parenthesis to the power 1 / 2")[$" " f ( 1 / 2 ) = ( − 4 )^(1 / 2) " "$] is an imaginary number. - We also exclude #math.equation(block: false, alt: "b equals 1")[$b = 1$] as a base because #math.equation(block: false, alt: "1 to the power x equals 1")[$1^(x) = 1$] for all values of #math.equation(block: false, alt: "x")[$x$]; hence the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 1 to the power x")[$" " f ( x ) = 1^(x)$] is actually the constant function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 1")[$" " f ( x ) = 1$]. ] === Graphs of Exponential Functions The graphs of exponential functions have two characteristic shapes, depending on whether the base, #math.equation(block: false, alt: "b")[$b$], is greater than #math.equation(block: false, alt: "1")[$1$] or less than #math.equation(block: false, alt: "1")[$1$]. As typical examples, consider the graphs of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 2 to the power x")[$f ( x ) = 2^(x)$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals open parenthesis the fraction 1 over 2 close parenthesis to the power x")[$g ( x ) = attach(( display(frac(1, 2)) ), t: x)$] shown below. Some values for #math.equation(block: false, alt: "f")[$f$] and #math.equation(block: false, alt: "g")[$g$] are recorded in the tables. The two characteristic shapes are one family: y = bˣ with the base on a slider. For b \> 1 the function is increasing — and the larger the base, the faster it climbs; try b = 2 against b = 3. Slide b below 1 and the graph flips to the decreasing shape: the smaller the base, the steeper the decay. Right at b = 1 the “growth” degenerates into the horizontal line y = 1. Two things never change as you drag: every graph passes through (0, 1) (the dashed line marks height 1), because b⁰ = 1 for any base, and the graph never touches the x-axis — a positive base to any power stays positive.{"functions":\[{"color":"\#1f4e79","expression":"b^x","expression2":"","id":"exponential","inequality":"lt","lineDash":\[\],"lineWidth":2.5,"mode":"cartesian","tMax":10,"tMin":-10,"variable":"x","visible":true},{"color":"\#6a6a6a","expression":"1","expression2":"","id":"unit-height","inequality":"lt","lineDash":\[6,4\],"lineWidth":1.5,"mode":"cartesian","tMax":10,"tMin":-10,"variable":"x","visible":true}\],"grid":{"majorSpacing":1,"showAxes":true,"showLabels":true},"parameters":\[{"id":"p-b","label":"Base b","max":4,"min":0.2,"name":"b","step":0.05,"value":2}\],"title":"y = b\\u02e3 as the base b varies","version":1,"viewport":{"centerX":0,"centerY":2.5,"scale":75}}#figure(table( columns: 2, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]]), [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "the fraction 1 over 8")[$frac(1, 8)$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$]], [#math.equation(block: false, alt: "minus 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: "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: "3")[$3$]], [#math.equation(block: false, alt: "8")[$8$]], )) #figure(table( columns: 2, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "g open parenthesis x close parenthesis")[$g ( x )$]]), [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "8")[$8$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "2")[$2$]], [#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: "the fraction 1 over 4")[$frac(1, 4)$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "the fraction 1 over 8")[$frac(1, 8)$]], )) #figure(figph[increasing and decreasing exponential graphs], alt: "increasing and decreasing exponential graphs", caption: none) Notice that #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 2 to the power x")[$" " f ( x ) = 2^(x) " "$] is an increasing function and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals open parenthesis the fraction 1 over 2 close parenthesis to the power x")[$" " g ( x ) = attach(( display(frac(1, 2)) ), t: x) " "$] is a decreasing function. Both are concave up. In general, exponential functions have the following properties. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Properties of Exponential Functions, #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals a b to the power x")[$f ( x ) = a b^(x)$], #math.equation(block: false, alt: "a greater than 0")[$a > 0$]] + Domain: all real numbers. + Range: all positive numbers. + If #math.equation(block: false, alt: "b greater than 1")[$b > 1$], the function is increasing and concave up; #linebreak() if #math.equation(block: false, alt: "0 less than b less than 1")[$0 < b < 1$], the function is decreasing and concave up. + The #math.equation(block: false, alt: "y")[$y$]-intercept is #math.equation(block: false, alt: "open parenthesis 0 , a close parenthesis")[$( 0 , a )$]. There is no #math.equation(block: false, alt: "x")[$x$]-intercept. ] In the table for #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$], you can see that as the #math.equation(block: false, alt: "x")[$x$]-values decrease toward negative infinity, the corresponding #math.equation(block: false, alt: "y")[$y$]-values decrease toward zero. As a result, the graph of #math.equation(block: false, alt: "f")[$f$] decreases toward the #math.equation(block: false, alt: "x")[$x$]-axis as we move to the left. Thus, the negative #math.equation(block: false, alt: "x")[$x$]-axis is a #strong[horizontal asymptote] for exponential functions with #math.equation(block: false, alt: "b greater than 1")[$b > 1$], as shown in figure (a). For exponential functions with #math.equation(block: false, alt: "0 less than b less than 1")[$0 < b < 1$], the positive #math.equation(block: false, alt: "x")[$x$]-axis is an asymptote, as illustrated in figure (b). (See Some Basic Functions to review asymptotes.) Which statement is true? \_\_\_\_\_ #solutionbox[ The outputs of an exponential function cannot be negative. ] Which statement is true? + An exponential function is not defined for negative inputs. + The outputs of an exponential function cannot be negative. + The #math.equation(block: false, alt: "y")[$y$]-intercept of the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 2 open parenthesis 3 close parenthesis to the power x")[$f ( x ) = 2 ( 3 )^(x)$] is #math.equation(block: false, alt: "open parenthesis 0 , 6 close parenthesis")[$( 0 , 6 )$]. + The function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 16 open parenthesis 0.5 close parenthesis to the power x")[$f ( x ) = 16 ( 0.5 )^(x)$] decreases by #math.equation(block: false, alt: "8")[$8$] each time we increase #math.equation(block: false, alt: "x")[$x$] by #math.equation(block: false, alt: "1")[$1$]. In Example, we compare two increasing exponential functions. The larger the value of the base, #math.equation(block: false, alt: "b")[$b$], the faster the function grows. In this example, both functions have #math.equation(block: false, alt: "a equals 1")[$a = 1$]. #examplebox("Example 1")[][ Compare the graphs of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 3 to the power x")[$f ( x ) = 3^(x)$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals 4 to the power x")[$g ( x ) = 4^(x)$]. #solutionbox[ We evaluate each function for several convenient values, as shown in the table. Then we plot the points for each function and connect them with smooth curves. For positive #math.equation(block: false, alt: "x")[$x$]-values, #math.equation(block: false, alt: "g open parenthesis x close parenthesis")[$g ( x )$] is always larger than #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$], and is increasing more rapidly. In the figure, we can see that #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals 4 to the power x")[$g ( x ) = 4^(x)$] climbs more rapidly than #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 3 to the power x")[$f ( x ) = 3^(x)$]. Both graphs cross the #math.equation(block: false, alt: "y")[$y$]-axis at (0, 1). #figure(table( columns: 3, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]], [#math.equation(block: false, alt: "g open parenthesis x close parenthesis")[$g ( x )$]]), [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#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: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "the fraction 1 over 3")[$display(frac(1, 3))$]], [#math.equation(block: false, alt: "the fraction 1 over 4")[$display(frac(1, 4))$]], [#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: "1")[$1$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "9")[$9$]], [#math.equation(block: false, alt: "16")[$16$]], )) #figure(figph[two increasing exponential functions], alt: "two increasing exponential functions", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ For decreasing exponential functions, those with bases between #math.equation(block: false, alt: "0")[$0$] and #math.equation(block: false, alt: "1")[$1$], the smaller the base, the more steeply the graph decreases. For example, compare the graphs of #math.equation(block: false, alt: "p open parenthesis x close parenthesis equals 0.8 to the power x")[$" " p ( x ) = 0.8^(x) " "$] and #math.equation(block: false, alt: "q open parenthesis x close parenthesis equals 0.5 to the power x")[$" " q ( x ) = 0.5^(x) " "$] shown in the figure at right. #figure(figph[two decreasing exponential functions], alt: "two decreasing exponential functions", caption: none) ] + State the ranges of the functions #math.equation(block: false, alt: "f")[$f$] and #math.equation(block: false, alt: "g")[$g$] from the previous Example on the domain #math.equation(block: false, alt: "open bracket minus 2 , 2 close bracket")[$[ − 2 , 2 ]$]. #linebreak() #math.equation(block: false, alt: "f :")[$f :$] \_\_\_\_\_ #linebreak() #math.equation(block: false, alt: "g :")[$g :$] \_\_\_\_\_ + State the ranges of the functions #math.equation(block: false, alt: "p")[$p$] and #math.equation(block: false, alt: "q")[$q$] shown in the Note above on the domain #math.equation(block: false, alt: "open bracket minus 2 , 2 close bracket")[$[ − 2 , 2 ]$]. Round your answers to two decimal places. #linebreak() #math.equation(block: false, alt: "p :")[$p :$] \_\_\_\_\_ #linebreak() #math.equation(block: false, alt: "q :")[$q :$] \_\_\_\_\_ #solutionbox[ + #math.equation(block: false, alt: "f : open bracket the fraction 1 over 9 , 9 close bracket")[$f : [ display(frac(1, 9)) , 9 ]$]; #math.equation(block: false, alt: "g : open bracket the fraction 1 over 16 , 16 close bracket")[$" " " " " " " " g : [ display(frac(1, 16)) , 16 ]$] + #math.equation(block: false, alt: "p : open bracket 0.64 , 1.56 close bracket")[$p : [ 0.64 , 1.56 ]$]; #math.equation(block: false, alt: "q : open bracket 0.25 , 4 close bracket")[$" " " " " " " " q : [ 0.25 , 4 ]$] ] + State the ranges of the functions #math.equation(block: false, alt: "f")[$f$] and #math.equation(block: false, alt: "g")[$g$] from the previous Example on the domain #math.equation(block: false, alt: "open bracket minus 2 , 2 close bracket")[$[ − 2 , 2 ]$]. + State the ranges of the functions #math.equation(block: false, alt: "p")[$p$] and #math.equation(block: false, alt: "q")[$q$] shown in the Note above on the domain #math.equation(block: false, alt: "open bracket minus 2 , 2 close bracket")[$[ − 2 , 2 ]$]. Round your answers to two decimal places. #solutionbox[ + #math.equation(block: false, alt: "f : open bracket the fraction 1 over 9 , 9 close bracket")[$f : [ display(frac(1, 9)) , 9 ]$]; #math.equation(block: false, alt: "g : open bracket the fraction 1 over 16 , 16 close bracket")[$" " " " " " " " g : [ display(frac(1, 16)) , 16 ]$] + #math.equation(block: false, alt: "p : open bracket 0.64 , 1.56 close bracket")[$p : [ 0.64 , 1.56 ]$]; #math.equation(block: false, alt: "q : open bracket 0.25 , 4 close bracket")[$" " " " " " " " q : [ 0.25 , 4 ]$] ] === Transformations of Exponential Functions In Modeling with Functions, we considered transformations of the basic graphs. For instance, the graphs of the functions #math.equation(block: false, alt: "y equals x squared minus 4")[$y = x^(2) − 4$] and #math.equation(block: false, alt: "y equals open parenthesis x minus 4 close parenthesis squared")[$y = ( x − 4 )^(2)$] are shifts of the basic parabola, #math.equation(block: false, alt: "y equals x squared")[$y = x^(2)$]. In a similar way, we can shift or stretch the graph of an exponential function while the basic shape is preserved. #examplebox("Example 2")[][ Use your calculator to graph the following functions. Describe how these graphs compare with the graph of #math.equation(block: false, alt: "h open parenthesis x close parenthesis equals 2 to the power x")[$h ( x ) = 2^(x)$]. + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 2 to the power x plus 3")[$f ( x ) = 2^(x) + 3$] + #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals 2 to the power x plus 3")[$g ( x ) = 2^(x + 3)$] #solutionbox[ Enter the formulas for the three functions as shown below. Note the parentheses around the exponent in the keying sequence for #math.equation(block: false, alt: "Y sub 3 equals g open parenthesis x close parenthesis .")[$Y_(3) = g ( x ) .