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== 7.1#h(0.6em)Transformations of Graphs


    In Chapter 4 we saw that the amplitude, period, and midline of a sinusoidal graph are determined by the coefficients in its formula. The #strong[circular functions] (sine and cosine of real numbers) behave the same way.


    
      
=== Period, Midline, and Amplitude


      Changes to the amplitude, period, and midline are called #strong[transformations] of the basic sine and cosine graphs.


      The transformations this section studies, applied live to y = a sin x + k against the basic graph y = sin x (gray). The slider a stretches the wave vertically: the amplitude is |a|, and dragging a negative reflects the graph about its midline — compare a = 2 with a = −2. The slider k slides the whole wave up or down: the dashed line y = k is the midline, and the graph always stays within |a| of it. Neither slider changes where the wave crosses its midline — the period stays 2π — which is exactly why amplitude, midline, and period are three independent transformations.{"functions":\[{"color":"\#6a6a6a","expression":"sin(x)","expression2":"","id":"basic-sine","inequality":"lt","lineDash":\[6,4\],"lineWidth":1.5,"mode":"cartesian","tMax":10,"tMin":-10,"variable":"x","visible":true},{"color":"\#1f4e79","expression":"a\*sin(x) + k","expression2":"","id":"transformed","inequality":"lt","lineDash":\[\],"lineWidth":2.5,"mode":"cartesian","tMax":10,"tMin":-10,"variable":"x","visible":true},{"color":"\#b8442a","expression":"k","expression2":"","id":"midline","inequality":"lt","lineDash":\[8,5\],"lineWidth":1.5,"mode":"cartesian","tMax":10,"tMin":-10,"variable":"x","visible":true}\],"grid":{"majorSpacing":1,"showAxes":true,"showLabels":true},"parameters":\[{"id":"p-a","label":"Amplitude a","max":3,"min":-3,"name":"a","step":0.1,"value":2},{"id":"p-k","label":"Midline k","max":2,"min":-2,"name":"k","step":0.1,"value":0}\],"title":"y = a sin x + k: amplitude a, midline y = k","version":1,"viewport":{"centerX":0,"centerY":0,"scale":46}}- Changing the midline shifts the graph vertically.
- Changing the amplitude stretches or compresses the graph vertically.
- Changing the period stretches or compresses the graph horizontally.


      First, we'll consider changes in amplitude.


      #examplebox("Example 1")[][
Compare the graphs of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 2 sin ⁡ open parenthesis x close parenthesis")[$" " f ( x ) = 2 sin ( x ) " "$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals 0.5 sin ⁡ open parenthesis x close parenthesis")[$" " g ( x ) = 0.5 sin ( x ) " "$] with the graph of #math.equation(block: false, alt: "y equals sin ⁡ open parenthesis x close parenthesis")[$" " y = sin ( x )$].


        
        #solutionbox[
Use technology to graph the three functions for #math.equation(block: false, alt: "minus 2 π less than or equal to x less than or equal to 2 π")[$− 2 π ≤ x ≤ 2 π$]. The graphs are shown below.


          #figure(figph[sineusoidal graphs], alt: "sineusoidal graphs", caption: none)


          All three graphs have the same period (#math.equation(block: false, alt: "2 π")[$2 π$]) and midline (#math.equation(block: false, alt: "y equals 0")[$y = 0$]), but the graph of #math.equation(block: false, alt: "f")[$f$] has amplitude 2, and the graph of #math.equation(block: false, alt: "g")[$g$] has amplitude 0.5.
]
]


      The amplitude of #math.equation(block: false, alt: "y equals A sin ⁡ open parenthesis t close parenthesis")[$" " y = A sin ( t ) " "$] is given by #math.equation(block: false, alt: "vertical bar A vertical bar")[$| A |$], and the same is true of #math.equation(block: false, alt: "y equals A cos ⁡ open parenthesis t close parenthesis")[$" " y = A cos ( t )$].  In the next exercise, remember that the amplitude is always a nonnegative number.


      
        Compare the graphs of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 3 cos ⁡ open parenthesis x close parenthesis")[$" " f ( x ) = 3 cos ( x ) " "$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals minus 3 cos ⁡ open parenthesis x close parenthesis")[$" " g ( x ) = − 3 cos ( x ) " "$] with the graph of #math.equation(block: false, alt: "y equals cos ⁡ open parenthesis x close parenthesis")[$" " y = cos ( x )$].


        
          Both graphs have amplitude 3. The graph of #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals minus 3 cos ⁡ open parenthesis x close parenthesis")[$" " g ( x ) = − 3 cos ( x ) " "$] is reflected about the #math.equation(block: false, alt: "x")[$x$]-axis.


          #figure(figph[cosines], alt: "cosines", caption: none)


        
      
      Next, we'll consider changes in the period of the graph.


      #examplebox("Example 2")[][
Compare the graphs of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals cos ⁡ open parenthesis 2 x close parenthesis")[$" " f ( x ) = cos ( 2 x ) " "$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals cos ⁡ open parenthesis the fraction 1 over 3 x close parenthesis")[$" " g ( x ) = cos ( display(frac(1, 3)) x ) " "$] with the graph of #math.equation(block: false, alt: "y equals cos ⁡ open parenthesis x close parenthesis")[$" " y = cos ( x )$].


        
        #solutionbox[
Use technology to graph #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals cos ⁡ open parenthesis 2 x close parenthesis")[$f ( x ) = cos ( 2 x )$] and #math.equation(block: false, alt: "y equals cos ⁡ open parenthesis x close parenthesis")[$y = cos ( x )$] for #math.equation(block: false, alt: "minus 2 π less than or equal to x less than or equal to 2 π")[$− 2 π ≤ x ≤ 2 π$], as shown below.


          #figure(figph[cos(x) and cos(2x)], alt: "cos(x) and cos(2x)", caption: none)


          Both graphs have the same amplitude (#math.equation(block: false, alt: "1")[$1$]) and midline (#math.equation(block: false, alt: "y equals 0")[$y = 0$]), but the graph of #math.equation(block: false, alt: "f")[$f$] completes two cycles from #math.equation(block: false, alt: "0")[$0$] to #math.equation(block: false, alt: "2 π")[$2 π$] instead of one. The period of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals cos ⁡ open parenthesis 2 x close parenthesis")[$" " f ( x ) = cos ( 2 x ) " "$] is #math.equation(block: false, alt: "the fraction 2 π over 2 equals π")[$display(frac(2 π, 2)) = π$].


          Now graph #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals cos ⁡ open parenthesis the fraction 1 over 3 x close parenthesis")[$" " g ( x ) = cos ( display(frac(1, 3)) x ) " "$] and #math.equation(block: false, alt: "y equals cos ⁡ open parenthesis x close parenthesis")[$" " y = cos ( x ) " "$] for #math.equation(block: false, alt: "0 less than or equal to x less than or equal to 6 π")[$0 ≤ x ≤ 6 π$].


          #figure(figph[sines], alt: "sines", caption: none)


          The graph of #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals cos ⁡ open parenthesis the fraction 1 over 3 x close parenthesis")[$" " g ( x ) = cos ( display(frac(1, 3)) x ) " "$] completes one cycle between #math.equation(block: false, alt: "0")[$0$] and #math.equation(block: false, alt: "6 π")[$6 π$]. Its period is #math.equation(block: false, alt: "the fraction 2 π over the fraction 1 over 3 equals 6 π")[$display(frac(2 π, frac(1, 3))) = 6 π$].
]
]


      The period of #math.equation(block: false, alt: "y equals cos ⁡ open parenthesis B t close parenthesis")[$" " y = cos ( B t ) " "$] is given by #math.equation(block: false, alt: "the fraction 2 π over vertical bar B vertical bar")[$display(frac(2 π, | B |))$], and the same is true of #math.equation(block: false, alt: "y equals sin ⁡ open parenthesis B t close parenthesis")[$" " y = sin ( B t )$].


      
        + Compare the graph of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals sin ⁡ open parenthesis 3 x close parenthesis")[$" " f ( x ) = sin ( 3 x ) " "$] with the graph of #math.equation(block: false, alt: "y equals sin ⁡ open parenthesis x close parenthesis")[$" " y = sin ( x ) " "$] for #math.equation(block: false, alt: "0 less than or equal to x less than or equal to 2 π")[$0 ≤ x ≤ 2 π$].
+ Compare the graph of #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals sin ⁡ open parenthesis the fraction 1 over 4 x close parenthesis")[$" " g ( x ) = sin ( display(frac(1, 4)) x ) " "$] with the graph of #math.equation(block: false, alt: "y equals sin ⁡ open parenthesis x close parenthesis")[$" " y = sin ( x ) " "$] for #math.equation(block: false, alt: "0 less than or equal to x less than or equal to 8 π")[$0 ≤ x ≤ 8 π$].


        
          + The graph of #math.equation(block: false, alt: "f")[$f$] completes 3 cycles from #math.equation(block: false, alt: "0")[$0$] to #math.equation(block: false, alt: "2 π")[$2 π$]. Its period is #math.equation(block: false, alt: "the fraction 2 π over 3")[$display(frac(2 π, 3))$].
+ The graph of #math.equation(block: false, alt: "g")[$g$] completes one cycle from #math.equation(block: false, alt: "0")[$0$] to #math.equation(block: false, alt: "8 π")[$8 π$]. Its period is #math.equation(block: false, alt: "8 π")[$8 π$].


        
      
      Next we'll consider changes in midline.


      #examplebox("Example 3")[][
Compare the graph of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 2 plus sin ⁡ open parenthesis x close parenthesis")[$" " f ( x ) = 2 + sin ( x ) " "$] with the graph of #math.equation(block: false, alt: "y equals sin ⁡ open parenthesis x close parenthesis")[$" " y = sin ( x ) " "$].


