7.1 Transformations of Graphs
Algebra Refresher
Complete the table.
-
| [latex]t[/latex] |
[latex]0[/latex] |
[latex]\dfrac{\pi}{2}[/latex] |
[latex]\pi[/latex] |
[latex]\dfrac{3\pi}{2}[/latex] |
[latex]2\pi[/latex] |
| [latex]t+\dfrac{\pi}{6}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
-
| [latex]t[/latex] |
[latex]0[/latex] |
[latex]\dfrac{\pi}{2}[/latex] |
[latex]\pi[/latex] |
[latex]\dfrac{3\pi}{2}[/latex] |
[latex]2\pi[/latex] |
| [latex]t-\dfrac{\pi}{6}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
-
| [latex]x[/latex] |
[latex]0[/latex] |
[latex]\dfrac{\pi}{2}[/latex] |
[latex]\pi[/latex] |
[latex]\dfrac{3\pi}{2}[/latex] |
[latex]2\pi[/latex] |
| [latex]x-\dfrac{\pi}{3}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
-
| [latex]x[/latex] |
[latex]0[/latex] |
[latex]\dfrac{\pi}{2}[/latex] |
[latex]\pi[/latex] |
[latex]\dfrac{3\pi}{2}[/latex] |
[latex]2\pi[/latex] |
| [latex]x+\dfrac{\pi}{3}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\underline{\qquad\qquad\qquad\qquad}[/latex]
Algebra Refresher Answers
-
| [latex]t[/latex] |
[latex]0[/latex] |
[latex]\dfrac{\pi}{2}[/latex] |
[latex]\pi[/latex] |
[latex]\dfrac{3\pi}{2}[/latex] |
[latex]2\pi[/latex] |
| [latex]t+\dfrac{\pi}{6}[/latex] |
[latex]\dfrac{\pi}{6}[/latex] |
[latex]\dfrac{2\pi}{3}[/latex] |
[latex]\dfrac{7\pi}{6}[/latex] |
[latex]\dfrac{5\pi}{3}[/latex] |
[latex]\dfrac{13\pi}{6}[/latex] |
-
| [latex]t[/latex] |
[latex]0[/latex] |
[latex]\dfrac{\pi}{2}[/latex] |
[latex]\pi[/latex] |
[latex]\dfrac{3\pi}{2}[/latex] |
[latex]2\pi[/latex] |
| [latex]t-\dfrac{\pi}{6}[/latex] |
[latex]\dfrac{-\pi}{6}[/latex] |
[latex]\dfrac{\pi}{3}[/latex] |
[latex]\dfrac{5\pi}{6}[/latex] |
[latex]\dfrac{4\pi}{3}[/latex] |
[latex]\dfrac{11\pi}{6}[/latex] |
-
| [latex]x[/latex] |
[latex]0[/latex] |
[latex]\dfrac{\pi}{2}[/latex] |
[latex]\pi[/latex] |
[latex]\dfrac{3\pi}{2}[/latex] |
[latex]2\pi[/latex] |
| [latex]x-\dfrac{\pi}{3}[/latex] |
[latex]\dfrac{-\pi}{3}[/latex] |
[latex]\dfrac{\pi}{6}[/latex] |
[latex]\dfrac{2\pi}{3}[/latex] |
[latex]\dfrac{7\pi}{6}[/latex] |
[latex]\dfrac{5\pi}{3}[/latex] |
-
| [latex]x[/latex] |
[latex]0[/latex] |
[latex]\dfrac{\pi}{2}[/latex] |
[latex]\pi[/latex] |
[latex]\dfrac{3\pi}{2}[/latex] |
[latex]2\pi[/latex] |
| [latex]x+\dfrac{\pi}{3}[/latex] |
[latex]\dfrac{\pi}{3}[/latex] |
[latex]\dfrac{5\pi}{6}[/latex] |
[latex]\dfrac{4\pi}{3}[/latex] |
[latex]\dfrac{11\pi}{6}[/latex] |
[latex]\dfrac{7\pi}{3}[/latex] |
- Identify the amplitude, period, and midline of a circular function
- Graph a circular function
- Find a formula for the graph of a circular function
- Model periodic phenomena with circular functions
- Graph transformations of the tangent functions
- Solve trigonometric equations graphically
In Chapter 4 we saw that the amplitude, period, and midline of a sinusoidal graph are determined by the coefficients in its formula. The 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 transformations of the basic sine and cosine graphs.
- 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.
Compare the graphs of [latex]f(x)=2\sin x[/latex] and [latex]g(x)=0.5\sin x[/latex] with the graph of [latex]y=\sin x{.}[/latex]
Solution
With your calculator set in radian mode, graph the three functions in the ZTrig window (press ZOOM 7). The graphs are shown below. All three graphs have the same period ([latex]2\pi[/latex]) and midline ([latex]y=0[/latex]), but the graph of [latex]f[/latex] has amplitude 2, and the graph of [latex]g[/latex] has amplitude 0.5.