$] #math.equation(block: false, alt: "Y sub 1 equals 2")[$#h(2em) Y_(1) = 2$] ^ #math.equation(block: false, alt: "X")[$X$] #math.equation(block: false, alt: "Y sub 2 equals 2")[$#h(2em) Y_(2) = 2$] ^ #math.equation(block: false, alt: "X")[$X$] + 3 #math.equation(block: false, alt: "Y sub 3 equals 2")[$#h(2em) Y_(3) = 2$] ^ ( #math.equation(block: false, alt: "X")[$X$] + 3 ) The graphs of #math.equation(block: false, alt: "h open parenthesis x close parenthesis equals 2 to the power x")[$h ( x ) = 2^(x)$], #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 2 to the power x plus 3")[$f ( x ) = 2^(x) + 3$], and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals 2 to the power x plus 3")[$g ( x ) = 2^(x + 3)$] in the standard window are shown below. #figure(figph[calculator graphs of transformations of exponential functions], alt: "calculator graphs of transformations of exponential functions", caption: none) + The graph of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 2 to the power x plus 3")[$f ( x ) = 2^(x) + 3$], shown in figure (b), has the same basic shape as that of #math.equation(block: false, alt: "h open parenthesis x close parenthesis equals 2 to the power x")[$h ( x ) = 2^(x)$], but it has a horizontal asymptote at #math.equation(block: false, alt: "y equals 3")[$y = 3$] instead of at #math.equation(block: false, alt: "y equals 0")[$y = 0$] (the #math.equation(block: false, alt: "x")[$x$]-axis). In fact, #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals h open parenthesis x close parenthesis plus 3")[$f ( x ) = h ( x ) + 3$], so the graph of #math.equation(block: false, alt: "f")[$f$] is a vertical translation of the graph of #math.equation(block: false, alt: "h")[$h$] by #math.equation(block: false, alt: "3")[$3$] units. If every point on the graph of #math.equation(block: false, alt: "h open parenthesis x close parenthesis equals 2 to the power x")[$h ( x ) = 2^(x)$] is moved #math.equation(block: false, alt: "3")[$3$] units upward, the result is the graph of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 2 to the power x plus 3")[$f ( x ) = 2^(x) + 3$]. + First note that #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals 2 to the power x plus 3 equals h open parenthesis x plus 3 close parenthesis")[$g ( x ) = 2^(x) + 3 = h ( x + 3 )$]. In fact, the graph of #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals 2 to the power x plus 3")[$g ( x ) = 2^(x + 3)$] shown in figure (c) has the same basic shape as #math.equation(block: false, alt: "h open parenthesis x close parenthesis equals 2 to the power x")[$h ( x ) = 2^(x)$] but has been translated #math.equation(block: false, alt: "3")[$3$] units to the left. ] ] Which function translates the graph of #math.equation(block: false, alt: "y equals 8 to the power x")[$" " y = 8^(x) " "$] two units to the right? \_\_\_\_\_ #solutionbox[ #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals 8 to the power x minus 2")[$f ( x ) = 8^(x − 2)$] ] Which function translates the graph of #math.equation(block: false, alt: "y equals 8 to the power x")[$y = 8^(x)$] two units to the right? + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 8 to the power x minus 2")[$f ( x ) = 8^(x − 2)$] + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 8 to the power 2 x")[$f ( x ) = 8^(2 x)$] + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 2 plus open parenthesis 8 close parenthesis to the power x")[$f ( x ) = 2 + ( 8 )^(x)$] + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 8 to the power x minus 2")[$f ( x ) = 8^(x) − 2$] What about reflections? Recall that the graph of #math.equation(block: false, alt: "y equals minus f open parenthesis x close parenthesis")[$y = − f ( x )$] is the reflection about the #math.equation(block: false, alt: "x")[$x$]-axis of the graph of #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$y = f ( x )$]. The graphs of #math.equation(block: false, alt: "y equals 2 to the power x")[$y = 2^(x)$] and #math.equation(block: false, alt: "y equals minus 2 to the power x")[$y = − 2^(x)$] are shown at left below. #figure(figph[vertical reflection of an exponential function], alt: "vertical reflection of an exponential function", caption: none) #figure(figph[horizontal reflection of an exponential function], alt: "horizontal reflection of an exponential function", caption: none) You may have also noticed a relationship between the graphs of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 2 to the power x")[$" " f ( x ) = 2^(x) " "$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals open parenthesis the fraction 1 over 2 close parenthesis to the power x")[$" " g ( x ) = attach(( display(frac(1, 2)) ), t: x)$], which are shown at right above. The graph of #math.equation(block: false, alt: "g")[$g$] is the reflection of the graph of #math.equation(block: false, alt: "f")[$f$] about the #math.equation(block: false, alt: "y")[$y$]-axis. We can see why this is true by writing the formula for #math.equation(block: false, alt: "g open parenthesis x close parenthesis")[$g ( x )$] in another way: #math.equation(block: true, alt: "g open parenthesis x close parenthesis equals open parenthesis the fraction 1 over 2 close parenthesis to the power x equals open parenthesis 2 to the power minus 1 close parenthesis to the power x equals 2 to the power minus x")[$g ( x ) = attach(( frac(1, 2) ), t: x) = attach(( 2^(− 1) ), t: x) = 2^(− x)$] We see that #math.equation(block: false, alt: "g open parenthesis x close parenthesis")[$g ( x )$] is the same function as #math.equation(block: false, alt: "f open parenthesis minus x close parenthesis")[$f ( − x )$]. Replacing #math.equation(block: false, alt: "x")[$x$] by #math.equation(block: false, alt: "minus x")[$− x$] in the formula for a function switches every point #math.equation(block: false, alt: "open parenthesis p , q close parenthesis")[$( p , q )$] on the graph with the point #math.equation(block: false, alt: "open parenthesis minus p , q close parenthesis")[$( − p , q )$] and thus reflects the graph about the #math.equation(block: false, alt: "y")[$y$]-axis. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Reflections of Graphs] + The graph of #math.equation(block: false, alt: "y equals minus f open parenthesis x close parenthesis")[$y = − f ( x )$] is the reflection of the graph of #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$y = f ( x )$] about the #math.equation(block: false, alt: "x")[$x$]-axis. + The graph of #math.equation(block: false, alt: "y equals f open parenthesis minus x close parenthesis")[$y = f ( − x )$] is the reflection of the graph of #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$y = f ( x )$] about the #math.equation(block: false, alt: "y")[$y$]-axis. ] Which of the functions below have the same graph? Explain why. \_\_\_\_\_ + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals open parenthesis the fraction 1 over 4 close parenthesis to the power x")[$f ( x ) = attach(( display(frac(1, 4)) ), t: x)$] + #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals minus 4 to the power x")[$g ( x ) = − 4^(x)$] + #math.equation(block: false, alt: "h open parenthesis x close parenthesis equals 4 to the power minus x")[$h ( x ) = 4^(− x)$] #solutionbox[ (a) and (c) are the same function. ] Which of the functions below have the same graph? Explain why. + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals open parenthesis the fraction 1 over 4 close parenthesis to the power x")[$f ( x ) = attach(( display(frac(1, 4)) ), t: x)$] + #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals minus 4 to the power x")[$g ( x ) = − 4^(x)$] + #math.equation(block: false, alt: "h open parenthesis x close parenthesis equals 4 to the power minus x")[$h ( x ) = 4^(− x)$] #solutionbox[ (a) and (c) are the same function. ] How are the graphs of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals b to the power x")[$f ( x ) = b^(x)$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals open parenthesis the fraction 1 over b close parenthesis to the power x")[$g ( x ) = attach(( display(frac(1, b)) ), t: x)$] related? \_\_\_\_\_ How are the graphs of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals b to the power x")[$" " f ( x ) = b^(x) " "$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals open parenthesis the fraction 1 over b close parenthesis to the power x")[$" " g ( x ) = attach(( display(frac(1, b)) ), t: x) " "$] related? === Comparing Exponential and Power Functions Exponential functions are not the same as the power functions we studied in Power Functions. Although both involve expressions with exponents, it is the location of the variable that makes the difference. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Power Functions vs Exponential Functions] #figure(table( columns: 3, align: left, inset: 6pt, table.header([$#hide($G e n e r a l f o r m u l a a n d m$)$], [#strong[Power Functions]], [#strong[Exponential Functions]]), [#emph[General formula]], [#math.equation(block: false, alt: "h open parenthesis x close parenthesis equals k x to the power p")[$h ( x ) = k x^(p)$]], [#math.equation(block: false, alt: "f open parenthesis x close parenthesis equals a b to the power x")[$f ( x ) = a b^(x)$]], [#emph[Description]], [variable base and constant exponent], [constant base and variable exponent], [#emph[Example]], [#math.equation(block: false, alt: "h open parenthesis x close parenthesis equals 2 x cubed")[$h ( x ) = 2 x^(3)$]], [#math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 2 open parenthesis 3 to the power x close parenthesis")[$f ( x ) = 2 ( 3^(x) )$]], )) ] These two families of functions have very different properties, as well. #examplebox("Example 3")[][ Compare the power function #math.equation(block: false, alt: "h open parenthesis x close parenthesis equals 2 x cubed")[$h ( x ) = 2 x^(3)$] and the exponential function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 2 open parenthesis 3 to the power x close parenthesis")[$f ( x ) = 2 ( 3^(x) )$]. #solutionbox[ First, compare the values for these two functions shown in the table. The scaling exponent for #math.equation(block: false, alt: "h open parenthesis x close parenthesis")[$h ( x )$] is #math.equation(block: false, alt: "3")[$3$], so that when #math.equation(block: false, alt: "x")[$x$] doubles, say, from #math.equation(block: false, alt: "1")[$1$] to #math.equation(block: false, alt: "2")[$2$], the output is multiplied by #math.equation(block: false, alt: "2 cubed")[$2^(3)$], or #math.equation(block: false, alt: "8")[$8$]. On the other hand, we can tell that #math.equation(block: false, alt: "f")[$f$] is exponential because its values increase by a factor of #math.equation(block: false, alt: "3")[$3$] for each unit increase in #math.equation(block: false, alt: "x")[$x$]. (To see this, divide any function value by the previous one.) #figure(table( columns: 3, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "h open parenthesis x close parenthesis equals 2 x cubed")[$h ( x ) = 2 x^(3)$]], [#math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 2 open parenthesis 3 to the power x close parenthesis")[$f ( x ) = 2 ( 3^(x) )$]]), [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "minus 54")[$− 54$]], [#math.equation(block: false, alt: "the fraction 2 over 27")[$display(frac(2, 27))$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "minus 16")[$− 16$]], [#math.equation(block: false, alt: "the fraction 1 over 4")[$display(frac(1, 4))$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "the fraction 2 over 3")[$display(frac(2, 3))$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "6")[$6$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "16")[$16$]], [#math.equation(block: false, alt: "18")[$18$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "54")[$54$]], [#math.equation(block: false, alt: "54")[$54$]], )) As you would expect, the graphs of the two functions are also quite different. For starters, note that the power function goes through the origin, while the exponential function has #math.equation(block: false, alt: "y")[$y$]-intercept #math.equation(block: false, alt: "open parenthesis 0 , 2 close parenthesis")[$( 0 , 2 )$]as shown at left below. #figure(figph[power function vs exponential from -2 to 3], alt: "power function vs exponential from -2 to 3", caption: none) #figure(figph[power function vs exponential from 0 to 6], alt: "power function vs exponential from 0 to 6", caption: none) From the table, we see that #math.equation(block: false, alt: "h open parenthesis 3 close parenthesis equals f open parenthesis 3 close parenthesis equals 54")[$h ( 3 ) = f ( 3 ) = 54$], so the two graphs intersect at #math.equation(block: false, alt: "x equals 3")[$x = 3$]. (They also intersect at approximately #math.equation(block: false, alt: "x equals 2.48")[$x = 2.48$].) However, if you compare the values of #math.equation(block: false, alt: "h open parenthesis x close parenthesis equals 2 x cubed")[$h ( x ) = 2 x^(3)$] and #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 2 open parenthesis 3 to the power x close parenthesis")[$f ( x ) = 2 ( 3^(x) )$] for larger values of #math.equation(block: false, alt: "x")[$x$], you will see that eventually