        
        #solutionbox[
Graph both functions #math.equation(block: false, alt: "minus 2 π less than or equal to x less than or equal to 2 π")[$" " − 2 π ≤ x ≤ 2 π$]. The graphs are shown below.


          #figure(figph[sin(x) and 2+sin(x)], alt: "sin(x) and 2+sin(x)", caption: none)


          Each point on the graph of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 2 plus sin ⁡ open parenthesis x close parenthesis")[$" " f ( x ) = 2 + sin ( x ) " "$] has #math.equation(block: false, alt: "y")[$y$]-coordinate 2 units higher than the corresponding point on the graph of #math.equation(block: false, alt: "y equals sin ⁡ open parenthesis x close parenthesis")[$" " y = sin ( x ) " "$]. Thus, the graph of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 2 plus sin ⁡ open parenthesis x close parenthesis")[$" " f ( x ) = 2 + sin ( x ) " "$] is shifted vertically by 2 units relative to the graph of #math.equation(block: false, alt: "y equals sin ⁡ open parenthesis x close parenthesis")[$" " y = sin ( x ) " "$]. In particular, the midline of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 2 plus sin ⁡ open parenthesis x close parenthesis")[$" " f ( x ) = 2 + sin ( x ) " "$] is the line #math.equation(block: false, alt: "y equals 2")[$y = 2$].
]
]


      
        Compare the graph of #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals minus 3 plus cos ⁡ open parenthesis x close parenthesis")[$" " g ( x ) = − 3 + cos ( x ) " "$] with the graph of #math.equation(block: false, alt: "y equals cos ⁡ open parenthesis x close parenthesis")[$" " y = cos ( x )$].


        
          The graph of #math.equation(block: false, alt: "g")[$g$] is shifted down 3 units. Its midline is #math.equation(block: false, alt: "y equals minus 3")[$y = − 3$].


        
      
      Here is a summary of our findings.


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Amplitude, Period, and Midline of Sinusoidal Functions]


        + The graph of
		#math.equation(block: true, alt: "y equals A cos ⁡ open parenthesis x close parenthesis or y equals A sin ⁡ open parenthesis x close parenthesis")[$y = A cos ( x ) " " " " " " " " "or" " " " " " " " " y = A sin ( x )$]
		has #strong[amplitude] #math.equation(block: false, alt: "vertical bar A vertical bar")[$| A |$].
+ The graph of
		#math.equation(block: true, alt: "y equals cos ⁡ open parenthesis B x close parenthesis or y equals sin ⁡ open parenthesis B x close parenthesis")[$y = cos ( B x ) " " " " " " " " "or" " " " " " " " " y = sin ( B x )$]
		has #strong[period] #math.equation(block: false, alt: "the fraction 2 π over B")[$display(frac(2 π, B))$].
+ The graph of
		#math.equation(block: true, alt: "y equals k plus cos ⁡ open parenthesis x close parenthesis or y equals k plus sin ⁡ open parenthesis x close parenthesis")[$y = k + cos ( x ) " " " " " " " " "or" " " " " " " " " y = k + sin ( x )$]
		has #strong[midline] #math.equation(block: false, alt: "y equals k")[$y = k$].
]


    
    
      
=== Graphs of Sinusoidal Functions


      The values of the parameters #math.equation(block: false, alt: "A , B , and k")[$A , " " B , " " "and" " " k$]  determine the shape of the graphs of


      #math.equation(block: true, alt: "y equals k plus A sin ⁡ open parenthesis B x close parenthesis or y equals k plus A cos ⁡ open parenthesis B x close parenthesis")[$y = k + A sin ( B x ) " " " " " " " " "or" " " " " " " " " y = k + A cos ( B x )$]
      By adjusting the amplitude, period, and midline of the sine or cosine graph, we can sketch these sinusoidal functions.


      #examplebox("Example 4")[][
+ State the amplitude, period, and midline of #math.equation(block: false, alt: "y equals 2 plus 3 cos ⁡ open parenthesis 4 t close parenthesis")[$" " y = 2 + 3 cos ( 4 t )$].
+ Sketch by hand a graph of #math.equation(block: false, alt: "y equals 2 plus 3 cos ⁡ open parenthesis 4 t close parenthesis")[$" " y = 2 + 3 cos ( 4 t )$].


        
        #solutionbox[
+ The amplitude of the graph is 3, its midline is #math.equation(block: false, alt: "y equals 2")[$y = 2$], and its period is #math.equation(block: false, alt: "the fraction 2 π over 4 equals the fraction π over 2")[$display(frac(2 π, 4)) = display(frac(π, 2))$].
+ One way to make a quick sketch of a sinusoidal graph is to use a table of values. The trick is to choose convenient values for the input variable. In the table below, notice that we choose the quadrantal angles as the input values for the trigonometric function.#figure(table(
  columns: 5, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "t")[$t$]], [#math.equation(block: false, alt: "4 t")[$4 t$]], [#math.equation(block: false, alt: "cos ⁡ open parenthesis 4 t close parenthesis")[$cos ( 4 t )$]], [#math.equation(block: false, alt: "3 cos ⁡ open parenthesis 4 t close parenthesis")[$3 cos ( 4 t )$]], [#math.equation(block: false, alt: "y equals 2 plus cos ⁡ open parenthesis 4 t close parenthesis")[$y = 2 + cos ( 4 t )$]]),
  [$#hide($0000$)$], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], [$#hide($0000$)$], [$#hide($0000$)$],
  [$#hide($0000$)$], [#math.equation(block: false, alt: "the fraction π over 2")[$display(frac(bold(italic(π)), 2))$]], [#math.equation(block: false, alt: "0")[$0$]], [$#hide($0000$)$], [$#hide($0000$)$],
  [$#hide($0000$)$], [#math.equation(block: false, alt: "π")[$bold(italic(π))$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [$#hide($0000$)$], [$#hide($0000$)$],
  [$#hide($0000$)$], [#math.equation(block: false, alt: "the fraction 3 π over 2")[$display(frac(3 bold(italic(π)), 2))$]], [#math.equation(block: false, alt: "0")[$0$]], [$#hide($0000$)$], [$#hide($0000$)$],
  [$#hide($0000$)$], [#math.equation(block: false, alt: "2 π")[$2 bold(italic(π))$]], [#math.equation(block: false, alt: "1")[$1$]], [$#hide($0000$)$], [$#hide($0000$)$],
))

 #linebreak() Now we work backwards from #math.equation(block: false, alt: "4 t")[$4 t$] to find the values of #math.equation(block: false, alt: "t")[$t$], and forwards from #math.equation(block: false, alt: "cos ⁡ open parenthesis 4 t close parenthesis")[$cos ( 4 t )$] to find the values of #math.equation(block: false, alt: "y")[$y$].#figure(table(
  columns: 5, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "t")[$t$]], [#math.equation(block: false, alt: "4 t")[$4 t$]], [#math.equation(block: false, alt: "cos ⁡ open parenthesis 4 t close parenthesis")[$cos ( 4 t )$]], [#math.equation(block: false, alt: "3 cos ⁡ open parenthesis 4 t close parenthesis")[$3 cos ( 4 t )$]], [#math.equation(block: false, alt: "y equals 2 plus cos ⁡ open parenthesis 4 t close parenthesis")[$y = 2 + cos ( 4 t )$]]),
  [#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: "3")[$3$]], [#math.equation(block: false, alt: "5")[$5$]],
  [#math.equation(block: false, alt: "the fraction π over 8")[$display(frac(bold(italic(π)), 8))$]], [#math.equation(block: false, alt: "the fraction π over 2")[$display(frac(π, 2))$]], [#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: "the fraction π over 4")[$display(frac(bold(italic(π)), 4))$]], [#math.equation(block: false, alt: "π")[$π$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]],
  [#math.equation(block: false, alt: "the fraction 3 π over 8")[$display(frac(3 bold(italic(π)), 8))$]], [#math.equation(block: false, alt: "the fraction 3 π over 2")[$display(frac(3 π, 2))$]], [#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: "the fraction π over 2")[$display(frac(bold(italic(π)), 2))$]], [#math.equation(block: false, alt: "2 π")[$2 π$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "5")[$5$]],
))

 #linebreak() Notice from the table that the graph completes one cycle from #math.equation(block: false, alt: "t equals 0")[$t = 0$] to #math.equation(block: false, alt: "t equals the fraction π over 2")[$t = display(frac(π, 2))$], which confirms that the period is #math.equation(block: false, alt: "the fraction π over 2")[$display(frac(π, 2))$]. Finally, we plot the points #math.equation(block: false, alt: "open parenthesis t , y close parenthesis")[$( t , y )$] from the table, and use them as "guidepoints" to sketch a sinusoidal graph, as shown below.#figure(figph[Graph of y=2+3cos(4t)], alt: "Graph of y=2+3cos(4t)", caption: none)
]
]