The amplitude of [latex]y=A\sin t[/latex] is given by [latex]|A|[/latex], and the same is true of [latex]y=A\cos t{.}[/latex] In the next exercise, remember that the amplitude is always a nonnegative number.
Checkpoint 7.2.
Compare the graphs of [latex]f(x)=3\cos x[/latex] and [latex]g(x)=-3\cos x[/latex] with the graph of [latex]y=\cos x{.}[/latex]
Solution
Both graphs have amplitude 3. The graph of [latex]g(x)=-3\cos x[/latex] is reflected about the [latex]x[/latex]-axis.
Next, we’ll consider changes in the period of the graph.
Compare the graphs of [latex]f(x)=\cos 2x[/latex] and [latex]g(x)=\cos \dfrac{1}{3}x[/latex] with the graph of [latex]y=\cos x{.}[/latex]
Solution
With your calculator set in radian mode, graph [latex]f(x)=\cos 2x[/latex] and [latex]y=\cos x[/latex] in the same window, as shown below. Both graphs have the same amplitude ([latex]1[/latex]) and midline ([latex]y=0[/latex]), but the graph of [latex]f[/latex] completes two cycles from [latex]0[/latex] to [latex]2\pi[/latex] instead of one. The period of [latex]f(x)=\cos {2}x[/latex] is [latex]\dfrac{2\pi}{{2}}=\pi{.}[/latex]
Now graph [latex]g(x)=\cos \dfrac{1}{3}x[/latex] and [latex]y=\cos x[/latex] in the same window. Set Xmin [latex]=0[/latex] and Xmax [latex]=6\pi{.}[/latex]
The graph of [latex]g(x)=\cos {\dfrac{1}{3}}x[/latex] completes one cycle between [latex]0[/latex] and [latex]6\pi{.}[/latex] Its period is [latex]\dfrac{2\pi}{{\dfrac{1}{3}}}=6\pi{.}[/latex]
The period of [latex]y=\cos Bt[/latex] is given by [latex]\dfrac{2\pi}{\lvert B \rvert}{,}[/latex] and the same is true of [latex]y=\sin Bt{.}[/latex]
Checkpoint 7.4.
- Compare the graph of [latex]f(x)=\sin 3x[/latex] with the graph of [latex]y=\sin x{.}[/latex] Use the window Xmin [latex]=0{,}[/latex] Xmax [latex]=2\pi{,}[/latex] Ymin [latex]=-2{,}[/latex] Ymax [latex]=2{.}[/latex]
- Compare the graph of [latex]g(x)=\sin \dfrac{1}{4}x[/latex] with the graph of [latex]y=\sin x{.}[/latex] Use the window Xmin [latex]=0{,}[/latex] Xmax [latex]=8\pi{,}[/latex] Ymin [latex]=-2{,}[/latex] Ymax [latex]=2{.}[/latex]
Solution
- The graph of [latex]f[/latex] completes 3 cycles from [latex]0[/latex] to [latex]2\pi{.}[/latex] Its period is [latex]\dfrac{2\pi}{3}{.}[/latex]
- The graph of [latex]g[/latex] completes one cycle from [latex]0[/latex] to [latex]8\pi{.}[/latex] Its period is [latex]8\pi{.}[/latex]
Next we’ll consider changes in midline.
Compare the graph of [latex]f(x)=2+\sin x[/latex] with the graph of [latex]y=\sin x{.}[/latex]
Solution
Graph both functions in the ZTrig window. The graphs are shown below. Each point on the graph of [latex]f(x)=2+\sin x[/latex] has [latex]y[/latex]-coordinate 2 units higher than the corresponding point on the graph of [latex]y=\sin x{.}[/latex] Thus, the graph of [latex]f(x)=2+\sin x[/latex] is shifted vertically by 2 units relative to the graph of [latex]y=\sin x{.}[/latex] In particular, the midline of [latex]f(x)={2}+\sin x[/latex] is the line [latex]y={2}{.}[/latex]
Checkpoint 7.6.
Compare the graph of [latex]g(x)=-3+\cos x[/latex] with the graph of [latex]y=\cos x{.}[/latex]
Solution
The graph of [latex]g[/latex] is shifted down 3 units. Its midline is [latex]y=-3{.}[/latex]
Here is a summary of our findings.
- The graph of
[latex]y=A\cos x ~~~~{or}~~~~ y=A\sin x[/latex]
has amplitude [latex]\lvert A \rvert{.}[/latex]
- The graph of
[latex]y=\cos Bx ~~~~{or}~~~~ y=\sin Bx[/latex]
has period [latex]\dfrac{2\pi}{B}{.}[/latex]
- The graph of
[latex]y=k+\cos x ~~~~{or}~~~~ y=k+\sin x[/latex]
has midline [latex]y=k{.}[/latex]
The values of the parameters [latex]A,~ B, ~ {and}~ k[/latex] determine the shape of the graphs of
[latex]y=k+A\sin Bx ~~~~{or}~~~~ y=k+A\cos Bx[/latex]
By adjusting the amplitude, period, and midline of the sine or cosine graph, we can sketch these sinusoidal functions.