the exponential function overtakes the power function, as shown at right above. ] ] #figure(figph[power function vs exponential from 0 to 50], alt: "power function vs exponential from 0 to 50", caption: none) The relationship in Example holds true for all increasing power and exponential functions: For large enough values of #math.equation(block: false, alt: "x")[$x$], the exponential function will always be greater than the power function, regardless of the parameters in the functions. The figure at left shows the graphs of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x to the power 6")[$f ( x ) = x^(6)$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals 1.8 to the power x")[$g ( x ) = 1.8^(x)$]. At first, #math.equation(block: false, alt: "f open parenthesis x close parenthesis greater than g open parenthesis x close parenthesis")[$f ( x ) > g ( x )$], but at around #math.equation(block: false, alt: "x equals 37")[$x = 37$], #math.equation(block: false, alt: "g open parenthesis x close parenthesis")[$g ( x )$] overtakes #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$], and #math.equation(block: false, alt: "g open parenthesis x close parenthesis greater than f open parenthesis x close parenthesis")[$g ( x ) > f ( x )$] for all #math.equation(block: false, alt: "x greater than 37")[$x > 37$]. Which function grows fastest in the long run? \_\_\_\_\_ #solutionbox[ #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals 2 open parenthesis 3 to the power x close parenthesis")[$f ( x ) = 2 ( 3^(x) )$] ] Which function grows fastest in the long run? + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 5 x squared")[$f ( x ) = 5 x^(2)$] + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 3 x to the power 5")[$f ( x ) = 3 x^(5)$] + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 5 open parenthesis 2 to the power x close parenthesis")[$f ( x ) = 5 ( 2^(x) )$] + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 2 open parenthesis 3 to the power x close parenthesis")[$f ( x ) = 2 ( 3^(x) )$] Which of the following functions are exponential functions, and which are power functions? + #math.equation(block: false, alt: "F open parenthesis x close parenthesis equals 1.5 to the power x")[$F ( x ) = 1.5^(x)$] \_\_\_\_\_ + #math.equation(block: false, alt: "G open parenthesis x close parenthesis equals 3 x to the power 1.5")[$G ( x ) = 3 x^(1.5)$] \_\_\_\_\_ + #math.equation(block: false, alt: "H open parenthesis x close parenthesis equals 3 to the power 1.5 x")[$H ( x ) = 3^(1.5 x)$] \_\_\_\_\_ + #math.equation(block: false, alt: "K open parenthesis x close parenthesis equals open parenthesis 3 x close parenthesis to the power 1.5")[$K ( x ) = ( 3 x )^(1.5)$] \_\_\_\_\_ #solutionbox[ Exponential: (a) and (c); power: (b) and (d) ] Which of the following functions are exponential functions, and which are power functions? + #math.equation(block: false, alt: "F open parenthesis x close parenthesis equals 1.5 to the power x")[$F ( x ) = 1.5^(x)$] + #math.equation(block: false, alt: "G open parenthesis x close parenthesis equals 3 x to the power 1.5")[$G ( x ) = 3 x^(1.5)$] + #math.equation(block: false, alt: "H open parenthesis x close parenthesis equals 3 to the power 1.5 x")[$H ( x ) = 3^(1.5 x)$] + #math.equation(block: false, alt: "K open parenthesis x close parenthesis equals open parenthesis 3 x close parenthesis to the power 1.5")[$K ( x ) = ( 3 x )^(1.5)$] #solutionbox[ Exponential: (a) and (c); power: (b) and (d) ] Discuss the differences between a power function and an exponential function. \_\_\_\_\_ Discuss the differences between a power function and an exponential function. === Exponential Equations An #strong[exponential equation] is one in which the variable is part of an exponent. For example, the equation #math.equation(block: true, alt: "3 to the power x equals 81")[$3^(x) = 81$] is exponential. Many exponential equations can be solved by writing both sides of the equation as powers with the same base. To solve the equation above, we write #math.equation(block: true, alt: "3 to the power x equals 3 to the power 4")[$3^(x) = 3^(4)$] which is true if and only if #math.equation(block: false, alt: "x equals 4")[$x = 4$]. In general, if two equivalent powers have the same base, then their exponents must be equal also, as long as the base is not #math.equation(block: false, alt: "0")[$0$] or #math.equation(block: false, alt: "plus or minus 1")[$± 1$]. Sometimes the laws of exponents can be used to express both sides of an equation as single powers of a common base. #examplebox("Example 4")[][ Solve the following equations. + #math.equation(block: false, alt: "3 to the power x minus 2 equals 9 cubed")[$3^(x − 2) = 9^(3)$] + #math.equation(block: false, alt: "27 times 3 to the power minus 2 x equals 9 to the power x plus 1")[$27 ⋅ 3^(− 2 x) = 9^(x + 1)$] #solutionbox[ + Using the fact that #math.equation(block: false, alt: "9 equals 3 squared")[$9 = 3^(2)$], we write each side of the equation as a power of #math.equation(block: false, alt: "3")[$3$]: #math.equation(block: true, alt: "3 to the power x minus 2, equals open parenthesis 3 squared close parenthesis cubed; 3 to the power x minus 2, equals 3 to the power 6")[$3^(x − 2) & = attach(( 3^(2) ), t: 3) \ 3^(x − 2) & = 3^(6)$] Now we equate the exponents to obtain #math.equation(block: true, alt: "x minus 2, equals 6; x, equals 8")[$x − 2 & = 6 \ x & = 8$] + We write each factor as a power of #math.equation(block: false, alt: "3")[$3$]. #math.equation(block: true, alt: "3 cubed times 3 to the power minus 2 x equals open parenthesis 3 squared close parenthesis to the power x plus 1")[$3^(3) ⋅ 3^(− 2 x) = attach(( 3^(2) ), t: x + 1)$] We use the laws of exponents to simplify each side: #math.equation(block: true, alt: "3 to the power 3 minus 2 x equals 3 to the power 2 x plus 2")[$3^(3 − 2 x) = 3^(2 x + 2)$] Now we equate the exponents to obtain #math.equation(block: true, alt: "3 minus 2 x, equals 2 x plus 2; minus 4 x equals, minus 1")[$3 − 2 x & = 2 x + 2 \ − 4 x = & − 1$] The solution is #math.equation(block: false, alt: "x equals the fraction 1 over 4")[$x = display(frac(1, 4))$]. ] ] Which is a good strategy for solving #math.equation(block: false, alt: "3 to the power x minus 2 equals 81")[$3^(x − 2) = 81$]? \_\_\_\_\_ #solutionbox[ Write the right side as a power of 3. ] Which is a good strategy for solving #math.equation(block: false, alt: "3 to the power x minus 2 equals 81")[$3^(x − 2) = 81$]? + Divide both sides by 3. + Add #math.equation(block: false, alt: "3 squared")[$3^(2)$] to both sides. + Simplify the left side. + Write the right side as a power of 3. Solve the equation #math.equation(block: false, alt: "2 to the power x plus 2 equals 128")[$2^(x + 2) = 128$]. #math.equation(block: false, alt: "x equals")[$x =$]\_\_\_\_\_ #math.equation(block: true, alt: "Write each side as a power of 2.")[$"Write each side as a power of 2."$] #math.equation(block: true, alt: "Equate exponents.")[$"Equate exponents."$] #solutionbox[ #math.equation(block: true, alt: "x equals 5")[$x = 5$] ] Solve the equation #math.equation(block: false, alt: "2 to the power x plus 2 equals 128")[$2^(x + 2) = 128$]. #solutionbox[ Write each side as a power of 2, then equate exponents to find #math.equation(block: false, alt: "x equals 5")[$x = 5$] ] Exponential equations arise frequently in the study of exponential growth. #examplebox("Example 5")[][ During the summer a population of fleas doubles in number every #math.equation(block: false, alt: "5")[$5$] days. If a population starts with #math.equation(block: false, alt: "10")[$10$] fleas, how long will it be before there are #math.equation(block: false, alt: "10 , 240")[$10 , 240$] fleas? #solutionbox[ Let #math.equation(block: false, alt: "P")[$P$] represent the number of fleas present after #math.equation(block: false, alt: "t")[$t$] days. The original population of #math.equation(block: false, alt: "10")[$10$] is multiplied by a factor of #math.equation(block: false, alt: "2")[$2$] every #math.equation(block: false, alt: "5")[$5$] days, or #math.equation(block: true, alt: "P open parenthesis t close parenthesis equals 10 times 2 to the power t / 5")[$P ( t ) = 10 ⋅ 2^(t / 5)$] We set #math.equation(block: false, alt: "P equals 10 , 240")[$P = 10 , 240$] and solve for #math.equation(block: false, alt: "t")[$t$]: #math.equation(block: true, alt: "10 , 240, equals 10 times 2 to the power t / 5, Divide both sides by 10.; 1024, equals 2 to the power t / 5, Write 1024 as a power of 2.; 2 to the power 10, equals 2 to the power t / 5")[$10 , 240 & = 10 ⋅ 2^(t / 5) & & "Divide both sides by 10." \ 1024 & = 2^(t / 5) & & "Write 1024 as a power of 2." \ 2^(10) & = 2^(t / 5)$] We equate the exponents to get #math.equation(block: false, alt: "10 equals the fraction t over 5")[$10 = display(frac(t, 5))$], or #math.equation(block: false, alt: "t equals 50")[$t = 50$]. The population will grow to #math.equation(block: false, alt: "10 , 240")[$10 , 240$] fleas in #math.equation(block: false, alt: "50")[$50$] days. ] ] During an advertising campaign in a large city, the makers of Chip-O's corn chips estimate that the number of people who have heard of Chip-O's increases by a factor of #math.equation(block: false, alt: "8")[$8$] every 4 days. + If 100 people are given trial bags of Chip-O's to start the campaign, write a function, #math.equation(block: false, alt: "N open parenthesis t close parenthesis")[$N ( t )$], for the number of people who have heard of Chip-O's after #math.equation(block: false, alt: "t")[$t$] days of advertising. #linebreak() #math.equation(block: false, alt: "N open parenthesis t close parenthesis equals")[$N ( t ) =$]\_\_\_\_\_ + Use your calculator to graph the function #math.equation(block: false, alt: "N open parenthesis t close parenthesis")[$N ( t )$] on the domain #math.equation(block: false, alt: "0 less than or equal to t less than or equal to 15")[$0 ≤ t ≤ 15$]. + How many days should the makers run the campaign in order for Chip-O's to be familiar to #math.equation(block: false, alt: "51 , 200")[$51 , 200$] people? Use algebraic methods to find your answer and verify on your graph. #linebreak() Answer: \_\_\_\_\_ days #solutionbox[ + #math.equation(block: false, alt: "N open parenthesis t close parenthesis equals 100 times 8 to the power t / 4")[$N ( t ) = 100 ⋅ 8^(t / 4)$] + A graph is below. + 12 days ] A graph for part (b): #figure(figph[GC graph], alt: "GC graph", caption: none) During an advertising campaign in a large city, the makers of Chip-O's corn chips estimate that the number of people who have heard of Chip-O's increases by a factor of #math.equation(block: false, alt: "8")[$8$] every 4 days. + If 100 people are given trial bags of Chip-O's to start the campaign, write a function, #math.equation(block: false, alt: "N open parenthesis t close parenthesis")[$N ( t )$], for the number of people who have heard of Chip-O's after #math.equation(block: false, alt: "t")[$t$] days of advertising. + Use your calculator to graph the function #math.equation(block: false, alt: "N open parenthesis t close parenthesis")[$N ( t )$] on the domain #math.equation(block: false, alt: "0 less than or equal to t less than or equal to 15")[$0 ≤ t ≤ 15$]. + How many days should the makers run the campaign in order for Chip-O's to be familiar to #math.equation(block: false, alt: "51 , 200")[$51 , 200$] people? Use algebraic methods to find your answer and verify on your graph. #solutionbox[ + #math.equation(block: false, alt: "N open parenthesis t close parenthesis equals 100 times 8 to the power t / 4")[$N ( t ) = 100 ⋅ 8^(t / 4)$] + #figure(figph[GC graph], alt: "GC graph", caption: none) + 12 days ] Suppose #math.equation(block: false, alt: "g")[$g$] is an exponential function, with #math.equation(block: false, alt: "g open parenthesis 0 close parenthesis equals 48")[$" " g ( 0 ) = 48 " "$] and #math.equation(block: false, alt: "g open parenthesis 1 close parenthesis equals 36")[$" " g ( 1 ) = 36$]. What is #math.equation(block: false, alt: "g open parenthesis 2 close parenthesis")[$g ( 2 )$]? \_\_\_\_\_ #solutionbox[ 27 ] Suppose #math.equation(block: false, alt: "g")[$g$] is an exponential function, with #math.equation(block: false, alt: "g open parenthesis 0 close parenthesis equals 48")[$" " g ( 0 ) = 48 " "$] and #math.equation(block: false, alt: "g open parenthesis 1 close parenthesis equals 36")[$" " g ( 1 ) = 36$]. What is #math.equation(block: false, alt: "g open parenthesis 2 close parenthesis")[$g ( 2 )$]? + #math.equation(block: false, alt: "24")[$24$] + #math.equation(block: false, alt: "27")[$27$] + #math.equation(block: false, alt: "12")[$12$] + #math.equation(block: false, alt: "18")[$18$] #notebox("Technology", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Graphical Solution of Exponential Equations] It is not always so easy to express both sides of the equation as powers of the same base. In the following sections, we will develop more general methods for finding exact solutions to exponential equations. But we can use a graphing utility to obtain approximate solutions. #examplebox("Example 6")[][ Use the graph of #math.equation(block: false, alt: "y equals 2 to the power x")[$y = 2^(x)$] to find an approximate solution to the equation #math.equation(block: false, alt: "2 to the power x equals 5")[$2^(x) = 5$] accurate to the nearest hundredth. #solutionbox[ Enter #math.equation(block: false, alt: "Y sub 1 equals 2")[$Y_(1) = 2$] ^ X and use the standard graphing window (ZOOM 6) to obtain the graph shown in figure (a). We are looking for a point on this graph with #math.equation(block: false, alt: "y")[$y$]-coordinate #math.equation(block: false, alt: "5")[$5$]. Using the TRACE feature, we see that the #math.equation(block: false, alt: "y")[$y$]-coordinates are too small when #math.equation(block: false, alt: "x less than 2.1")[$x < 2.1$] and too large when #math.equation(block: false, alt: "x greater than 2.4")[$x > 2.4$]. The solution we want lies somewhere between #math.equation(block: false, alt: "x equals 2.1")[$x = 2.1$] and #math.equation(block: false, alt: "x equals 2.4")[$x = 2.4$], but this approximation is not accurate enough. #figure(figph[two calculator displays of exponential function with point], alt: "two calculator displays of exponential function with point", caption: none) To improve our approximation, we will use the #strong[intersect] feature. Set #math.equation(block: false, alt: "Y sub 2 equals 5")[$Y_(2) = 5$] and press GRAPH. The #math.equation(block: false, alt: "x")[$x$]-coordinate of the intersection point of the two graphs is the solution of the equation #math.equation(block: false, alt: "2 to the power x equals 5")[$2^(x) = 5$] Activating the #strong[intersect] command results in figure (b), and we see that, to the nearest hundredth, the solution is #math.equation(block: false, alt: "2.32")[$2.32$]. We can verify that our estimate is reasonable by substituting into the equation: #math.equation(block: true, alt: "2 to the power 2.32 equals ? 