      
        + State the amplitude, period, and midline of the graph of #math.equation(block: false, alt: "y equals 4 minus 2 sin ⁡ open parenthesis the fraction t over 3 close parenthesis")[$" " y = 4 − 2 sin ( display(frac(t, 3)) )$].
+ Complete the table and sketch a graph of #math.equation(block: false, alt: "y equals 4 minus 2 sin ⁡ open parenthesis the fraction t over 3 close parenthesis")[$" " y = 4 − 2 sin ( display(frac(t, 3)) )$].#figure(table(
  columns: 5, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "t")[$t$]], [#math.equation(block: false, alt: "the fraction t over 3")[$display(frac(t, 3))$]], [#math.equation(block: false, alt: "sin ⁡ open parenthesis the fraction t over 3 close parenthesis")[$sin ( display(frac(t, 3)) )$]], [#math.equation(block: false, alt: "minus 2 sin ⁡ open parenthesis the fraction t over 3 close parenthesis")[$− 2 sin ( display(frac(t, 3)) )$]], [#math.equation(block: false, alt: "y equals 4 minus 2 sin ⁡ open parenthesis the fraction t over 3 close parenthesis")[$y = 4 − 2 sin ( display(frac(t, 3)) )$]]),
  [$#hide($00$)$], [#math.equation(block: false, alt: "0")[$0$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
  [], [#math.equation(block: false, alt: "the fraction π over 2")[$display(frac(bold(italic(π)), 2))$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
  [], [#math.equation(block: false, alt: "π")[$bold(italic(π))$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
  [], [#math.equation(block: false, alt: "the fraction 3 π over 2")[$display(frac(3 bold(italic(π)), 2))$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
  [], [#math.equation(block: false, alt: "2 π")[$2 bold(italic(π))$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
))

#figure(figph[grid], alt: "grid", caption: none)


        
          + Amplitude: 2, period: #math.equation(block: false, alt: "6 π")[$6 π$], midline: #math.equation(block: false, alt: "y equals 4")[$y = 4$].
+ #figure(table(
  columns: 5, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "t")[$t$]], [#math.equation(block: false, alt: "the fraction t over 3")[$display(frac(t, 3))$]], [#math.equation(block: false, alt: "sin ⁡ open parenthesis the fraction t over 3 close parenthesis")[$sin ( display(frac(t, 3)) )$]], [#math.equation(block: false, alt: "minus 2 sin ⁡ open parenthesis the fraction t over 3 close parenthesis")[$− 2 sin ( display(frac(t, 3)) )$]], [#math.equation(block: false, alt: "y equals 4 minus 2 sin ⁡ open parenthesis the fraction t over 3 close parenthesis")[$y = 4 − 2 sin ( display(frac(t, 3)) )$]]),
  [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "4")[$4$]],
  [#math.equation(block: false, alt: "the fraction 3 π over 2")[$display(frac(3 π, 2))$]], [#math.equation(block: false, alt: "the fraction π over 2")[$display(frac(π, 2))$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "2")[$2$]],
  [#math.equation(block: false, alt: "3 π")[$3 π$]], [#math.equation(block: false, alt: "π")[$π$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "4")[$4$]],
  [#math.equation(block: false, alt: "the fraction 9 π over 2")[$display(frac(9 π, 2))$]], [#math.equation(block: false, alt: "the fraction 3 π over 2")[$display(frac(3 π, 2))$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "6")[$6$]],
  [#math.equation(block: false, alt: "6 π")[$6 π$]], [#math.equation(block: false, alt: "2 π")[$2 π$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "4")[$4$]],
))

#figure(figph[Graph of y=4-2sin(t/3)], alt: "Graph of y=4-2sin(t/3)", caption: none)


        
      
    
    
      
=== Modeling with Sinusoidal Functions


      Sinusoidal functions are used to model a great variety of physical phenomena, including sound and light waves, tides and planetary orbits, and the life cycles of plants and animals. They are also often used to approximate periodic functions that are not exactly sinusoidal, such as blood pressure.


      #examplebox("Example 5")[][
A typical blood pressure for a healthy adult, measured in millimeters of mercury, varies between 70 and 110, and a typical heart rate is 60 beats per minute. Write a sinusoidal function that approximates blood pressure, and sketch its graph.


        
        #solutionbox[
We would like a function of the form #math.equation(block: false, alt: "y equals k plus a sin ⁡ open parenthesis B t close parenthesis")[$" " y = k + a sin ( B t )$], so we must find the values of the parameters #math.equation(block: false, alt: "A , B")[$A , " " B$] and #math.equation(block: false, alt: "k")[$k$].


          - The midline of the graph is #math.equation(block: false, alt: "y equals the fraction 70 plus 110 over 2 equals 90")[$y = display(frac(70 + 110, 2)) = 90$], and the amplitude is #math.equation(block: false, alt: "110 minus 90 equals 20")[$110 − 90 = 20$], so #math.equation(block: false, alt: "A equals 20")[$A = 20$] and #math.equation(block: false, alt: "k equals 90")[$k = 90$].
- The graph repeats 60 times per minute, so the period is #math.equation(block: false, alt: "the fraction 1 over 60")[$display(frac(1, 60))$] minute, and #math.equation(block: false, alt: "B equals the fraction 2 π over the fraction 1 over 60 equals 120 π")[$B = display(frac(2 π, frac(1, 60))) = 120 π$].


          Thus,


          #math.equation(block: true, alt: "y equals 90 plus 20 sin ⁡ open parenthesis 120 π t close parenthesis")[$y = 90 + 20 sin ( 120 π t )$]
          The graph of the function is shown below.


          #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)
]
]


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
In the preceding Example, we could have chosen either a sine or a cosine function to model blood pressure; both describe the periodic behavior described. However, if we are given, or would like to specify, the starting point for a sinusoidal function, one choice can be more suitable than the other. Consider the functions graphed below.


        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


        All four functions have the same amplitude and period, but they start at different points on the cycle.


        - The graphs in (a) and (b) start on the midline, so they are best modeled by sine functions.
- The graph in (b) starts out decreasing instead of increasing, so the coefficient #math.equation(block: false, alt: "A")[$A$] is negative.
- The graph in (c) is modeled by a cosine, because it starts at the maximum point, and the graph in (d) starts at the minimum point, so we choose a negative cosine to model it.
]


      (In The General Sinusoidal Function, we'll consider sinusoidal functions that start at other positions on the cycle.)


      In the next Checkpoint, note the starting point of the graph, and choose the most appropriate sinusoidal function to model the function.


      
        The graph below shows the voltage of a generator, as seen on an oscilloscope.


        + Write a sinusoidal function for the voltage level.
+ What is the frequency of the signal, in cycles per second?


        #figure(figph[sinusoidal function], alt: "sinusoidal function", caption: none)


        
          + #math.equation(block: false, alt: "y equals minus 35 cos ⁡ open parenthesis 100 π t close parenthesis")[$y = − 35 cos ( 100 π t )$]
+ 50 cycles per second


        
      
    
    
      
=== The Tangent Function


      The transformations of shifting and stretching can be applied to the tangent function as well. The graph of #math.equation(block: false, alt: "y equals tan ⁡ x")[$" " y = tan x " "$] does not have an amplitude, but we can see any vertical stretch by comparing the function values at the guidepoints.


      #examplebox("Example 6")[][
+ Graph #math.equation(block: false, alt: "y equals 1 plus 3 tan ⁡ open parenthesis 2 x close parenthesis")[$" " y = 1 + 3 tan ( 2 x )$].
+ Describe the transformations of the graph, compared to #math.equation(block: false, alt: "y equals tan ⁡ open parenthesis x close parenthesis")[$" " y = tan ( x )$].


        
        #solutionbox[
+ Recall that the period of the tangent function is #math.equation(block: false, alt: "π")[$π$].  We make a table of values for one cycle of the function, choosing multiples of #math.equation(block: false, alt: "the fraction π over 4")[$display(frac(π, 4))$] as the inputs for the tangent function. Then we plot the guidepoints, and sketch a tangent function through them. The graph is shown below.#figure(table(
  columns: 4, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "2 x")[$2 x$]], [#math.equation(block: false, alt: "tan ⁡ open parenthesis 2 x close parenthesis")[$tan ( 2 x )$]], [#math.equation(block: false, alt: "1 plus 3 tan ⁡ open parenthesis 2 x close parenthesis")[$1 + 3 tan ( 2 x )$]]),
  [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]],
  [#math.equation(block: false, alt: "the fraction π over 8")[$display(frac(π, 8))$]], [#math.equation(block: false, alt: "the fraction π over 4")[$display(frac(bold(italic(π)), 4))$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "4")[$4$]],
  [#math.equation(block: false, alt: "the fraction π over 4")[$display(frac(π, 4))$]], [#math.equation(block: false, alt: "the fraction π over 2")[$display(frac(bold(italic(π)), 2))$]], [#math.equation(block: false, alt: "minus minus minus")[$− − −$]], [#math.equation(block: false, alt: "minus minus minus")[$− − −$]],
  [#math.equation(block: false, alt: "the fraction 3 π over 8")[$display(frac(3 π, 8))$]], [#math.equation(block: false, alt: "the fraction 3 π over 4")[$display(frac(3 bold(italic(π)), 4))$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]],
  [#math.equation(block: false, alt: "the fraction π over 2")[$display(frac(π, 2))$]], [#math.equation(block: false, alt: "π")[$bold(italic(π))$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]],
))

#figure(figph[transformed tangent graph], alt: "transformed tangent graph", caption: none)
+ Writing the formula as 
            #math.equation(block: true, alt: "y equals A tan ⁡ open parenthesis B x close parenthesis plus k equals 3 tan ⁡ open parenthesis 2 x close parenthesis plus 1")[$y = A tan ( B x ) + k = 3 tan ( 2 x ) + 1$] 
    we see that the graph is stretched vertically by a factor of #math.equation(block: false, alt: "A equals 3")[$A = 3$]. The midline is #math.equation(block: false, alt: "y equals 1")[$y = 1$], so the graph is shifted up by 1 unit. Finally, the coefficient #math.equation(block: false, alt: "B equals 2")[$B = 2$] compresses the graph horizontally by a factor of 2, so the period of the graph is #math.equation(block: false, alt: "the fraction π over 2")[$display(frac(π, 2))$], and there are four cycles between #math.equation(block: false, alt: "0")[$0$] and #math.equation(block: false, alt: "2 π")[$2 π$]
]
]


      
        + Identify the midline and period of the tangent graph shown below.
+ Find an equation of the form

#math.equation(block: true, alt: "y equals A tan ⁡ open parenthesis B x close parenthesis plus k")[$y = A tan ( B x ) + k$]for the graph.

#figure(figph[transformed tangent graph], alt: "transformed tangent graph", caption: none)


        
          + Midline: #math.equation(block: false, alt: "y equals minus 2")[$y = − 2$], period #math.equation(block: false, alt: "equals 1")[$= 1$]
+ #math.equation(block: false, alt: "y equals minus 3 plus 2 tan ⁡ open parenthesis π x close parenthesis")[$y = − 3 + 2 tan ( π x )$]


        
      
      Review the following skills you will need for this section.


      #notebox("Algebra Refresher", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
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: "0")[$0$]], [#math.equation(block: false, alt: "the fraction π over 2")[$display(frac(π, 2))$]], [#math.equation(block: false, alt: "π")[$π$]], [#math.equation(block: false, alt: "the fraction 3 π over 2")[$display(frac(3 π, 2))$]], [#math.equation(block: false, alt: "2 π")[$2 π$]]),
  [#math.equation(block: false, alt: "t plus the fraction π over 6")[$t + display(frac(π, 6))$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
))
+ #figure(table(
  columns: 6, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "t")[$t$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "the fraction π over 2")[$display(frac(π, 2))$]], [#math.equation(block: false, alt: "π")[$π$]], [#math.equation(block: false, alt: "the fraction 3 π over 2")[$display(frac(3 π, 2))$]], [#math.equation(block: false, alt: "2 π")[$2 π$]]),
  [#math.equation(block: false, alt: "t minus the fraction π over 6")[$t − display(frac(π, 6))$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
))
+ #figure(table(
  columns: 6, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "the fraction π over 2")[$display(frac(π, 2))$]], [#math.equation(block: false, alt: "π")[$π$]], [#math.equation(block: false, alt: "the fraction 3 π over 2")[$display(frac(3 π, 2))$]], [#math.equation(block: false, alt: "2 π")[$2 π$]]),
  [#math.equation(block: false, alt: "x minus the fraction π over 3")[$x − display(frac(π, 3))$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
))
+ #figure(table(
  columns: 6, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "the fraction π over 2")[$display(frac(π, 2))$]], [#math.equation(block: false, alt: "π")[$π$]], [#math.equation(block: false, alt: "the fraction 3 π over 2")[$display(frac(3 π, 2))$]], [#math.equation(block: false, alt: "2 π")[$2 π$]]),
  [#math.equation(block: false, alt: "x plus the fraction π over 3")[$x + display(frac(π, 3))$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
))


        #math.equation(block: true, alt: "bar")[$underline(#h(2em) #h(2em) #h(2em) #h(2em))$]


        Algebra Refresher Answers


        + #figure(table(
  columns: 6, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "t")[$t$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "the fraction π over 2")[$display(frac(π, 2))$]], [#math.equation(block: false, alt: "π")[$π$]], [#math.equation(block: false, alt: "the fraction 3 π over 2")[$display(frac(3 π, 2))$]], [#math.equation(block: false, alt: "2 π")[$2 π$]]),
  [#math.equation(block: false, alt: "t plus the fraction π over 6")[$t + display(frac(π, 6))$]], [#math.equation(block: false, alt: "the fraction π over 6")[$display(frac(π, 6))$]], [#math.equation(block: false, alt: "the fraction 2 π over 3")[$display(frac(2 π, 3))$]], [#math.equation(block: false, alt: "the fraction 7 π over 6")[$display(frac(7 π, 6))$]], [#math.equation(block: false, alt: "the fraction 5 π over 3")[$display(frac(5 π, 3))$]], [#math.equation(block: false, alt: "the fraction 13 π over 6")[$display(frac(13 π, 6))$]],
))
+ #figure(table(
  columns: 6, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "t")[$t$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "the fraction π over 2")[$display(frac(π, 2))$]], [#math.equation(block: false, alt: "π")[$π$]], [#math.equation(block: false, alt: "the fraction 3 π over 2")[$display(frac(3 π, 2))$]], [#math.equation(block: false, alt: "2 π")[$2 π$]]),
  [#math.equation(block: false, alt: "t minus the fraction π over 6")[$t − display(frac(π, 6))$]], [#math.equation(block: false, alt: "the fraction minus π over 6")[$display(frac(− π, 6))$]], [#math.equation(block: false, alt: "the fraction π over 3")[$display(frac(π, 3))$]], [#math.equation(block: false, alt: "the fraction 5 π over 6")[$display(frac(5 π, 6))$]], [#math.equation(block: false, alt: "the fraction 4 π over 3")[$display(frac(4 π, 3))$]], [#math.equation(block: false, alt: "the fraction 11 π over 6")[$display(frac(11 π, 6))$]],
))
+ #figure(table(
  columns: 6, 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: "the fraction π over 2")[$display(frac(π, 2))$]], [#math.equation(block: false, alt: "π")[$π$]], [#math.equation(block: false, alt: "the fraction 3 π over 2")[$display(frac(3 π, 2))$]], [#math.equation(block: false, alt: "2 π")[$2 π$]]),
  [#math.equation(block: false, alt: "x minus the fraction π over 3")[$x − display(frac(π, 3))$]], [#math.equation(block: false, alt: "the fraction minus π over 3")[$display(frac(− π, 3))$]], [#math.equation(block: false, alt: "the fraction π over 6")[$display(frac(π, 6))$]], [#math.equation(block: false, alt: "the fraction 2 π over 3")[$display(frac(2 π, 3))$]], [#math.equation(block: false, alt: "the fraction 7 π over 6")[$display(frac(7 π, 6))$]], [#math.equation(block: false, alt: "the fraction 5 π over 3")[$display(frac(5 π, 3))$]],
))
+ #figure(table(
  columns: 6, 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: "the fraction π over 2")[$display(frac(π, 2))$]], [#math.equation(block: false, alt: "π")[$π$]], [#math.equation(block: false, alt: "the fraction 3 π over 2")[$display(frac(3 π, 2))$]], [#math.equation(block: false, alt: "2 π")[$2 π$]]),
  [#math.equation(block: false, alt: "x plus the fraction π over 3")[$x + display(frac(π, 3))$]], [#math.equation(block: false, alt: "the fraction π over 3")[$display(frac(π, 3))$]], [#math.equation(block: false, alt: "the fraction 5 π over 6")[$display(frac(5 π, 6))$]], [#math.equation(block: false, alt: "the fraction 4 π over 3")[$display(frac(4 π, 3))$]], [#math.equation(block: false, alt: "the fraction 11 π over 6")[$display(frac(11 π, 6))$]], [#math.equation(block: false, alt: "the fraction 7 π over 3")[$display(frac(7 π, 3))$]],
))
]


    
    
      
=== Section 7.1 Summary


      
        
==== Vocabulary


        - Transformation
- Amplitude
- Period
- Midline


      
      
        
==== Concepts


        + Changes to the amplitude, period, and midline of the basic sine and cosine graphs are called #strong[transformations].  Changing the midline shifts the graph vertically, changing the amplitude stretches or compresses the graph vertically, and changing the period stretches or compresses the graph horizontally.
+ #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Amplitude, Period, and Midline of Sinusoidal Functions]

+ The graph of
		          #math.equation(block: true, alt: "y equals A cos ⁡ open parenthesis x close parenthesis or y equals A sin ⁡ open parenthesis x close parenthesis")[$y = A cos ( x ) " " " " " " " " "or" " " " " " " " " y = A sin ( x )$]
		      has #strong[amplitude] #math.equation(block: false, alt: "vertical bar A vertical bar")[$| A |$].
+ The graph of
		          #math.equation(block: true, alt: "y equals cos ⁡ open parenthesis B close parenthesis x or y equals sin ⁡ open parenthesis B x close parenthesis")[$y = cos ( B ) x " " " " " " " " "or" " " " " " " " " y = sin ( B x )$]
		      has #strong[period] #math.equation(block: false, alt: "the fraction 2 π over B")[$display(frac(2 π, B))$].
+ The graph of
		          #math.equation(block: true, alt: "y equals k plus cos ⁡ open parenthesis x close parenthesis or y equals k plus sin ⁡ open parenthesis x close parenthesis")[$y = k + cos ( x ) " " " " " " " " "or" " " " " " " " " y = k + sin ( x )$]
		      has #strong[midline] #math.equation(block: false, alt: "y equals k")[$y = k$].
]
+ One way to make a quick sketch of a sinusoidal graph is to use a table of values.  The trick is to choose convenient values for the input variable.
+ The transformations of shifting and stretching can be applied to the tangent function as well.


      
      
        
==== Study Questions


        + Count from #math.equation(block: false, alt: "0")[$0$] to #math.equation(block: false, alt: "2 π")[$2 π$] by multiples of #math.equation(block: false, alt: "the fraction π over 4")[$display(frac(π, 4))$].
+ Count from #math.equation(block: false, alt: "0")[$0$] to #math.equation(block: false, alt: "2 π")[$2 π$] by multiples of #math.equation(block: false, alt: "the fraction π over 6")[$display(frac(π, 6))$].
+ Transformationhe maximum value of a certain sinusoidal function is #math.equation(block: false, alt: "M")[$M$], and its minimum value is #math.equation(block: false, alt: "m")[$m$]. What is the midline of the function? What is its amplitude?
+ #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals k plus A tan ⁡ open parenthesis x close parenthesis")[$f ( x ) = k + A tan ( x )$], and #math.equation(block: false, alt: "f open parenthesis 0 close parenthesis equals 4 , f open parenthesis the fraction π over 4 close parenthesis equals 6")[$f ( 0 ) = 4 , " " " " f ( display(frac(π, 4)) ) = 6$]. What are the values of #math.equation(block: false, alt: "k")[$k$] and #math.equation(block: false, alt: "A")[$A$]?