5")[$2^(2.32) limits(=)^(?) 5$] We enter 2 ^ 2.32 ENTER to get #math.equation(block: false, alt: "4.993322196")[$4.993322196$]. This number is not equal to #math.equation(block: false, alt: "5")[$5$], but it is close, so we believe that #math.equation(block: false, alt: "x equals 2.32")[$x = 2.32$] is a reasonable approximation to the solution of the equation #math.equation(block: false, alt: "2 to the power x equals 5")[$2^(x) = 5$]. ] ] ] Use the graph of #math.equation(block: false, alt: "y equals 5 to the power x")[$y = 5^(x)$] to find an approximate solution to #math.equation(block: false, alt: "5 to the power x equals 285")[$5^(x) = 285$], accurate to two decimal places. Answer: #math.equation(block: false, alt: "x approximately equals")[$x ≈$]\_\_\_\_\_ #solutionbox[ The point on the graph where #math.equation(block: false, alt: "y equals 285")[$y = 285$] has #math.equation(block: false, alt: "x approximately equals 3.51")[$x ≈ 3.51$] ] Use the graph of #math.equation(block: false, alt: "y equals 5 to the power x")[$y = 5^(x)$] to find an approximate solution to #math.equation(block: false, alt: "5 to the power x equals 285")[$5^(x) = 285$], accurate to two decimal places. #solutionbox[ The point on the graph where #math.equation(block: false, alt: "y equals 285")[$y = 285$] has #math.equation(block: false, alt: "x approximately equals 3.51")[$x ≈ 3.51$] ] Give an example of an exponential equation, and describe how to solve it. \_\_\_\_\_ Give an example of an exponential equation, and describe how to solve it. === Section Summary ==== Vocabulary Look up the definitions of new terms in the Glossary. - Exponential function - Base - Exponential equation ==== CONCEPTS + An exponential function has the form #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals a b to the power x , where b greater than 0 and b not equal to 1 , a not equal to 0")[$f ( x ) = a b^(x) ", where " " " b > 0 " " " " " and " " " " " b ≠ 1 , " " a ≠ 0$] + Quantities that increase or decrease by a constant percent in each time period grow or decay exponentially. + #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Properties of Exponential Functions #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals a b to the power x , a greater than 0")[$f ( x ) = a b^(x) , " " " " a > 0$]] + Domain: all real numbers. + Range: all positive numbers. + If #math.equation(block: false, alt: "b greater than 1")[$b > 1$], the function is increasing and concave up; if #math.equation(block: false, alt: "0 less than b less than 1")[$0 < b < 1$], the function is decreasing and concave up. + The #math.equation(block: false, alt: "y")[$y$]-intercept is #math.equation(block: false, alt: "open parenthesis 0 , a close parenthesis")[$( 0 , a )$]. There is no #math.equation(block: false, alt: "x")[$x$]-intercept. ] + The graphs of exponential functions can be transformed by shifts, stretches, and reflections. + #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Reflections of Graphs] + The graph of #math.equation(block: false, alt: "y equals minus f open parenthesis x close parenthesis")[$y = − f ( x )$] is the reflection of the graph of #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$y = f ( x )$] about the #math.equation(block: false, alt: "x")[$x$]-axis. + The graph of #math.equation(block: false, alt: "y equals f open parenthesis minus x close parenthesis")[$y = f ( − x )$] is the reflection of the graph of #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$y = f ( x )$] about the #math.equation(block: false, alt: "y")[$y$]-axis. ] + Exponential functions #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals a b to the power x")[$f ( x ) = a b^(x)$] have different properties than power functions #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals k x to the power p")[$f ( x ) = k x^(p)$]. + We can solve some #strong[exponential equations] by writing both sides with the same #strong[base] and equating the exponents. + We can use graphs to find approximate solutions to exponential equations. ==== STUDY QUESTIONS + Give the general form for an exponential function. What restrictions do we place on the base of the function? + Explain why the output of an exponential function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals b to the power x")[$f ( x ) = b^(x)$] is always positive, even if #math.equation(block: false, alt: "x")[$x$] is negative. + How are the graphs of the functions #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals b to the power x")[$f ( x ) = b^(x)$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals open parenthesis the fraction 1 over b close parenthesis to the power x")[$g ( x ) = attach(( display(frac(1, b)) ), t: x)$] related? + How is an exponential function different from a power function? + Delbert says that #math.equation(block: false, alt: "8 open parenthesis the fraction 1 over 2 close parenthesis to the power x")[$8 attach(( display(frac(1, 2)) ), t: x)$] is equivalent to #math.equation(block: false, alt: "4 to the power x")[$4^(x)$]. Convince him that he is mistaken. + Explain the algebraic technique for solving exponential equations described in this section. ==== SKILLS Practice each skill in the Homework problems listed. + Describe the graph of an exponential function: \#1–14 + Graph transformations of exponential functions: \#15–18, 53–60 + Evaluate exponential functions: \#19–22 + Find the equation of an exponential function from its graph: \#23–26 + Solve exponential equations: \#27–44 + Distinguish between power and exponential functions: \#45–52, 65, and 66 === Homework 4.2 For Problems 1 and 2, find the #math.equation(block: false, alt: "y")[$y$]-intercept of each exponential function and decide whether the graph is increasing or decreasing. + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 26 open parenthesis 1.4 close parenthesis to the power x")[$f ( x ) = 26 ( 1.4 )^(x)$] + #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals 1.2 open parenthesis 0.84 close parenthesis to the power x")[$g ( x ) = 1.2 ( 0.84 )^(x)$] + #math.equation(block: false, alt: "h open parenthesis x close parenthesis equals 75 open parenthesis the fraction 4 over 5 close parenthesis to the power x")[$h ( x ) = 75 attach(( display(frac(4, 5)) ), t: x)$] + #math.equation(block: false, alt: "k open parenthesis x close parenthesis equals the fraction 2 over 3 open parenthesis the fraction 9 over 8 close parenthesis to the power x")[$k ( x ) = display(frac(2, 3)) attach(( display(frac(9, 8)) ), t: x)$] + #math.equation(block: false, alt: "26")[$26$]; increasing + #math.equation(block: false, alt: "1.2")[$1.2$]; decreasing + #math.equation(block: false, alt: "75")[$75$]; decreasing + #math.equation(block: false, alt: "the fraction 2 over 3")[$frac(2, 3)$]; increasing + #math.equation(block: false, alt: "M open parenthesis x close parenthesis equals 1.5 open parenthesis 0.05 close parenthesis to the power x")[$M ( x ) = 1.5 ( 0.05 )^(x)$] + #math.equation(block: false, alt: "N open parenthesis x close parenthesis equals 0.05 open parenthesis 1.05 close parenthesis to the power x")[$N ( x ) = 0.05 ( 1.05 )^(x)$] + #math.equation(block: false, alt: "P open parenthesis x close parenthesis equals open parenthesis the fraction 5 over 8 close parenthesis to the power x")[$P ( x ) = attach(( display(frac(5, 8)) ), t: x)$] + #math.equation(block: false, alt: "Q open parenthesis x close parenthesis equals open parenthesis the fraction 4 over 3 close parenthesis to the power x")[$Q ( x ) = attach(( display(frac(4, 3)) ), t: x)$] For Problems 3–6, make a table of values and graph each pair of functions by hand on the domain #math.equation(block: false, alt: "open bracket minus 3 , 3 close bracket")[$[ − 3 , 3 ]$]. Describe the similarities and differences between the two graphs. + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 3 to the power x")[$f ( x ) = 3^(x)$] + #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals open parenthesis the fraction 1 over 3 close parenthesis to the power x")[$g ( x ) = attach(( display(frac(1, 3)) ), t: x)$] #figure(table( columns: 8, 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: "3")[$3$]]), [#math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 3 to the power x")[$f ( x ) = 3^(x)$]], [#math.equation(block: false, alt: "the fraction 1 over 27")[$frac(1, 27)$]], [#math.equation(block: false, alt: "the fraction 1 over 9")[$frac(1, 9)$]], [#math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "9")[$9$]], [#math.equation(block: false, alt: "27")[$27$]], [#math.equation(block: false, alt: "g open parenthesis x close parenthesis equals open parenthesis the fraction 1 over 3 close parenthesis to the power x")[$g ( x ) = attach(( frac(1, 3) ), t: x)$]], [#math.equation(block: false, alt: "27")[$27$]], [#math.equation(block: false, alt: "9")[$9$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$]], [#math.equation(block: false, alt: "the fraction 1 over 9")[$frac(1, 9)$]], [#math.equation(block: false, alt: "the fraction 1 over 27")[$frac(1, 27)$]], )) #figure(figph[exponential growth and decay], alt: "exponential growth and decay", caption: none) The two graphs are reflections of each other across the #math.equation(block: false, alt: "y")[$y$]-axis. #math.equation(block: false, alt: "f")[$f$] is increasing, #math.equation(block: false, alt: "g")[$g$] is decreasing. #math.equation(block: false, alt: "f")[$f$] has the negative #math.equation(block: false, alt: "x")[$x$]-axis as an asymptote, and #math.equation(block: false, alt: "g")[$g$] has the positive #math.equation(block: false, alt: "x")[$x$]-axis as its asymptote. + #math.equation(block: false, alt: "F open parenthesis x close parenthesis equals open parenthesis the fraction 1 over 10 close parenthesis to the power x")[$F ( x ) = attach(( display(frac(1, 10)) ), t: x)$] + #math.equation(block: false, alt: "G open parenthesis x close parenthesis equals 10 to the power x")[$G ( x ) = 10^(x)$] + #math.equation(block: false, alt: "h open parenthesis t close parenthesis equals 4 to the power minus t")[$h ( t ) = 4^(− t)$] + #math.equation(block: false, alt: "q open parenthesis t close parenthesis equals minus 4 to the power t")[$q ( t ) = − 4^(t)$] #figure(table( columns: 8, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "t")[$t$]], [#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: "3")[$3$]]), [#math.equation(block: false, alt: "h open parenthesis t close parenthesis equals 4 to the power minus t")[$h ( t ) = 4^(− t)$]], [#math.equation(block: false, alt: "64")[$64$]], [#math.equation(block: false, alt: "16")[$16$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$]], [#math.equation(block: false, alt: "the fraction 1 over 16")[$frac(1, 16)$]], [#math.equation(block: false, alt: "the fraction 1 over 64")[$frac(1, 64)$]], [#math.equation(block: false, alt: "q open parenthesis t close parenthesis equals minus 4 to the power t")[$q ( t ) = − 4^(t)$]], [#math.equation(block: false, alt: "the fraction minus 1 over 64")[$frac(− 1, 64)$]], [#math.equation(block: false, alt: "the fraction minus 1 over 16")[$frac(− 1, 16)$]], [#math.equation(block: false, alt: "the fraction minus 1 over 4")[$frac(− 1, 4)$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "minus 4")[$− 4$]], [#math.equation(block: false, alt: "minus 16")[$− 16$]], [#math.equation(block: false, alt: "minus 64")[$− 64$]], )) #figure(figph[exponential decay and negative of growth], alt: "exponential decay and negative of growth", caption: none) The graphs are reflections of each other across the origin. Both are decreasing, but #math.equation(block: false, alt: "h")[$h$] has the negative #math.equation(block: false, alt: "t")[$t$]-axis as an asymptote, and #math.equation(block: false, alt: "q")[$q$] has the positive t-axis as its asymptote. + #math.equation(block: false, alt: "P open parenthesis t close parenthesis equals minus 5 to the power t")[$P ( t ) = − 5^(t)$] + #math.equation(block: false, alt: "R open parenthesis t close parenthesis equals 5 to the power minus t")[$R ( t ) = 5^(− t)$] For Problems 7–12, match each function with its graph. #figure(figph[four exponentials], alt: "four exponentials", caption: none) + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 3 open parenthesis 2 to the power x close parenthesis")[$f ( x ) = 3 ( 2^(x) )$] + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 3 open parenthesis the fraction 1 over 2 close parenthesis to the power x")[$f ( x ) = 3 attach(( display(frac(1, 2)) ), t: x)$] + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 3 open parenthesis the fraction 1 over 3 close parenthesis to the power x")[$f ( x ) = 3 attach(( display(frac(1, 3)) ), t: x)$] + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 3 open parenthesis 3 to the power x close parenthesis")[$f ( x ) = 3 ( 3^(x) )$] + I + IV + III + II #figure(figph[four exponentials], alt: "four exponentials", caption: none) + #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals 2 open parenthesis 1.5 to the power x close parenthesis")[$g ( x ) = 2 ( 1.5^(x) )$] + #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals 2 open parenthesis 1.25 close parenthesis to the power x")[$g ( x ) = 2 attach(( 1.25 ), t: x)$] + #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals 2 open parenthesis 0.75 close parenthesis to the power x")[$g ( x ) = 2 attach(( 0.75 ), t: x)$] + #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals 2 open parenthesis 0.25 close parenthesis to the power x")[$g ( x ) = 2 ( 0.25 )^(x)$] For Problems 9–12, + Use a graphing calculator to graph the functions on the domain #math.equation(block: false, alt: "open bracket minus 5 , 5 close bracket")[$[ − 5 , 5 ]$]. + Give the range of the function on that domain, accurate to hundredths. #math.equation(block: true, alt: "g open parenthesis t close parenthesis equals 4 open parenthesis 1.3 to the power t close parenthesis")[$g ( t ) = 4 ( 1.3^(t) )$] + #figure(figph[growth], alt: "growth", caption: none) + #math.equation(block: false, alt: "open bracket 1.08 , 14.85 close bracket")[$[ 1.08 , 14.85 ]$] #math.equation(block: true, alt: "h open parenthesis t close parenthesis equals 3 open parenthesis 2.4 to the power t close parenthesis")[$h ( t ) = 3 ( 2.4^(t) )$] #math.equation(block: true, alt: "N open parenthesis x close parenthesis equals 50 open parenthesis 0.8 to the power x close parenthesis")[$N ( x ) = 50 ( 0.8^(x) )$] + #figure(figph[decay], alt: "decay", caption: none) + #math.equation(block: false, alt: "open bracket 16.38 , 152.59 close bracket")[$[ 16.38 , 152.59 ]$] #math.equation(block: true, alt: "P open parenthesis x close parenthesis equals 80 open parenthesis 0.7 to the power x close parenthesis")[$P ( x ) = 80 ( 0.7^(x) )$] For Problems 13 and 14, in each group of functions, which have identical graphs? Explain why. + #math.equation(block: false, alt: "h open parenthesis x close parenthesis equals 6 to the power x")[$h ( x ) = 6^(x)$] + #math.equation(block: false, alt: "k open parenthesis x close parenthesis equals open parenthesis the fraction 1 over 6 close parenthesis to the power x")[$k ( x ) = attach(( display(frac(1, 6)) ), t: x)$] + #math.equation(block: false, alt: "m open parenthesis x close parenthesis equals 6 to the power minus x")[$m ( x ) = 6^(− x)$] + #math.equation(block: false, alt: "n open parenthesis x close parenthesis equals the fraction 1 over 6 to the power x")[$n ( x ) = display(frac(1, 6^(x)))$] Because they are defined by equivalent expressions, (b), (c), and (d) have identical graphs + #math.equation(block: false, alt: "Q open parenthesis t close parenthesis equals 5 to the power t")[$Q ( t ) = 5^(t)$] + #math.equation(block: false, alt: "R open parenthesis t close parenthesis equals open parenthesis the fraction 1 over 5 close parenthesis to the power t")[$R ( t ) = attach(( display(frac(1, 5)) ), t: t)$] + #math.equation(block: false, alt: "F open parenthesis t close parenthesis equals open parenthesis the fraction 1 over 5 close parenthesis to the power minus t")[$F ( t ) = attach(( display(frac(1, 5)) ), t: − t)$] + #math.equation(block: false, alt: "G open parenthesis t close parenthesis equals the fraction 1 over 5 to the power minus t")[$G ( t ) = display(frac(1, 5^(− t)))$] For Problems 15–18, + Use the order of operations to explain why the two functions are different. + Complete the table of values and graph both functions in the same window. + Describe each as a transformation of #math.equation(block: false, alt: "y equals 2 to the power x")[$y = 2^(x)$] or #math.equation(block: false, alt: "y equals 3 to the power x")[$y = 3^(x)$]. #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals 2 to the power x minus 1")[$f ( x ) = 2^(x − 1)$], #math.equation(block: true, alt: "g open parenthesis x close parenthesis equals 2 to the power x minus 1")[$" " g ( x ) = 2^(x) − 1$] #figure(table( columns: 4, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y equals 2 to the power x")[$y = 2^(x)$]], [#math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]], [#math.equation(block: false, alt: "g open parenthesis x close parenthesis")[$g ( x )$]]), [#math.equation(block: false, alt: "minus 2")[$− 2$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [], [], [], [#math.equation(block: false, alt: "0")[$0$]], [], [], [], [#math.equation(block: false, alt: "1")[$1$]], [], [], [], [#math.equation(block: false, alt: "2")[$2$]], [], [], [], )) + To evaluate #math.equation(block: false, alt: "f")[$f$] we subtract #math.equation(block: false, alt: "1")[$1$] from the input before evaluating the exponential function; to evaluate #math.equation(block: false, alt: "g")[$g$] we subtract #math.equation(block: false, alt: "1")[$1$] from the output of the exponential function. + #figure(table( columns: 4, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y equals 2 to the power x")[$y = 2^(x)$]], [#math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]], [#math.equation(block: false, alt: "g open parenthesis x close parenthesis")[$g ( x )$]]), [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "the fraction 1 over 4")[$display(frac(1, 4))$]], [#math.equation(block: false, alt: "the fraction 1 over 8")[$display(frac(1, 8))$]], [#math.equation(block: false, alt: "the fraction minus 3 over 4")[$display(frac(− 3, 4))$]], [#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: "the fraction 1 over 4")[$display(frac(1, 4))$]], [#math.equation(block: false, alt: "the fraction minus 1 over 2")[$display(frac(− 1, 2))$]], [#math.equation(block: false, alt: "0")[$0$]], [#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: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]], [#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: "4")[$4$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "3")[$3$]], )) #figure(figph[two shifts of growth], alt: "two shifts of growth", caption: none) + The graph of #math.equation(block: false, alt: "f")[$f$] is translated #math.equation(block: false, alt: "1")[$1$] unit to the right; the graph of #math.equation(block: false, alt: "g")[$g$] is shifted #math.equation(block: false, alt: "1")[$1$] unit down. #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals 3 to the power x plus 2")[$f ( x ) = 3^(x) + 2$], #math.equation(block: true, alt: "g open parenthesis x close parenthesis equals 3 to the power x plus 2")[$" " g ( x ) = 3^(x + 2)$] #figure(table( columns: 4, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y equals 3 to the power x")[$y = 3^(x)$]], [#math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]], [#math.equation(block: false, alt: "g open parenthesis x close parenthesis")[$g ( x )$]]), [#math.equation(block: false, alt: "minus 2")[$− 2$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [#math.equation(block: false, alt: "minus 1")[$− 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: true, alt: "f open parenthesis x close parenthesis equals minus 3 to the power x")[$f ( x ) = − 3^(x)$], #math.equation(block: true, alt: "g open parenthesis x close parenthesis equals 3 to the power minus x")[$" " g ( x ) = 3^(− x)$] #figure(table( columns: 4, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y equals 3 to the power x")[$y = 3^(x)$]], [#math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]], [#math.equation(block: false, alt: "g open parenthesis x close parenthesis")[$g ( x )$]]), [#math.equation(block: false, alt: "minus 2")[$− 2$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [], [], [], [#math.equation(block: false, alt: "0")[$0$]], [], [], [], [#math.equation(block: false, alt: "1")[$1$]], [], [], [], [#math.equation(block: false, alt: "2")[$2$]], [], [], [], )) + To evaluate #math.equation(block: false, alt: "f")[$f$] we take the negative of the output of the exponential function; to evaluate #math.equation(block: false, alt: "g")[$g$] we take the negative of the input. + #figure(table( columns: 4, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y equals 3 to the power x")[$y = 3^(x)$]], [#math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]], [#math.equation(block: false, alt: "g open parenthesis x close parenthesis")[$g ( x )$]]), [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "the fraction 1 over 9")[$display(frac(1, 9))$]], [#math.equation(block: false, alt: "the fraction minus 1 over 9")[$display(frac(− 1, 9))$]], [#math.equation(block: false, alt: "9")[$9$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "the fraction 1 over 3")[$display(frac(1, 3))$]], [#math.equation(block: false, alt: "the fraction minus 1 over 3")[$display(frac(− 1, 3))$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "the fraction 1 over 3")[$display(frac(1, 3))$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "9")[$9$]], [#math.equation(block: false, alt: "minus 9")[$− 9$]], [#math.equation(block: false, alt: "the fraction 1 over 9")[$display(frac(1, 9))$]], )) #figure(figph[two shifts of growth], alt: "two shifts of growth", caption: none) + The graph of #math.equation(block: false, alt: "f")[$f$] is reflected about the #math.equation(block: false, alt: "x")[$x$]-axis; the graph of #math.equation(block: false, alt: "g")[$g$] is reflected about the #math.equation(block: false, alt: "y")[$y$]-axis. #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals 2 to the power minus x")[$f ( x ) = 2^(− x)$], #math.equation(block: true, alt: "g open parenthesis x close parenthesis equals minus 2 to the power x")[$" " g ( x ) = − 2^(x)$] #figure(table( columns: 4, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y equals 2 to the power x")[$y = 2^(x)$]], [#math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]], [#math.equation(block: false, alt: "g open parenthesis x close parenthesis")[$g ( x )$]]), [#math.equation(block: false, alt: "minus 2")[$− 2$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [], [], [], [#math.equation(block: false, alt: "0")[$0$]], [], [], [], [#math.equation(block: false, alt: "1")[$1$]], [], [], [], [#math.equation(block: false, alt: "2")[$2$]], [], [], [], )) In Problems 19–22, for the given function, evaluate each pair of expressions. Are they equivalent? #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals 3 open parenthesis 5 to the power x close parenthesis")[$f ( x ) = 3 ( 5^(x) )$] + #math.equation(block: false, alt: "f open parenthesis a plus 2 close parenthesis")[$f ( a + 2 )$] and #math.equation(block: false, alt: "9 f open parenthesis a close parenthesis")[$9 f ( a )$] + #math.equation(block: false, alt: "f open parenthesis 2 a close parenthesis")[$f ( 2 a )$] and #math.equation(block: false, alt: "2 f open parenthesis a close parenthesis")[$2 f ( a )$] + #math.equation(block: false, alt: "3 open parenthesis 5 to the power a plus 2 close parenthesis")[$3 ( 5^(a + 2) )$] is not equivalent to #math.equation(block: false, alt: "9 times 3 open parenthesis 5 to the power a close parenthesis")[$9 ⋅ 3 ( 5^(a) )$]. + #math.equation(block: false, alt: "3 open parenthesis 5 to the power 2 a close parenthesis")[$3 ( 5^(2 a) )$] is not equivalent to #math.equation(block: false, alt: "2 times 3 open parenthesis 5 to the power a close parenthesis")[$2 ⋅ 3 ( 5^(a) )$]. #math.equation(block: true, alt: "g open parenthesis x close parenthesis equals 1.8 to the power x")[$g ( x ) = 1.8^(x)$] + #math.equation(block: false, alt: "g open parenthesis h plus 3 close parenthesis")[$g ( h + 3 )$] and #math.equation(block: false, alt: "g open parenthesis