      
      
        
==== Skills


        + Identify the amplitude, period, and midline of a circular function    \#1–8, 23–30
+ Graph a circular function    \#9–16, 31–44
+ Find a formula for the graph of a circular function    \#17–30
+ Model periodic phenomena with circular functions    \#45–52
+ Graph transformations of the tangent function    \#535–8
+ Solve trigonometric equations graphically    \#59–70


      
    
    
      
=== Homework 7-1


      For Problems 1–8, state the amplitude, period, and midline of the graph.


      
        #math.equation(block: true, alt: "y equals minus 3 plus 2 sin ⁡ open parenthesis x close parenthesis")[$y = − 3 + 2 sin ( x )$]


        
          amplitude #math.equation(block: false, alt: "2")[$2$], period #math.equation(block: false, alt: "2 π")[$2 π$], midline #math.equation(block: false, alt: "y equals minus 3")[$y = − 3$]


        
      
      
        #math.equation(block: true, alt: "y equals 4 minus 3 cos ⁡ open parenthesis x close parenthesis")[$y = 4 − 3 cos ( x )$]


      
      
        #math.equation(block: true, alt: "y equals minus cos ⁡ open parenthesis 4 x close parenthesis")[$y = − cos ( 4 x )$]


        
          amplitude #math.equation(block: false, alt: "1")[$1$], period #math.equation(block: false, alt: "the fraction π over 2")[$display(frac(π, 2))$], midline #math.equation(block: false, alt: "y equals 0")[$y = 0$]


        
      
      
        #math.equation(block: true, alt: "y equals minus sin ⁡ open parenthesis 3 x close parenthesis")[$y = − sin ( 3 x )$]


      
      
        #math.equation(block: true, alt: "y equals minus 5 sin ⁡ open parenthesis the fraction x over 3 close parenthesis")[$y = − 5 sin ( display(frac(x, 3)) )$]


        
          amplitude #math.equation(block: false, alt: "5")[$5$], period #math.equation(block: false, alt: "6 π")[$6 π$], midline #math.equation(block: false, alt: "y equals 0")[$y = 0$]


        
      
      
        #math.equation(block: true, alt: "y equals 6 cos ⁡ open parenthesis the fraction x over 2 close parenthesis")[$y = 6 cos ( display(frac(x, 2)) )$]


      
      
        #math.equation(block: true, alt: "y equals 1 minus cos ⁡ open parenthesis π x close parenthesis")[$y = 1 − cos ( π x )$]


        
          amplitude #math.equation(block: false, alt: "1")[$1$], period #math.equation(block: false, alt: "2")[$2$], midline #math.equation(block: false, alt: "y equals 1")[$y = 1$]


        
      
      
        #math.equation(block: true, alt: "y equals 2 plus sin ⁡ open parenthesis 2 π x close parenthesis")[$y = 2 + sin ( 2 π x )$]


      
      In Problems 9–16, we use transformations to sketch graphs of the functions in Problems 1–8.  Sketch one cycle of each graph by hand and label scales on the axes.


      
        + #math.equation(block: false, alt: "y equals sin ⁡ open parenthesis x close parenthesis")[$y = sin ( x )$]
+ #math.equation(block: false, alt: "y equals 2 sin ⁡ open parenthesis x close parenthesis")[$y = 2 sin ( x )$]
+ #math.equation(block: false, alt: "y equals minus 3 plus 2 sin ⁡ open parenthesis x close parenthesis")[$y = − 3 + 2 sin ( x )$]


        
          #figure(figph[transformations of sine graph], alt: "transformations of sine graph", caption: none)


        
      
      
        + #math.equation(block: false, alt: "y equals cos ⁡ open parenthesis x close parenthesis")[$y = cos ( x )$]
+ #math.equation(block: false, alt: "y equals minus 3 cos ⁡ open parenthesis x close parenthesis")[$y = − 3 cos ( x )$]
+ #math.equation(block: false, alt: "y equals 4 minus 3 cos ⁡ open parenthesis x close parenthesis")[$y = 4 − 3 cos ( x )$]


      
      
        + #math.equation(block: false, alt: "y equals cos ⁡ open parenthesis x close parenthesis")[$y = cos ( x )$]
+ #math.equation(block: false, alt: "y equals cos ⁡ open parenthesis 4 x close parenthesis")[$y = cos ( 4 x )$]
+ #math.equation(block: false, alt: "y equals minus cos ⁡ open parenthesis x close parenthesis")[$y = − cos ( x )$]


        
          #figure(figph[transformations of cosine], alt: "transformations of cosine", caption: none)


        
      
      
        + #math.equation(block: false, alt: "y equals sin ⁡ open parenthesis x close parenthesis")[$y = sin ( x )$]
+ #math.equation(block: false, alt: "y equals sin ⁡ open parenthesis 3 x close parenthesis")[$y = sin ( 3 x )$]
+ #math.equation(block: false, alt: "y equals minus sin ⁡ open parenthesis 3 x close parenthesis")[$y = − sin ( 3 x )$]


      
      
        + #math.equation(block: false, alt: "y equals sin ⁡ open parenthesis x close parenthesis")[$y = sin ( x )$]
+ #math.equation(block: false, alt: "y equals sin ⁡ open parenthesis the fraction x over 3 close parenthesis")[$y = sin ( display(frac(x, 3)) )$]
+ #math.equation(block: false, alt: "y equals minus 5 sin ⁡ open parenthesis the fraction x over 3 close parenthesis")[$y = − 5 sin ( display(frac(x, 3)) )$]


        
          #figure(figph[transformations of sine], alt: "transformations of sine", caption: none)


        
      
      
        + #math.equation(block: false, alt: "y equals cos ⁡ open parenthesis x close parenthesis")[$y = cos ( x )$]
+ #math.equation(block: false, alt: "y equals cos ⁡ open parenthesis the fraction x over 2 close parenthesis")[$y = cos ( display(frac(x, 2)) )$]
+ #math.equation(block: false, alt: "y equals 6 cos ⁡ open parenthesis the fraction x over 2 close parenthesis")[$y = 6 cos ( display(frac(x, 2)) )$]


      
      
        + #math.equation(block: false, alt: "y equals cos ⁡ open parenthesis x close parenthesis")[$y = cos ( x )$]
+ #math.equation(block: false, alt: "y equals cos ⁡ open parenthesis π x close parenthesis")[$y = cos ( π x )$]
+ #math.equation(block: false, alt: "y equals 1 minus cos ⁡ open parenthesis π x close parenthesis")[$y = 1 − cos ( π x )$]


        
          #figure(figph[cosine transformations], alt: "cosine transformations", caption: none)


        
      
      
        + #math.equation(block: false, alt: "y equals sin ⁡ open parenthesis x close parenthesis")[$y = sin ( x )$]
+ #math.equation(block: false, alt: "y equals sin ⁡ open parenthesis 2 π x close parenthesis")[$y = sin ( 2 π x )$]
+ #math.equation(block: false, alt: "y equals 2 plus sin ⁡ open parenthesis 2 π x close parenthesis")[$y = 2 + sin ( 2 π x )$]


      
      For Problems 17–22, write an equation for the graph using sine or cosine.


      
        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


        
          #math.equation(block: true, alt: "y equals minus 2 sin ⁡ open parenthesis x close parenthesis")[$y = − 2 sin ( x )$]


        
      
      
        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


      
      
        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


        
          #math.equation(block: true, alt: "y equals minus 2 cos ⁡ open parenthesis x close parenthesis")[$y = − 2 cos ( x )$]


        
      
      
        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


      
      
        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


        
          #math.equation(block: true, alt: "y equals minus 0.75 cos ⁡ open parenthesis x close parenthesis")[$y = − 0.75 cos ( x )$]


        
      
      
        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


      
      For Problems 23–30,


      + State the amplitude, period, and midline of the graph.
+ Write an equation for the graph using sine or cosine.


      
        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


        
          + amplitude #math.equation(block: false, alt: "2")[$2$], period #math.equation(block: false, alt: "the fraction 2 π over 3")[$display(frac(2 π, 3))$], midline #math.equation(block: false, alt: "y equals 0")[$y = 0$]
+ #math.equation(block: false, alt: "y equals minus 2 sin ⁡ open parenthesis 3 x close parenthesis")[$y = − 2 sin ( 3 x )$]


        
      
      
        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


      
      
        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


        
          + amplitude #math.equation(block: false, alt: "3")[$3$], period #math.equation(block: false, alt: "2 π")[$2 π$], midline #math.equation(block: false, alt: "y equals 0")[$y = 0$]
+ #math.equation(block: false, alt: "y equals 3 sin ⁡ open parenthesis the fraction x over 2 close parenthesis")[$y = 3 sin ( display(frac(x, 2)) )$]


        
      
      
        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


      
      
        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


        
          + amplitude #math.equation(block: false, alt: "0.5")[$0.5$], period #math.equation(block: false, alt: "4 π")[$4 π$], midline #math.equation(block: false, alt: "y equals 3.5")[$y = 3.5$]
+ #math.equation(block: false, alt: "y equals 0.5 cos ⁡ open parenthesis the fraction x over 2 close parenthesis plus 3.5")[$y = 0.5 cos ( display(frac(x, 2)) ) + 3.5$]


        
      
      
        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


      
      
        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


        
          + amplitude #math.equation(block: false, alt: "2")[$2$], period #math.equation(block: false, alt: "4")[$4$], midline #math.equation(block: false, alt: "y equals minus 1")[$y = − 1$]
+ #math.equation(block: false, alt: "y equals minus 1 plus 2 sin ⁡ open parenthesis the fraction π x over 2 close parenthesis")[$y = − 1 + 2 sin ( display(frac(π x, 2)) )$]


        
      
      
        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


      
      In Problems 31–36, we use a table of values to sketch circular functions.


      + Complete the table of values for the function.
+ Sketch a graph of the function and label the scales on the axes.


      
        #math.equation(block: true, alt: "y equals 2 minus 5 cos ⁡ open parenthesis 2 t close parenthesis")[$y = 2 − 5 cos ( 2 t )$]


        #figure(table(
  columns: 5, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "t")[$t$]], [#math.equation(block: false, alt: "2 t")[$2 t$]], [#math.equation(block: false, alt: "cos ⁡ open parenthesis 2 t close parenthesis")[$cos ( 2 t )$]], [#math.equation(block: false, alt: "minus 5 cos ⁡ open parenthesis 2 t close parenthesis")[$− 5 cos ( 2 t )$]], [#math.equation(block: false, alt: "2 minus 5 cos ⁡ open parenthesis 2 t close parenthesis")[$2 − 5 cos ( 2 t )$]]),
  [$#hide($0000$)$], [#math.equation(block: false, alt: "0")[$0$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
  [$#hide($0000$)$], [#math.equation(block: false, alt: "the fraction π over 2")[$display(frac(π, 2))$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
  [$#hide($0000$)$], [#math.equation(block: false, alt: "π")[$π$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
  [$#hide($0000$)$], [#math.equation(block: false, alt: "the fraction 3 π over 2")[$display(frac(3 π, 2))$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
  [$#hide($0000$)$], [#math.equation(block: false, alt: "2 π")[$2 π$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
))


        #figure(figph[grid], alt: "grid", caption: none)


        
          + #figure(table(
  columns: 5, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "t")[$t$]], [#math.equation(block: false, alt: "2 t")[$2 t$]], [#math.equation(block: false, alt: "cos ⁡ open parenthesis 2 t close parenthesis")[$cos ( 2 t )$]], [#math.equation(block: false, alt: "minus 5 cos ⁡ open parenthesis 2 t close parenthesis")[$− 5 cos ( 2 t )$]], [#math.equation(block: false, alt: "2 minus 5 cos ⁡ open parenthesis 2 t close parenthesis")[$2 − 5 cos ( 2 t )$]]),
  [#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: "minus 5")[$− 5$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]],
  [#math.equation(block: false, alt: "the fraction π over 4")[$display(frac(π, 4))$]], [#math.equation(block: false, alt: "the fraction π over 2")[$display(frac(π, 2))$]], [#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: "the fraction π over 2")[$display(frac(π, 2))$]], [#math.equation(block: false, alt: "π")[$π$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "5")[$5$]], [#math.equation(block: false, alt: "7")[$7$]],
  [#math.equation(block: false, alt: "the fraction 3 π over 4")[$display(frac(3 π, 4))$]], [#math.equation(block: false, alt: "the fraction 3 π over 2")[$display(frac(3 π, 2))$]], [#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: "π")[$π$]], [#math.equation(block: false, alt: "2 π")[$2 π$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "minus 5")[$− 5$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]],
))
+ #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


        
      
      
        #math.equation(block: true, alt: "y equals minus 2 plus 4 sin ⁡ open parenthesis 3 t close parenthesis")[$y = − 2 + 4 sin ( 3 t )$]


        #figure(table(
  columns: 5, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "t")[$t$]], [#math.equation(block: false, alt: "3 t")[$3 t$]], [#math.equation(block: false, alt: "sin ⁡ open parenthesis 3 t close parenthesis")[$sin ( 3 t )$]], [#math.equation(block: false, alt: "4 sin ⁡ open parenthesis 3 t close parenthesis")[$4 sin ( 3 t )$]], [#math.equation(block: false, alt: "minus 2 plus 4 sin ⁡ open parenthesis 3 t close parenthesis")[$− 2 + 4 sin ( 3 t )$]]),
  [$#hide($0000$)$], [#math.equation(block: false, alt: "0")[$0$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
  [$#hide($0000$)$], [#math.equation(block: false, alt: "the fraction π over 2")[$display(frac(π, 2))$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
  [$#hide($0000$)$], [#math.equation(block: false, alt: "π")[$π$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
  [$#hide($0000$)$], [#math.equation(block: false, alt: "the fraction 3 π over 2")[$display(frac(3 π, 2))$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
  [$#hide($0000$)$], [#math.equation(block: false, alt: "2 π")[$2 π$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
))


        #figure(figph[grid], alt: "grid", caption: none)


      
      
        #math.equation(block: true, alt: "y equals 1 plus 3 cos ⁡ open parenthesis the fraction t over 2 close parenthesis")[$y = 1 + 3 cos ( display(frac(t, 2)) )$]


        #figure(table(
  columns: 5, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "t")[$t$]], [#math.equation(block: false, alt: "the fraction t over 2")[$display(frac(t, 2))$]], [#math.equation(block: false, alt: "cos ⁡ open parenthesis the fraction t over 2 close parenthesis")[$cos ( display(frac(t, 2)) )$]], [#math.equation(block: false, alt: "3 cos ⁡ open parenthesis the fraction t over 2 close parenthesis")[$3 cos ( display(frac(t, 2)) )$]], [#math.equation(block: false, alt: "1 plus 3 cos ⁡ open parenthesis the fraction t over 2 close parenthesis")[$1 + 3 cos ( display(frac(t, 2)) )$]]),
  [$#hide($0000$)$], [#math.equation(block: false, alt: "0")[$0$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
  [$#hide($0000$)$], [#math.equation(block: false, alt: "the fraction π over 2")[$display(frac(π, 2))$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
  [$#hide($0000$)$], [#math.equation(block: false, alt: "π")[$π$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
  [$#hide($0000$)$], [#math.equation(block: false, alt: "the fraction 3 π over 2")[$display(frac(3 π, 2))$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
  [$#hide($0000$)$], [#math.equation(block: false, alt: "2 π")[$2 π$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
))


        #figure(figph[grid], alt: "grid", caption: none)


        
          + #figure(table(
  columns: 5, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "t")[$t$]], [#math.equation(block: false, alt: "the fraction t over 2")[$display(frac(t, 2))$]], [#math.equation(block: false, alt: "cos ⁡ open parenthesis the fraction t over 2 close parenthesis")[$cos ( display(frac(t, 2)) )$]], [#math.equation(block: false, alt: "3 cos ⁡ open parenthesis the fraction t over 2 close parenthesis")[$3 cos ( display(frac(t, 2)) )$]], [#math.equation(block: false, alt: "1 plus 3 cos ⁡ open parenthesis the fraction t over 2 close parenthesis")[$1 + 3 cos ( display(frac(t, 2)) )$]]),
  [#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: "3")[$3$]], [#math.equation(block: false, alt: "4")[$4$]],
  [#math.equation(block: false, alt: "π")[$π$]], [#math.equation(block: false, alt: "the fraction π over 2")[$display(frac(π, 2))$]], [#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 π")[$2 π$]], [#math.equation(block: false, alt: "π")[$π$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]],
  [#math.equation(block: false, alt: "3 π")[$3 π$]], [#math.equation(block: false, alt: "the fraction 3 π over 2")[$display(frac(3 π, 2))$]], [#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: "4 π")[$4 π$]], [#math.equation(block: false, alt: "2 π")[$2 π$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "4")[$4$]],
))
+ #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


        
      
      
        #math.equation(block: true, alt: "y equals minus 2 minus 3 sin ⁡ open parenthesis the fraction t over 4 close parenthesis")[$y = − 2 − 3 sin ( display(frac(t, 4)) )$]


        #figure(table(
  columns: 5, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "t")[$t$]], [#math.equation(block: false, alt: "the fraction t over 4")[$display(frac(t, 4))$]], [#math.equation(block: false, alt: "sin ⁡ open parenthesis the fraction t over 4 close parenthesis")[$sin ( display(frac(t, 4)) )$]], [#math.equation(block: false, alt: "3 sin ⁡ open parenthesis the fraction t over 4 close parenthesis")[$3 sin ( display(frac(t, 4)) )$]], [#math.equation(block: false, alt: "minus 2 minus 3 sin ⁡ open parenthesis the fraction t over 4 close parenthesis")[$− 2 − 3 sin ( display(frac(t, 4)) )$]]),
  [$#hide($0000$)$], [#math.equation(block: false, alt: "0")[$0$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
  [$#hide($0000$)$], [#math.equation(block: false, alt: "the fraction π over 2")[$display(frac(π, 2))$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
  [$#hide($0000$)$], [#math.equation(block: false, alt: "π")[$π$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
  [$#hide($0000$)$], [#math.equation(block: false, alt: "the fraction 3 π over 2")[$display(frac(3 π, 2))$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
  [$#hide($0000$)$], [#math.equation(block: false, alt: "2 π")[$2 π$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
))


        #figure(figph[grid], alt: "grid", caption: none)


      
      
        #math.equation(block: true, alt: "y equals minus 3 plus 2 sin ⁡ open parenthesis the fraction t over 3 close parenthesis")[$y = − 3 + 2 sin ( display(frac(t, 3)) )$]