h close parenthesis g open parenthesis 3 close parenthesis")[$g ( h ) g ( 3 )$] + #math.equation(block: false, alt: "g open parenthesis 2 h close parenthesis")[$g ( 2 h )$] and #math.equation(block: false, alt: "open bracket g open parenthesis h close parenthesis close bracket squared")[$[ g ( h ) ]^(2)$] #math.equation(block: true, alt: "P open parenthesis t close parenthesis equals 8 to the power t")[$P ( t ) = 8^(t)$] + #math.equation(block: false, alt: "P open parenthesis w close parenthesis minus P open parenthesis z close parenthesis")[$P ( w ) − P ( z )$] and #math.equation(block: false, alt: "P open parenthesis w minus z close parenthesis")[$P ( w − z )$] + #math.equation(block: false, alt: "P open parenthesis minus x close parenthesis")[$P ( − x )$] and #math.equation(block: false, alt: "the fraction 1 over P open parenthesis x close parenthesis")[$display(frac(1, P ( x )))$] + #math.equation(block: false, alt: "8 to the power w minus 8 to the power z")[$8^(w) − 8^(z)$] is not equivalent to #math.equation(block: false, alt: "8 to the power w minus z")[$8^(w − z)$]. + #math.equation(block: false, alt: "8 to the power minus x")[$8^(− x)$] is equivalent to #math.equation(block: false, alt: "the fraction 1 over 8 to the power x")[$display(frac(1, 8^(x)))$]. #math.equation(block: true, alt: "Q open parenthesis t close parenthesis equals 5 open parenthesis 0.2 close parenthesis to the power t")[$Q ( t ) = 5 ( 0.2 )^(t)$] + #math.equation(block: false, alt: "Q open parenthesis b minus 1 close parenthesis")[$Q ( b − 1 )$] and #math.equation(block: false, alt: "5 Q open parenthesis b close parenthesis")[$5 Q ( b )$] + #math.equation(block: false, alt: "Q open parenthesis a close parenthesis Q open parenthesis b close parenthesis")[$Q ( a ) Q ( b )$] and #math.equation(block: false, alt: "5 Q open parenthesis a plus b close parenthesis")[$5 Q ( a + b )$] The graph of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals P sub 0 b to the power x")[$f ( x ) = P_(0) b^(x)$] is shown in the figure. #figure(figph[growth], alt: "growth", caption: none) + Read the value of #math.equation(block: false, alt: "P sub 0")[$P_(0)$] from the graph. + Make a short table of values for the function by reading values from the graph. Does your table confirm that the function is exponential? + Use your table to calculate the growth factor, #math.equation(block: false, alt: "b")[$b$]. + Using your answers to parts (a) and (c), write a formula for #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]. + #math.equation(block: false, alt: "P sub 0 equals 300")[$P_(0) = 300$] + #figure(table( columns: 4, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#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: "300")[$300$]], [#math.equation(block: false, alt: "600")[$600$]], [#math.equation(block: false, alt: "1200")[$1200$]], )) + #math.equation(block: false, alt: "b equals 2")[$b = 2$] + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 300 open parenthesis 2 close parenthesis to the power x")[$f ( x ) = 300 ( 2 )^(x)$] The graph of #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals P sub 0 b to the power x")[$g ( x ) = P_(0) b^(x)$] is shown in the figure. #figure(figph[decay], alt: "decay", caption: none) + Read the value of #math.equation(block: false, alt: "P sub 0")[$P_(0)$] from the graph. + Make a short table of values for the function by reading values from the graph. Does your table confirm that the function is exponential? + Use your table to calculate the decay factor, #math.equation(block: false, alt: "b")[$b$]. + Using your answers to parts (a) and (c), write a formula for #math.equation(block: false, alt: "g open parenthesis x close parenthesis")[$g ( x )$]. For several days after the Northridge earthquake on January 17, 1994, the area received a number of significant aftershocks. The red graph shows that the number of aftershocks decreased exponentially over time. The graph of the function #math.equation(block: false, alt: "S open parenthesis d close parenthesis equals S sub 0 b to the power d")[$S ( d ) = S_(0) b^(d)$], shown in black, approximates the data. (Source: #emph[Los Angeles Times], June 27, 1995) #figure(figph[decay], alt: "decay", caption: none) + Read the value of #math.equation(block: false, alt: "S sub 0")[$S_(0)$] from the graph. + Find an approximation for the decay factor, #math.equation(block: false, alt: "b")[$b$], by comparing two points on the graph. (Some of the points on the graph of #math.equation(block: false, alt: "S open parenthesis d close parenthesis")[$S ( d )$] are approximately #math.equation(block: false, alt: "open parenthesis 1 , 82 close parenthesis")[$( 1 , 82 )$], #math.equation(block: false, alt: "open parenthesis 2 , 45 close parenthesis")[$( 2 , 45 )$], #math.equation(block: false, alt: "open parenthesis 3 , 25 close parenthesis")[$( 3 , 25 )$], and #math.equation(block: false, alt: "open parenthesis 4 , 14 close parenthesis")[$( 4 , 14 )$].) + Using your answers to (a) and (b), write a formula for #math.equation(block: false, alt: "S open parenthesis d close parenthesis")[$S ( d )$]. + #math.equation(block: false, alt: "S sub 0 equals 150")[$S_(0) = 150$] + #math.equation(block: false, alt: "b approximately equals 0.55")[$b ≈ 0.55$] + #math.equation(block: false, alt: "S open parenthesis d close parenthesis equals 150 open parenthesis 0.55 close parenthesis to the power d")[$S ( d ) = 150 ( 0.55 )^(d)$] The frequency of a musical note depends on its pitch. The graph shows that the frequency increases exponentially. The function #math.equation(block: false, alt: "F open parenthesis p close parenthesis equals F sub 0 b to the power p")[$F ( p ) = F_(0) b^(p)$] gives the frequency as a function of the number of half-tones, #math.equation(block: false, alt: "p")[$p$], above the starting point on the scale. #figure(figph[growth], alt: "growth", caption: none) + Read the value of #math.equation(block: false, alt: "F sub 0")[$F_(0)$] from the graph. (This is the frequency of the note A above middle C.) + Find an approximation for the growth factor, #math.equation(block: false, alt: "b")[$b$], by comparing two points on the graph. (Some of the points on the graph of #math.equation(block: false, alt: "F open parenthesis p close parenthesis")[$F ( p )$] are approximately #math.equation(block: false, alt: "open parenthesis 1 , 466 close parenthesis")[$( 1 , 466 )$], #math.equation(block: false, alt: "open parenthesis 2 , 494 close parenthesis")[$( 2 , 494 )$], #math.equation(block: false, alt: "open parenthesis 3 , 523 close parenthesis")[$( 3 , 523 )$], and #math.equation(block: false, alt: "open parenthesis 4 , 554 close parenthesis")[$( 4 , 554 )$].) + Using your answers to (a) and (b), write a formula for #math.equation(block: false, alt: "F open parenthesis p close parenthesis")[$F ( p )$]. + The frequency doubles when you raise a note by one octave, which is equivalent to #math.equation(block: false, alt: "12")[$12$] half-tones. Use this information to find an exact value for #math.equation(block: false, alt: "b")[$b$]. Solve the equation algebraically. #math.equation(block: true, alt: "5 to the power x plus 2 equals 25 to the power 4 / 3")[$5^(x + 2) = 25^(4 / 3)$] #math.equation(block: true, alt: "the fraction 2 over 3")[$display(frac(2, 3))$] #math.equation(block: true, alt: "3 to the power x minus 1 equals 27 to the power 1 / 2")[$3^(x − 1) = 27^(1 / 2)$] #math.equation(block: true, alt: "3 to the power 2 x minus 1 equals the fraction the square root of 3 over 9")[$3^(2 x − 1) = display(frac(sqrt(3), 9))$] #math.equation(block: true, alt: "the fraction minus 1 over 4")[$display(frac(− 1, 4))$] #math.equation(block: true, alt: "2 to the power 3 x minus 1 equals the fraction the square root of 2 over 16")[$2^(3 x − 1) = display(frac(sqrt(2), 16))$] #math.equation(block: true, alt: "4 times 2 to the power x minus 3 equals 8 to the power minus 2 x")[$4 ⋅ 2^(x − 3) = 8^(− 2 x)$] #math.equation(block: true, alt: "the fraction 1 over 7")[$display(frac(1, 7))$] #math.equation(block: true, alt: "9 times 3 to the power x plus 2 equals 81 to the power minus x")[$9 ⋅ 3^(x + 2) = 81^(− x)$] #math.equation(block: true, alt: "27 to the power 4 x plus 2 equals 81 to the power x minus 1")[$27^(4 x + 2) = 81^(x − 1)$] #math.equation(block: true, alt: "the fraction minus 5 over 4")[$display(frac(− 5, 4))$] #math.equation(block: true, alt: "16 to the power 2 minus 3 x equals 64 to the power x plus 5")[$16^(2 − 3 x) = 64^(x + 5)$] #math.equation(block: true, alt: "10 to the power x squared minus 1 equals 1000")[$10^(x^(2) − 1) = 1000$] #math.equation(block: true, alt: "plus or minus 2")[$± 2$] #math.equation(block: true, alt: "5 to the power x squared minus x minus 4 equals 25")[$5^(x^(2) − x − 4) = 25$] Before the advent of antibiotics, an outbreak of cholera might spread through a city so that the number of cases doubled every #math.equation(block: false, alt: "6")[$6$] days. + Twenty-six cases were discovered on July 5. Write a function for the number of cases of cholera #math.equation(block: false, alt: "t")[$t$] days later. + Use your calculator to graph your function on the interval #math.equation(block: false, alt: "0 less than or equal to t less than or equal to 90")[$0 ≤ t ≤ 90$]. + When should hospitals expect to be treating #math.equation(block: false, alt: "106 , 496")[$106 , 496$] cases? Use algebraic methods to find your answer, and verify it on your graph. + #math.equation(block: false, alt: "N open parenthesis t close parenthesis equals 26 open parenthesis 2 close parenthesis to the power t / 6")[$N ( t ) = 26 ( 2 )^(t / 6)$] + #figure(figph[GC growth], alt: "GC growth", caption: none) + #math.equation(block: false, alt: "72")[$72$] days later An outbreak of ungulate fever can sweep through the livestock in a region so that the number of animals affected triples every #math.equation(block: false, alt: "4")[$4$] days. + A rancher discovers #math.equation(block: false, alt: "4")[$4$] cases of ungulate fever among his herd. Write a function for the number of cases of ungulate fever #math.equation(block: false, alt: "t")[$t$] days later. + Use your calculator to graph your function on the interval #math.equation(block: false, alt: "0 less than or equal to t less than or equal to 20")[$0 ≤ t ≤ 20$]. + If the rancher does not act quickly, how long will it be until #math.equation(block: false, alt: "324")[$324$] head are affected? Use algebraic methods to find your answer, and verify it on your graph. A smart television set loses #math.equation(block: false, alt: "30 %")[$30 upright(%)$] of its value every #math.equation(block: false, alt: "2")[$2$] years. + Write a function for the value of a television set #math.equation(block: false, alt: "t")[$t$] years after it was purchased if it cost #math.equation(block: false, alt: "$ 700")[$upright(\$) 700$] originally. + Use your calculator to graph your function on the interval #math.equation(block: false, alt: "0 less than or equal to t less than or equal to 20")[$0 ≤ t ≤ 20$]. + How long will it be before a #math.equation(block: false, alt: "$ 700")[$upright(\$) 700$] television set depreciates to #math.equation(block: false, alt: "$ 343")[$upright(\$) 343$]? Use algebraic methods to find your answer, and verify it on your graph. + #math.equation(block: false, alt: "V open parenthesis t close parenthesis equals 700 open parenthesis 0.7 close parenthesis to the power t / 2")[$V ( t ) = 700 ( 0.7 )^(t / 2)$] + #figure(figph[GC decay], alt: "GC decay", caption: none) + #math.equation(block: false, alt: "4")[$4$] yr A mobile home loses #math.equation(block: false, alt: "20 %")[$20 upright(%)$] of its value every #math.equation(block: false, alt: "3")[$3$] years. + A certain mobile home costs #math.equation(block: false, alt: "$ 20 , 000")[$upright(\$) 20 , 000$]. Write a function for its value after #math.equation(block: false, alt: "t")[$t$] years. + Use your calculator to graph your function on the interval #math.equation(block: false, alt: "0 less than or equal to t less than or equal to 30")[$0 ≤ t ≤ 30$]. + How long will it be before a #math.equation(block: false, alt: "$ 20 , 000")[$upright(\$) 20 , 000$] mobile home depreciates to #math.equation(block: false, alt: "$ 12 , 800")[$upright(\$) 12 , 800$]? Use algebraic methods to find your answer, and verify it on your graph. For Problems 41–44, use a graph to find an approximate solution accurate to the nearest hundredth. #math.equation(block: true, alt: "3 to the power x minus 1 equals 4")[$3^(x − 1) = 4$] #math.equation(block: true, alt: "x equals 2.26")[$x = 2.26$] #math.equation(block: true, alt: "2 to the power x plus 3 equals 5")[$2^(x + 3) = 5$] #math.equation(block: true, alt: "4 to the power minus x equals 7")[$4^(− x) = 7$] #math.equation(block: true, alt: "x equals minus 1.40")[$x = − 1.40$] #math.equation(block: true, alt: "6 to the power minus x equals 3")[$6^(− x) = 3$] For Problems 45 and 46, decide whether each function is an exponential function, a power function, or neither. + #math.equation(block: false, alt: "g open parenthesis t close parenthesis equals 3 t to the power 0.4")[$g ( t ) = 3 t^(0.4)$] + #math.equation(block: false, alt: "h open parenthesis t close parenthesis equals 4 open parenthesis 0.3 close parenthesis to the power t")[$h ( t ) = 4 ( 0.3 )^(t)$] + #math.equation(block: false, alt: "D open parenthesis x close parenthesis equals 6 x to the power 1 / 2")[$D ( x ) = 6 x^(1 / 2)$] + #math.equation(block: false, alt: "E open parenthesis x close parenthesis equals 4 x plus x to the power 4")[$E ( x ) = 4 x + x^(4)$] + Power + Exponential + Power + Neither + #math.equation(block: false, alt: "R open parenthesis w close parenthesis equals 5 open parenthesis 5 close parenthesis to the power w minus 1")[$R ( w ) = 5 ( 5 )^(w − 1)$] + #math.equation(block: false, alt: "Q open parenthesis w close parenthesis equals 2 to the power w minus w squared")[$Q ( w ) = 2^(w) − w^(2)$] + #math.equation(block: false, alt: "M open parenthesis z close parenthesis equals 0.2 z to the power 1.3")[$M ( z ) = 0.2 z^(1.3)$] + #math.equation(block: false, alt: "N open parenthesis z close parenthesis equals z to the power minus 3")[$N ( z ) = z^(− 3)$] For Problems 47–50, decide whether the table could describe a linear function, a power function, an exponential function, or none of these. Find a formula for each linear, power, or exponential function. + #figure(table( columns: 2, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "y")[$y$]]), [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "6")[$6$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "12")[$12$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "24")[$24$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "48")[$48$]], )) + #figure(table( columns: 2, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "t")[$t$]], [#math.equation(block: false, alt: "P")[$P$]]), [#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: "0.5")[$0.5$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "4.5")[$4.5$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "8")[$8$]], )) + Exponential #math.equation(block: false, alt: "y equals 3 times 2 to the power x")[$y = 3 ⋅ 2^(x)$] + Power #math.equation(block: false, alt: "P equals 0.5 t squared")[$P = 0.5 t^(2)$] + #figure(table( columns: 2, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "N")[$N$]]), [#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: "2")[$2$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "16")[$16$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "54")[$54$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "128")[$128$]], )) + #figure(table( columns: 2, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "p")[$p$]], [#math.equation(block: false, alt: "R")[$R$]]), [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "405")[$405$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "135")[$135$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "45")[$45$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "15")[$15$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "5")[$5$]], )) + #figure(table( columns: 2, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "t")[$t$]], [#math.equation(block: false, alt: "y")[$y$]]), [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "100")[$100$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "50")[$50$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "33 the fraction 1 over 3")[$33 frac(1, 3)$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "25")[$25$]], [#math.equation(block: false, alt: "5")[$5$]], [#math.equation(block: false, alt: "20")[$20$]], )) + #figure(table( columns: 2, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "P")[$P$]]), [#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: "1")[$1$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "5")[$5$]], [#math.equation(block: false, alt: "8")[$8$]], )) + Power #math.equation(block: false, alt: "y equals 100 x to the power minus 1")[$y = 100 x^(− 1)$] + Exponential #math.equation(block: false, alt: "P equals the fraction 1 over 4 times 2 to the power x")[$P = frac(1, 4) ⋅ 2^(x)$] + #figure(table( columns: 2, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "h")[$h$]], [#math.equation(block: false, alt: "a")[$a$]]), [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "70")[$70$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "7")[$7$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "0.7")[$0.7$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "0.07")[$0.07$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "0.007")[$0.007$]], )) + #figure(table( columns: 2, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "t")[$t$]], [#math.equation(block: false, alt: "Q")[$Q$]]), [#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: "the fraction 1 over 4")[$frac(1, 4)$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "the fraction 9 over 4")[$frac(9, 4)$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "4")[$4$]], )) For Problems 51 and 52, fill in the tables. Graph each pair of functions in the same window. Then answer the questions below. + Give the range of #math.equation(block: false, alt: "f")[$f$] and the range of #math.equation(block: false, alt: "g")[$g$]. + For how many values of #math.equation(block: false, alt: "x")[$x$] does #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals g open parenthesis x close parenthesis")[$f ( x ) = g ( x )$]? + Estimate the value(s) of #math.equation(block: false, alt: "x")[$x$] for which #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals g open parenthesis x close parenthesis")[$f ( x ) = g ( x )$]. + For what values of #math.equation(block: false, alt: "x")[$x$] is #math.equation(block: false, alt: "f open parenthesis x close parenthesis less than g open parenthesis x close parenthesis")[$f ( x ) < g ( x )$]? + Which function grows more rapidly for large values of #math.equation(block: false, alt: "x")[$x$]? #figure(table( columns: 3, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x squared")[$f ( x ) = x^(2)$]], [#math.equation(block: false, alt: "g open parenthesis x close parenthesis equals 2 to the power x")[$g ( x ) = 2^(x)$]]), [#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: "4")[$4$]], [], [], [#math.equation(block: false, alt: "5")[$5$]], [], [], )) #figure(table( columns: 3, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x squared")[$f ( x ) = x^(2)$]], [#math.equation(block: false, alt: "g open parenthesis x close parenthesis equals 2 to the power x")[$g ( x ) = 2^(x)$]]), [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$]], [#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")[$0$]], [1], [#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: "4")[$4$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "9")[$9$]], [#math.equation(block: false, alt: "8")[$8$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "16")[$16$]], [#math.equation(block: false, alt: "16")[$16$]], [#math.equation(block: false, alt: "5")[$5$]], [#math.equation(block: false, alt: "25")[$25$]], [#math.equation(block: false, alt: "32")[$32$]], )) #figure(figph[exponential growth and qudratic], alt: "exponential growth and qudratic", caption: none) + Range of #math.equation(block: false, alt: "f")[$f$]: #math.equation(block: false, alt: "open bracket 0 , ∞ close parenthesis")[$[ 0 , upright(∞) )$]; Range of #math.equation(block: false, alt: "g")[$g$]: #math.equation(block: false, alt: "open parenthesis 0 , ∞ close parenthesis")[$( 0 , upright(∞) )$] + #math.equation(block: false, alt: "3")[$3$] + #math.equation(block: false, alt: "minus 0.7667")[$− 0.7667$], #math.equation(block: false, alt: "2")[$2$], #math.equation(block: false, alt: "4")[$4$] + #math.equation(block: false, alt: "open parenthesis minus 0.7667 , 2 close parenthesis")[$( − 0.7667 , 2 )$] and #math.equation(block: false, alt: "open parenthesis 4 , ∞ close parenthesis")[$( 4 , upright(∞) )$] + g #figure(table( columns: 3, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x cubed")[$f ( x ) = x^(3)$]], [#math.equation(block: false, alt: "g open parenthesis x close parenthesis equals 3 to the power x")[$g ( x ) = 3^(x)$]]), [#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: "4")[$4$]], [], [], [#math.equation(block: false, alt: "5")[$5$]], [], [], )) For Problems 53–60, sketch the graph of each transformation of the given function, then write a formula and check your sketch with a graphing calculator. State the domain and range of each transformation, its intercept(s), and any asymptotes. #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals 3 to the power x")[$f ( x ) = 3^(x)$] + #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis minus 4")[$y = f ( x ) − 4$] + #math.equation(block: false, alt: "y equals f open parenthesis x minus 4 close parenthesis")[$y = f ( x − 4 )$] + #math.equation(block: false, alt: "y equals minus 4 f open parenthesis x close parenthesis")[$y = − 4 f ( x )$] + #math.equation(block: false, alt: "y equals 3 to the power x minus 4")[$y = 3^(x) − 4$]#figure(figph[shifted growth], alt: "shifted growth", caption: none) #linebreak() Domain: #math.equation(block: false, alt: "open parenthesis minus ∞ , ∞ close parenthesis")[$( − upright(∞) , upright(∞) )$]; range: #math.equation(block: false, alt: "open parenthesis minus 4 , ∞ close parenthesis")[$( − 4 , upright(∞) )$], #math.equation(block: false, alt: "x")[$x$]-intercept #math.equation(block: false, alt: "open parenthesis 1.26 , 0 close parenthesis")[$( 1.26 , 0 )$]; #math.equation(block: false, alt: "y")[$y$]-intercept #math.equation(block: false, alt: "open parenthesis 0 , minus 3 close parenthesis")[$( 0 , − 3 )$]; horizontal asymptote #math.equation(block: false, alt: "y equals minus 4")[$y = − 4$] + #math.equation(block: false, alt: "y equals 3 to the power x minus 4")[$y = 3^(x − 4)$],#figure(figph[growth], alt: "growth", caption: none) #linebreak() Domain: #math.equation(block: false, alt: "open parenthesis minus ∞ , ∞ close parenthesis")[$( − upright(∞) , upright(∞) )$]; range: #math.equation(block: false, alt: "open parenthesis 0 , ∞ close parenthesis")[$( 0 , upright(∞) )$], no #math.equation(block: false, alt: "x")[$x$]-intercept; #math.equation(block: false, alt: "y")[$y$]-intercept #math.equation(block: false, alt: "open parenthesis 0 , the fraction 1 over 81 close parenthesis")[$( 0 , display(frac(1, 81)) )$]; the #math.equation(block: false, alt: "x")[$x$]-axis is the horizontal asymptote. + #math.equation(block: false, alt: "y equals minus 4 times 3 to the power x")[$y = − 4 ⋅ 3^(x)$],#figure(figph[growth reflected], alt: "growth reflected", caption: none) #linebreak() Domain: #math.equation(block: false, alt: "open parenthesis minus ∞ , ∞ close parenthesis")[$( − upright(∞) , upright(∞) )$]; range: #math.equation(block: false, alt: "open parenthesis minus ∞ , 0 close parenthesis")[$( − upright(∞) , 0 )$], no #math.equation(block: false, alt: "x")[$x$]-intercept; #math.equation(block: false, alt: "y")[$y$]-intercept #math.equation(block: false, alt: "open parenthesis 0 , minus 4 close parenthesis")[$( 0 , − 4 )$]; the #math.equation(block: false, alt: "x")[$x$]-axis is the horizontal asymptote. #math.equation(block: true, alt: "g open parenthesis x close parenthesis equals 4 to the power x")[$g ( x ) = 4^(x)$] + #math.equation(block: false, alt: "y equals g open parenthesis x close parenthesis plus 2")[$y = g ( x ) + 2$] + #math.equation(block: false, alt: "y equals g open parenthesis x plus 2 close parenthesis")[$y = g ( x + 2 )$] + #math.equation(block: false, alt: "y equals 2 g open parenthesis x close parenthesis")[$y = 2 g ( x )$] #math.equation(block: true, alt: "h open parenthesis t close parenthesis equals 6 to the power t")[$h ( t ) = 6^(t)$] + #math.equation(block: false, alt: "y equals minus h open parenthesis t close parenthesis")[$y = − h ( t )$] + #math.equation(block: false, alt: "y equals h open parenthesis minus t close parenthesis")[$y = h ( − t )$] + #math.equation(block: false, alt: "y equals minus h open parenthesis minus t close parenthesis")[$y = − h ( − t )$] + #math.equation(block: false, alt: "y equals minus 6 to the power t")[$y = − 6^(t)$]#figure(figph[reflected growth], alt: "reflected growth", caption: none) #linebreak() Domain: #math.equation(block: false, alt: "open parenthesis minus ∞ , ∞ close parenthesis")[$( − upright(∞) , upright(∞) )$]; range: #math.equation(block: false, alt: "open parenthesis minus ∞ , 0 close parenthesis")[$( − upright(∞) , 0 )$], no #math.equation(block: false, alt: "t")[$t$]-intercept; #math.equation(block: false, alt: "y")[$y$]-intercept #math.equation(block: false, alt: "open parenthesis 0 , minus 1 close parenthesis")[$( 0 , − 1 )$]; the #math.equation(block: false, alt: "t")[$t$]-axis is the horizontal asymptote. + #math.equation(block: false, alt: "y equals 6 to the power minus t")[$y = 6^(− t)$],#figure(figph[decay], alt: "decay", caption: none) #linebreak() Domain: #math.equation(block: false, alt: "open parenthesis minus ∞ , ∞ close parenthesis")[$( − upright(∞) , upright(∞) )$]; range: #math.equation(block: false, alt: "open parenthesis 0 , ∞ close parenthesis")[$( 0 , upright(∞) )$], no #math.equation(block: false, alt: "t")[$t$]-intercept; #math.equation(block: false, alt: "y")[$y$]-intercept #math.equation(block: false, alt: "open parenthesis 0 , 1 close parenthesis")[$( 0 , 1 )$]; the #math.equation(block: false, alt: "t")[$t$]-axis is the horizontal asymptote. + #math.equation(block: false, alt: "y equals minus 6 to the power minus t")[$y = − 6^(− t)$],#figure(figph[decay reflected], alt: "decay reflected", caption: none) #linebreak() Domain: #math.equation(block: false, alt: "open parenthesis minus ∞ , ∞ close parenthesis")[$( − upright(∞) , upright(∞) )$]; range: #math.equation(block: false, alt: "open parenthesis minus ∞ , 0 close parenthesis")[$( − upright(∞) , 0 )$], no #math.equation(block: false, alt: "t")[$t$]-intercept; #math.equation(block: false, alt: "y")[$y$]-intercept #math.equation(block: false, alt: "open parenthesis 0 , minus 1 close parenthesis")[$( 0 , − 1 )$]; the #math.equation(block: false, alt: "t")[$t$]-axis is the horizontal asymptote. #math.equation(block: true, alt: "j open parenthesis t close parenthesis equals open parenthesis the fraction 1 over 3 close parenthesis to the power t")[$j ( t ) = attach(( display(frac(1, 3)) ), t: t)$] + #math.equation(block: false, alt: "y equals j open parenthesis minus t close parenthesis")[$y = j ( − t )$] + #math.equation(block: false, alt: "y equals minus j open parenthesis t close parenthesis")[$y = − j ( t )$] + #math.equation(block: false, alt: "y equals minus j open parenthesis minus t close parenthesis")[$y = − j ( − t )$] #math.equation(block: true, alt: "g open parenthesis x close parenthesis equals 2 to the power x")[$g ( x ) = 2^(x)$] + #math.equation(block: false, alt: "y equals g open parenthesis x minus 3 close parenthesis")[$y = g ( x − 3 )$] + #math.equation(block: false, alt: "y equals g open parenthesis x minus 3 close parenthesis plus 4")[$y = g ( x − 3 ) + 4$] + #math.equation(block: false, alt: "y equals 2 to the power x minus 3")[$y = 2^(x − 3)$]#figure(figph[growth], alt: "growth", caption: none) #linebreak() Domain: #math.equation(block: false, alt: "open parenthesis minus ∞ , ∞ close parenthesis")[$( − upright(∞) , upright(∞) )$]; range: #math.equation(block: false, alt: "open parenthesis 0 , ∞ close parenthesis")[$( 0 , upright(∞) )$], no #math.equation(block: false, alt: "x")[$x$]-intercept; #math.equation(block: false, alt: "y")[$y$]-intercept #math.equation(block: false, alt: "open parenthesis 0 , the fraction 1 over 8 close parenthesis")[$( 0 , frac(1, 8) )$]; the #math.equation(block: false, alt: "x")[$x$]-axis is the horizontal asymptote. + #math.equation(block: false, alt: "y equals 2 to the power x minus 3 plus 4")[$y = 2^(x − 3) + 4$],#figure(figph[shifted growth], alt: "shifted growth", caption: none) #linebreak() Domain: #math.equation(block: false, alt: "open parenthesis minus ∞ , ∞ close parenthesis")[$( − upright(∞) , upright(∞) )$]; range: #math.equation(block: false, alt: "open parenthesis 4 , ∞ close parenthesis")[$( 4 , upright(∞) )$], no #math.equation(block: false, alt: "x")[$x$]-intercept; #math.equation(block: false, alt: "y")[$y$]-intercept #math.equation(block: false, alt: "open parenthesis 0 , the fraction 33 over 8 close parenthesis")[$( 0 , frac(33, 8) )$]; horizontal asymptote #math.equation(block: false, alt: "y equals 4")[$y = 4$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals 10 to the power x")[$f ( x ) = 10^(x)$] + #math.equation(block: false, alt: "y equals f open parenthesis x plus 5 close parenthesis")[$y = f ( x + 5 )$] + #math.equation(block: false, alt: "y equals f open parenthesis x plus 5 close parenthesis minus 20")[$y = f ( x + 5 ) − 20$] #math.equation(block: true, alt: "N open parenthesis t close parenthesis equals open parenthesis the fraction 1 over 2 close parenthesis to the power t")[$N ( t ) = attach(( display(frac(1, 2)) ), t: t)$] + #math.equation(block: false, alt: "y equals minus N open parenthesis t close parenthesis")[$y = − N ( t )$] + #math.equation(block: false, alt: "y equals 6 minus N open parenthesis t close parenthesis")[$y = 6 − N ( t )$] + #math.equation(block: false, alt: "y equals minus open parenthesis the fraction 1 over 2 close parenthesis to the power t")[$y = − attach(( display(frac(1, 2)) ), t: t)$]#figure(figph[reflected decay], alt: "reflected decay", caption: none) #linebreak() Domain: #math.equation(block: false, alt: "open parenthesis minus ∞ , ∞ close parenthesis")[$( − upright(∞) , upright(∞) )$]; range: #math.equation(block: false, alt: "open parenthesis minus ∞ , 0 close parenthesis")[$( − upright(∞) , 0 )$], no #math.equation(block: false, alt: "t")[$t$]-intercept; #math.equation(block: false, alt: "y")[$y$]-intercept #math.equation(block: false, alt: "open parenthesis 0 , minus 1 close parenthesis")[$( 0 , − 1 )$]; the #math.equation(block: false, alt: "t")[$t$]-axis is the horizontal asymptote. + #math.equation(block: false, alt: "y equals 6 minus open parenthesis the fraction 1 over 2 close parenthesis to the power t")[$y = 6 − attach(( display(frac(1, 2)) ), t: t)$],#figure(figph[shifted reflected decay], alt: "shifted reflected decay", caption: none) #linebreak() Domain: #math.equation(block: false, alt: "open parenthesis minus ∞ , ∞ close parenthesis")[$( − upright(∞) , upright(∞) )$]; range: #math.equation(block: false, alt: "open parenthesis minus ∞ , 6 close parenthesis")[$( − upright(∞) , 6 )$], #math.equation(block: false, alt: "t")[$t$]-intercept approximately #math.equation(block: false, alt: "open parenthesis minus 2.58 , 0 close parenthesis")[$( − 2.58 , 0 )$]; #math.equation(block: false, alt: "y")[$y$]-intercept #math.equation(block: false, alt: "open parenthesis 0 , 5 close parenthesis")[$( 0 , 5 )$]; horizontal asymptote is #math.equation(block: false, alt: "y equals 6")[$y = 6$] #math.equation(block: true, alt: "P open parenthesis t close parenthesis equals 0.4 to the power t")[$P ( t ) = 0.4^(t)$] + #math.equation(block: false, alt: "y equals minus P open parenthesis t close parenthesis")[$y = − P ( t )$] + #math.equation(block: false, alt: "y equals 8 minus P open parenthesis t close parenthesis")[$y = 8 − P ( t )$] For Problems 61–64, + Describe the graph as a transformation of #math.equation(block: false, alt: "y equals 2 to the power x")[$y = 2^(x)$]. + Give an equation for the function graphed. #figure(figph[shifted decay], alt: "shifted decay", caption: none) + The graph of #math.equation(block: false, alt: "y equals 2 to the power x")[$y = 2^(x)$] has been reflected about the #math.equation(block: false, alt: "y")[$y$]-axis and shifted up #math.equation(block: false, alt: "2")[$2$] units. + #math.equation(block: false, alt: "y equals 2 to the power minus x plus 2")[$y = 2^(− x) + 2$] #figure(figph[decay], alt: "decay", caption: none) #figure(figph[reflected and translated growth], alt: "reflected and translated growth", caption: none) + The graph of #math.equation(block: false, alt: "y equals 2 to the power x")[$y = 2^(x)$] has been reflected about the #math.equation(block: false, alt: "x")[$x$]-axis and shifted up #math.equation(block: false, alt: "10")[$10$] units. + #math.equation(block: false, alt: "y equals minus 2 to the power x plus 10")[$y = − 2^(x) + 10$] #figure(figph[growth], alt: "growth", caption: none) For Problems 65 and 66, match the graph of each function to its formula. In each formula, #math.equation(block: false, alt: "a greater than 0")[$a > 0$] and #math.equation(block: false, alt: "b greater than 1")[$b > 1$]. + #math.equation(block: false, alt: "y equals a b to the power x")[$y = a b^(x)$] + #math.equation(block: false, alt: "y equals a b to the power minus x")[$y = a b^(− x)$] + #math.equation(block: false, alt: "y equals a x to the power b")[$y = a x^(b)$] #figure(figph[three curves], alt: "three curves", caption: none) + I + III + II + #math.equation(block: false, alt: "y equals a x to the power minus b")[$y = a x^(− b)$] + #math.equation(block: false, alt: "y equals minus a b to the power x")[$y = − a b^(x)$] + #math.equation(block: false, alt: "y equals a x to the power 1 / b")[$y = a x^(1 / b)$] #figure(figph[three curves], alt: "three curves", caption: none) The function #math.equation(block: false, alt: "f open parenthesis t close parenthesis")[$f ( t )$] describes a volunteer's heart rate during a treadmill test. #math.equation(block: true, alt: "f open parenthesis t close parenthesis equals open brace 100, 0 less than or equal to t less than 3; 56 t minus 68, 3 less than or equal to t less than 4; 186 minus 500 open parenthesis 0.5 close parenthesis to the power t, 4 less than or equal to t less than 9; 100 plus 6.6 open parenthesis 0.6 close parenthesis to the power t minus 14, 9 less than or equal to t less than 20")[$f ( t ) = \{ 100 & 0 ≤ t < 3 \ 56 t − 68 & 3 ≤ t < 4 \ 186 − 500 ( 0.5 )^(t) & 4 ≤ t < 9 \ 100 + 6.6 ( 0.6 )^(t − 14) & 9 ≤ t < 20$] The heart rate is given in beats per minute and #math.equation(block: false, alt: "t")[$t$] is in minutes. (See Some Basic Functions to review functions defined piecewise.) (Source: Davis, Kimmet, and Autry, 1986) + Evaluate the function to complete the table.#figure(table( columns: 6, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "t")[$t$]], [#math.equation(block: false, alt: "3.5")[$3.5$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "8")[$8$]], [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "15")[$15$]]), [#math.equation(block: false, alt: "f open parenthesis t close parenthesis")[$f ( t )$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], )) + Sketch the graph of the function. + The treadmill test began with walking at #math.equation(block: false, alt: "5.5")[$5.5$] kilometers per hour, then jogging, starting at #math.equation(block: false, alt: "12")[$12$] kilometers per hour and increasing to #math.equation(block: false, alt: "14")[$14$] kilometers per hour, and finished with a cool-down walking period. Identify each of these activities on the graph and describe the volunteer's heart rate during each phase. + #figure(table( columns: 6, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "t")[$t$]], [#math.equation(block: false, alt: "3.5")[$3.5$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "8")[$8$]], [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "15")[$15$]]), [#math.equation(block: false, alt: "f open parenthesis t close parenthesis")[$f ( t )$]], [#math.equation(block: false, alt: "128")[$128$]], [#math.equation(block: false, alt: "154.75")[$154.75$]], [#math.equation(block: false, alt: "184.05")[$184.05$]], [#math.equation(block: false, alt: "150.93")[$150.93$]], [#math.equation(block: false, alt: "103.96")[$103.96$]], )) + #figure(figph[piecewise], alt: "piecewise", caption: none) + From #math.equation(block: false, alt: "0")[$0$] to #math.equation(block: false, alt: "3")[$3$] minutes, the volunteer is walking with heart rate #math.equation(block: false, alt: "100")[$100$] beats per minute. The volunteer jogged at a steady pace from #math.equation(block: false, alt: "3")[$3$] to #math.equation(block: false, alt: "4")[$4$] minutes, and the heart rate increased to about #math.equation(block: false, alt: "155")[$155$] beats per minutes. From #math.equation(block: false, alt: "4")[$4$] to #math.equation(block: false, alt: "9")[$9$] minutes, the jogging pace increased, and the heart rate rose to about #math.equation(block: false, alt: "185")[$185$] beats per minute. The cooldown started at #math.equation(block: false, alt: "9")[$9$] minutes, and the heart rate decreased rapidly and leveled off to about #math.equation(block: false, alt: "100")[$100$] beats per minute. Carbon dioxide (#math.equation(block: false, alt: "CO sub 2")[$"CO"_(2)$]) is called a greenhouse gas because it traps part of the Earth's outgoing energy. Animals release #math.equation(block: false, alt: "CO sub 2")[$"CO"_(2)$] into the atmosphere, and plants remove #math.equation(block: false, alt: "CO sub 2")[$"CO"_(2)$] through photosynthesis. In modern times, deforestation and the burning of fossil fuels both contribute to #math.equation(block: false, alt: "CO sub 2")[$"CO"_(2)$] levels. The figure shows atmospheric concentrations of #math.equation(block: false, alt: "CO sub 2")[$"CO"_(2)$], in parts per million, measured at the Mauna Loa Observatory in Hawaii. + The red curve shows annual oscillations in #math.equation(block: false, alt: "CO sub 2")[$"CO"_(2)$] levels. Can you explain why #math.equation(block: false, alt: "CO sub 2")[$"CO"_(2)$] levels vary throughout the year? Hint: Why would photosynthesis vary throughout the year? + The blue curve shows the average annual #math.equation(block: false, alt: "CO sub 2")[$"CO"_(2)$] readings. By approximately how much does the #math.equation(block: false, alt: "CO sub 2")[$"CO"_(2)$] level vary from its average value during the year? + In 1960, the average #math.equation(block: false, alt: "CO sub 2")[$"CO"_(2)$] level was #math.equation(block: false, alt: "316.75")[$316.75$] parts per million, and the average level has been rising by #math.equation(block: false, alt: "0.4 %")[$0.4 upright(%)$] per year. If the level continues to rise at this rate, what #math.equation(block: false, alt: "CO sub 2")[$"CO"_(2)$] readings can we expect in the year 2100? #figure(figph[CO2 concentrations], alt: "CO2 concentrations", caption: none)