        #figure(table(
  columns: 5, align: left, inset: 6pt,
  table.header([$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$]),
  [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
  [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
  [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
  [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
  [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
))


        #figure(figph[grid], alt: "grid", caption: none)


        
          + #figure(table(
  columns: 5, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "t")[$t$]], [#math.equation(block: false, alt: "the fraction t over 3")[$display(frac(t, 3))$]], [#math.equation(block: false, alt: "sin ⁡ open parenthesis the fraction t over 3 close parenthesis")[$sin ( display(frac(t, 3)) )$]], [#math.equation(block: false, alt: "2 sin ⁡ open parenthesis the fraction t over 3 close parenthesis")[$2 sin ( display(frac(t, 3)) )$]], [#math.equation(block: false, alt: "minus 3 plus 2 sin ⁡ open parenthesis the fraction t over 3 close parenthesis")[$− 3 + 2 sin ( display(frac(t, 3)) )$]]),
  [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]],
  [#math.equation(block: false, alt: "the fraction 3 π over 2")[$display(frac(3 π, 2))$]], [#math.equation(block: false, alt: "the fraction π over 2")[$display(frac(π, 2))$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]],
  [#math.equation(block: false, alt: "3 π")[$3 π$]], [#math.equation(block: false, alt: "π")[$π$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]],
  [#math.equation(block: false, alt: "the fraction 9 π over 2")[$display(frac(9 π, 2))$]], [#math.equation(block: false, alt: "the fraction 3 π over 2")[$display(frac(3 π, 2))$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "minus 5")[$− 5$]],
  [#math.equation(block: false, alt: "6 π")[$6 π$]], [#math.equation(block: false, alt: "2 π")[$2 π$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]],
))
+ #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


        
      
      
        #math.equation(block: true, alt: "y equals minus 1 plus 4 cos ⁡ open parenthesis the fraction t over 6 close parenthesis")[$y = − 1 + 4 cos ( display(frac(t, 6)) )$]


        #figure(table(
  columns: 5, align: left, inset: 6pt,
  table.header([$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$]),
  [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
  [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
  [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
  [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
  [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
))


        #figure(figph[grid], alt: "grid", caption: none)


      
      For Problems 37–44, label the scales on the axes for the graph.


      
        #math.equation(block: true, alt: "y equals 3 minus 4 sin ⁡ open parenthesis 2 x close parenthesis")[$y = 3 − 4 sin ( 2 x )$]


        #figure(figph[sinusoidal graph, no scale on axes], alt: "sinusoidal graph, no scale on axes", caption: none)


        
          #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


        
      
      
        #math.equation(block: true, alt: "y equals 2 cos ⁡ open parenthesis 5 x close parenthesis plus 2")[$y = 2 cos ( 5 x ) + 2$]


        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


      
      
        #math.equation(block: true, alt: "y equals the fraction 1 over 2 sin ⁡ open parenthesis 3 x close parenthesis plus the fraction 3 over 2")[$y = display(frac(1, 2)) sin ( 3 x ) + display(frac(3, 2))$]


        #figure(figph[sinusoidal graph, no scale], alt: "sinusoidal graph, no scale", caption: none)


        
          #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


        
      
      
        #math.equation(block: true, alt: "y equals the fraction 2 over 5 cos ⁡ open parenthesis 6 x close parenthesis plus the fraction 4 over 5")[$y = display(frac(2, 5)) cos ( 6 x ) + display(frac(4, 5))$]


        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


      
      
        #math.equation(block: true, alt: "50 minus 30 sin ⁡ open parenthesis the fraction x over 4 close parenthesis")[$50 − 30 sin ( display(frac(x, 4)) )$]


        #figure(figph[sinusoidal graph, no scale], alt: "sinusoidal graph, no scale", caption: none)


        
          #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


        
      
      
        #math.equation(block: true, alt: "25 cos ⁡ open parenthesis the fraction x over 3 close parenthesis plus 15")[$25 cos ( display(frac(x, 3)) ) + 15$]


        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


      
      
        #math.equation(block: true, alt: "y equals 4 sin ⁡ open parenthesis π x close parenthesis minus 3")[$y = 4 sin ( π x ) − 3$]


        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


        
          #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


        
      
      
        #math.equation(block: true, alt: "y equals the fraction 1 over 2 cos ⁡ open parenthesis the fraction π x over 2 close parenthesis plus 2")[$y = display(frac(1, 2)) cos ( display(frac(π x, 2)) ) + 2$]


        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


      
      
        The height of the tide in Cabot Cove can be approximated by a sinusoidal function.  At 5 am on July 23, the water level reached its high mark at the 20-foot line on the pier, and at 11 am, the water level was at its lowest at the 4-foot line.


        + Sketch a graph of #math.equation(block: false, alt: "W open parenthesis t close parenthesis")[$W ( t )$], the water level as a function of time, from 5 am on July 23 to 5 am on July 24.
+ Write an equation for the function.


        
          + #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)
+ #math.equation(block: false, alt: "W open parenthesis t close parenthesis equals 12 plus 8 cos ⁡ open parenthesis the fraction π t over 6 close parenthesis")[$W ( t ) = 12 + 8 cos ( display(frac(π t, 6)) )$]


        
      
      
        The population of mosquitoes at Marsh Lake is a sinusoidal function of time.  The population peaks around June 1 at about 6000 mosquitoes per square kilometer, and is smallest on December 1, at 1000 mosquitoes per square kilometer.


        + Sketch a graph of #math.equation(block: false, alt: "M open parenthesis t close parenthesis")[$M ( t )$], the number of mosquitoes as a function of the month, where #math.equation(block: false, alt: "t equals 0")[$t = 0$] on June 1.
+ Write an equation for the function.


      
      
        The paddlewheel on the Delta Queen steamboat is 28 feet in diameter, and is rotating once every ten seconds. The bottom of the paddlewheel is 4 feet below the surface of the water.


        + The ship's logo is painted on one of the paddlewheel blades. At #math.equation(block: false, alt: "t equals 0")[$t = 0$], the blade with the logo is at the top of the wheel. Sketch a graph of the logo's heightabove the water as a function of #math.equation(block: false, alt: "t")[$t$].
+ Write an equation for the function.


        
          + #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)
+ #math.equation(block: false, alt: "h equals 10 plus 14 cos ⁡ open parenthesis the fraction π t over 5 close parenthesis")[$h = 10 + 14 cos ( display(frac(π t, 5)) )$]


        
      
      
        Delbert's bicycle wheel is 24 inches in diameter, and he has a light attached to the spokes 10 inches from the center of the wheel. It is dark, and he is cycling home slowly from work. The bicycle wheel makes one revolution every second.


        + At #math.equation(block: false, alt: "t equals 0")[$t = 0$], the light is at its highest point the bicycle wheel. Sketch a graph of the light's height as a function of #math.equation(block: false, alt: "t")[$t$].
+ Write an equation for the function.


      
      For Problems 49–52, write an equation for the sinusoidal function whose graph is shown.


      
        The number of hours of daylight in Salt Lake City varies from a minimum of 9.6 hours on the winter solstice to a maximum of 14.4 hours on the summer solstice. Time is measured in months, starting at the winter solstice.


        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


        
          #math.equation(block: true, alt: "H equals 12 minus 2.4 cos ⁡ open parenthesis the fraction π t over 6 close parenthesis")[$H = 12 − 2.4 cos ( display(frac(π t, 6)) )$]


        
      
      
        A weight is 6.5 feet above the floor, suspended from the ceiling by a spring.  The weight is pulled down to 5 feet above the floor and released, rising past 6.5 feet in 0.5 second before attaining its maximum height of 8 feet.  The weight oscillates between its minimum and maximum height.


        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


      
      
        The voltage used in U.S. electrical current changes from 155V to 155V and back 60 times each second.


        #figure(figph[voltage], alt: "voltage", caption: none)


        
          #math.equation(block: true, alt: "y equals 155 cos ⁡ open parenthesis 120 π t close parenthesis")[$y = 155 cos ( 120 π t )$]


        
      
      
        Although the moon is spherical, what we see from earth looks like a disk, sometimes only partly visible. The percentage of the moon's disk that is visible varies between 0 (at new moon) to 100 (at full moon), over a 28-day cycle.


        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


      
      For Problems 53–58,


      + Make a table of values and sketch a graph of the function.
+ Give its period and midline.


      
        #math.equation(block: true, alt: "y equals tan ⁡ open parenthesis 2 x close parenthesis")[$y = tan ( 2 x )$]


        
          + #figure(table(
  columns: 6, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "the fraction minus π over 4")[$display(frac(− π, 4))$]], [#math.equation(block: false, alt: "the fraction minus π over 8")[$display(frac(− π, 8))$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "the fraction π over 8")[$display(frac(π, 8))$]], [#math.equation(block: false, alt: "the fraction π over 4")[$display(frac(π, 4))$]]),
  [#math.equation(block: false, alt: "tan ⁡ 2 x")[$tan 2 x$]], [undef], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], [undef],
))

#figure(figph[transformed tangent function], alt: "transformed tangent function", caption: none)
+ period #math.equation(block: false, alt: "the fraction π over 2")[$display(frac(π, 2))$], midline #math.equation(block: false, alt: "y equals 0")[$y = 0$]


        
      
      
        #math.equation(block: true, alt: "y equals tan ⁡ open parenthesis 4 x close parenthesis")[$y = tan ( 4 x )$]


      
      
        #math.equation(block: true, alt: "y equals 4 plus 2 tan ⁡ open parenthesis 3 x close parenthesis")[$y = 4 + 2 tan ( 3 x )$]


        
          + #figure(table(
  columns: 6, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "the fraction minus π over 6")[$display(frac(− π, 6))$]], [#math.equation(block: false, alt: "the fraction minus π over 12")[$display(frac(− π, 12))$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "the fraction π over 12")[$display(frac(π, 12))$]], [#math.equation(block: false, alt: "the fraction π over 6")[$display(frac(π, 6))$]]),
  [#math.equation(block: false, alt: "4 plus 2 tan ⁡ 3 x")[$4 + 2 tan 3 x$]], [undef], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "6")[$6$]], [undef],
))

#figure(figph[transformed tangent graph], alt: "transformed tangent graph", caption: none)
+ period #math.equation(block: false, alt: "the fraction π over 3")[$display(frac(π, 3))$], midline #math.equation(block: false, alt: "y equals 4")[$y = 4$]


        
      
      
        #math.equation(block: true, alt: "y equals 3 plus the fraction 1 over 2 tan ⁡ open parenthesis 2 x close parenthesis")[$y = 3 + display(frac(1, 2)) tan ( 2 x )$]


      
      
        #math.equation(block: true, alt: "y equals 3 minus tan ⁡ open parenthesis the fraction x over 4 close parenthesis")[$y = 3 − tan ( display(frac(x, 4)) )$]


        
          + #figure(table(
  columns: 6, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "minus 2 π")[$− 2 π$]], [#math.equation(block: false, alt: "minus π")[$− π$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "π")[$π$]], [#math.equation(block: false, alt: "2 π")[$2 π$]]),
  [#math.equation(block: false, alt: "3 minus tan ⁡ open parenthesis the fraction x over 4 close parenthesis")[$3 − tan ( display(frac(x, 4)) )$]], [undef], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "2")[$2$]], [undef],
))

#figure(figph[transformed tangent graph], alt: "transformed tangent graph", caption: none)
+ period #math.equation(block: false, alt: "4 π")[$4 π$], midline #math.equation(block: false, alt: "y equals 3")[$y = 3$]


        
      
      
        #math.equation(block: true, alt: "y equals 1 minus 2 tan ⁡ open parenthesis the fraction x over 3 close parenthesis")[$y = 1 − 2 tan ( display(frac(x, 3)) )$]


      
      For Problems 59–64, use the graph to find all solutions between #math.equation(block: false, alt: "0")[$0$] and #math.equation(block: false, alt: "2 π")[$2 π$].


      
        #math.equation(block: true, alt: "3 cos ⁡ open parenthesis 4 x close parenthesis equals 1.5")[$3 cos ( 4 x ) = 1.5$]


        #figure(figph[sinusoidal graph and horizontal line], alt: "sinusoidal graph and horizontal line", caption: none)


        
          #math.equation(block: true, alt: "the fraction π over 12")[$display(frac(π, 12))$], #math.equation(block: true, alt: "the fraction 5 π over 12")[$" " display(frac(5 π, 12))$], #math.equation(block: true, alt: "the fraction 7 π over 12")[$" " display(frac(7 π, 12))$], #math.equation(block: true, alt: "the fraction 11 π over 12")[$" " display(frac(11 π, 12))$], #math.equation(block: true, alt: "the fraction 13 π over 12")[$" " display(frac(13 π, 12))$], #math.equation(block: true, alt: "the fraction 17 π over 12")[$" " display(frac(17 π, 12))$], #math.equation(block: true, alt: "the fraction 19 π over 12")[$" " display(frac(19 π, 12))$], #math.equation(block: true, alt: "the fraction 23 π over 12")[$" " display(frac(23 π, 12))$]


        
      
      
        #math.equation(block: true, alt: "2 sin ⁡ open parenthesis 3 x close parenthesis equals minus the square root of 2")[$2 sin ( 3 x ) = − sqrt(2)$]


        #figure(figph[sinusoidal graph and horizontal line], alt: "sinusoidal graph and horizontal line", caption: none)


      
      
        #math.equation(block: true, alt: "2 plus 3 sin ⁡ open parenthesis 2 x close parenthesis equals 0.5")[$2 + 3 sin ( 2 x ) = 0.5$]


        #figure(figph[sinusoidal graph and horizontal line], alt: "sinusoidal graph and horizontal line", caption: none)


        
          #math.equation(block: true, alt: "the fraction 7 π over 12")[$display(frac(7 π, 12))$], #math.equation(block: true, alt: "the fraction 11 π over 12")[$" " display(frac(11 π, 12))$], #math.equation(block: true, alt: "the fraction 19 π over 12")[$" " display(frac(19 π, 12))$], #math.equation(block: true, alt: "the fraction 23 π over 12")[$" " display(frac(23 π, 12))$]


        
      
      
        #math.equation(block: true, alt: "2 plus 4 cos ⁡ open parenthesis 2 x close parenthesis equals 4")[$2 + 4 cos ( 2 x ) = 4$]


        #figure(figph[sinusoidal graph and horizontal line], alt: "sinusoidal graph and horizontal line", caption: none)


      
      
        #math.equation(block: true, alt: "minus 3 plus tan ⁡ open parenthesis 3 x close parenthesis equals minus 2")[$− 3 + tan ( 3 x ) = − 2$]


        #figure(figph[transformed tangent and horizontal line], alt: "transformed tangent and horizontal line", caption: none)


        
          #math.equation(block: true, alt: "the fraction π over 12")[$display(frac(π, 12))$], #math.equation(block: true, alt: "the fraction 5 π over 12")[$" " display(frac(5 π, 12))$], #math.equation(block: true, alt: "the fraction 3 π over 4")[$" " display(frac(3 π, 4))$], #math.equation(block: true, alt: "the fraction 13 π over 12")[$" " display(frac(13 π, 12))$], #math.equation(block: true, alt: "the fraction 17 π over 12")[$" " display(frac(17 π, 12))$], #math.equation(block: true, alt: "the fraction 7 π over 4")[$" " display(frac(7 π, 4))$]


        
      
      
        #math.equation(block: true, alt: "2 plus tan ⁡ open parenthesis 4 x close parenthesis equals 3")[$2 + tan ( 4 x ) = 3$]


        #figure(figph[transformed tangent and horizontal line], alt: "transformed tangent and horizontal line", caption: none)


      
      For Problems 65–70,


      + Use technology to graph the function for #math.equation(block: false, alt: "0 less than or equal to x less than or equal to 2 π")[$0 ≤ x ≤ 2 π$].
+ Use the intersect feature to find all solutions between #math.equation(block: false, alt: "0")[$0$] and #math.equation(block: false, alt: "2 π")[$2 π$]. Round your answers to hundredths.


      
        + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 3 sin ⁡ open parenthesis 2 x close parenthesis")[$f ( x ) = 3 sin ( 2 x )$]
+ #math.equation(block: false, alt: "3 sin ⁡ open parenthesis 2 x close parenthesis equals minus 1.5")[$3 sin ( 2 x ) = − 1.5$]


        
          #math.equation(block: true, alt: "1.83 , 2.88 , 4.97 , 6.02")[$1.83 , " " 2.88 , " " 4.97 , " " 6.02$]


        
      
      
        + #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals minus 2 cos ⁡ open parenthesis 3 x close parenthesis")[$g ( x ) = − 2 cos ( 3 x )$]
+ #math.equation(block: false, alt: "minus 2 cos ⁡ open parenthesis 3 x close parenthesis equals 1")[$− 2 cos ( 3 x ) = 1$]


      
      
        + #math.equation(block: false, alt: "h open parenthesis x close parenthesis equals 2 minus 4 cos ⁡ open parenthesis the fraction x over 4 close parenthesis")[$h ( x ) = 2 − 4 cos ( display(frac(x, 4)) )$]
+ #math.equation(block: false, alt: "2 minus 4 cos ⁡ open parenthesis the fraction x over 4 close parenthesis equals 0")[$2 − 4 cos ( display(frac(x, 4)) ) = 0$]


        
          #math.equation(block: true, alt: "4.19")[$4.19$]


        
      
      
        + #math.equation(block: false, alt: "H open parenthesis x close parenthesis equals 3 plus 2 sin ⁡ open parenthesis the fraction x over 2 close parenthesis")[$H ( x ) = 3 + 2 sin ( display(frac(x, 2)) )$]
+ #math.equation(block: false, alt: "3 plus 2 sin ⁡ open parenthesis the fraction x over 2 close parenthesis equals 5")[$3 + 2 sin ( display(frac(x, 2)) ) = 5$]


      
      
        + #math.equation(block: false, alt: "G open parenthesis x close parenthesis equals minus 1 plus 3 cos ⁡ open parenthesis 3 x close parenthesis")[$G ( x ) = − 1 + 3 cos ( 3 x )$]
+ #math.equation(block: false, alt: "minus 1 plus 3 cos ⁡ open parenthesis 3 x close parenthesis equals 1")[$− 1 + 3 cos ( 3 x ) = 1$]


        
          #math.equation(block: true, alt: "0.28 , 1.81 , 2.37 , 3.91 , 4.47 , 6.00")[$0.28 , " " 1.81 , " " 2.37 , " " 3.91 , " " 4.47 , " " 6.00$]


        
      
      
        + #math.equation(block: false, alt: "F open parenthesis x close parenthesis equals 4 minus 3 sin ⁡ open parenthesis 2 x close parenthesis")[$F ( x ) = 4 − 3 sin ( 2 x )$]
+ #math.equation(block: false, alt: "4 minus 3 sin ⁡ open parenthesis 2 x close parenthesis equals 2.5")[$4 − 3 sin ( 2 x ) = 2.5$]