- State the amplitude, period, and midline of [latex]y=2+3\cos 4t{.}[/latex]
- Sketch by hand a graph of [latex]y=2+3\cos 4t{.}[/latex]
Solution
- The amplitude of the graph is 3, its midline is [latex]y=2{,}[/latex] and its period is [latex]\dfrac{2\pi}{4}=\dfrac{\pi}{2}{.}[/latex]
- 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.
| [latex]t[/latex] |
[latex]4t[/latex] |
[latex]\cos 4t[/latex] |
[latex]3\cos 4t[/latex] |
[latex]y=2+\cos 4t[/latex] |
| [latex]\hphantom{0000}[/latex] |
[latex]{0}[/latex] |
[latex]1[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
| [latex]\hphantom{0000}[/latex] |
[latex]{\dfrac{\pi}{2}}[/latex] |
[latex]0[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
| [latex]\hphantom{0000}[/latex] |
[latex]{\pi}[/latex] |
[latex]-1[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
| [latex]\hphantom{0000}[/latex] |
[latex]{\dfrac{3\pi}{2}}[/latex] |
[latex]0[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
| [latex]\hphantom{0000}[/latex] |
[latex]{2\pi}[/latex] |
[latex]1[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
Now we work backward from [latex]4t[/latex] to find the values of [latex]t{,}[/latex] and forward from [latex]\cos 4t[/latex] to find the values of [latex]y{.}[/latex]
| [latex]t[/latex] |
[latex]4t[/latex] |
[latex]\cos 4t[/latex] |
[latex]3\cos 4t[/latex] |
[latex]y=2+\cos 4t[/latex] |
| [latex]0[/latex] |
[latex]0[/latex] |
[latex]1[/latex] |
[latex]3[/latex] |
[latex]5[/latex] |
| [latex]{\dfrac{\pi}{8}}[/latex] |
[latex]\dfrac{\pi}{2}[/latex] |
[latex]0[/latex] |
[latex]0[/latex] |
[latex]{2}[/latex] |
| [latex]{\dfrac{\pi}{4}}[/latex] |
[latex]\pi[/latex] |
[latex]-1[/latex] |
[latex]-3[/latex] |
[latex]{-1}[/latex] |
| [latex]{\dfrac{3\pi}{8}}[/latex] |
[latex]\dfrac{3\pi}{2}[/latex] |
[latex]0[/latex] |
[latex]0[/latex] |
[latex]{2}[/latex] |
| [latex]{\dfrac{\pi}{2}}[/latex] |
[latex]2\pi[/latex] |
[latex]1[/latex] |
[latex]3[/latex] |
[latex]{5}[/latex] |
Notice from the table that the graph completes one cycle from [latex]t=0[/latex] to [latex]t=\dfrac{\pi}{2}{,}[/latex] which confirms that the period is [latex]\dfrac{\pi}{2}{.}[/latex] Finally, we plot the points [latex](t,y)[/latex] from the table and use them as “guide points” to sketch a sinusoidal graph, as shown below.
Checkpoint 7.8.
- State the amplitude, period, and midline of the graph of [latex]y=4-2\sin \dfrac{t}{3}{.}[/latex]
- Complete the table and sketch a graph of [latex]y=4-2\sin \dfrac{t}{3}{.}[/latex]
| [latex]t[/latex] |
[latex]\dfrac{t}{3}[/latex] |
[latex]\sin \dfrac{t}{3}[/latex] |
[latex]-2\sin \dfrac{t}{3}[/latex] |
[latex]y=4-2\sin \dfrac{t}{3}[/latex] |
| [latex]\hphantom{00}[/latex] |
[latex]{0}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
|
[latex]{\dfrac{\pi}{2}}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
|
[latex]{\pi}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
|
[latex]{\dfrac{3\pi}{2}}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
|
[latex]{2\pi}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
Solution
- Amplitude: 2, period: [latex]6\pi{,}[/latex] midline: [latex]y=4{.}[/latex]
-
| [latex]t[/latex] |
[latex]\frac{t}{3}[/latex] |
[latex]\sin \frac{t}{3}[/latex] |
[latex]-2\sin \frac{t}{3}[/latex] |
[latex]y=4-2\sin \frac{t}{3}[/latex] |
| [latex]0[/latex] |
[latex]0[/latex] |
[latex]0[/latex] |
[latex]0[/latex] |
[latex]4[/latex] |
| [latex]\frac{3\pi}{2}[/latex] |
[latex]\frac{\pi}{2}[/latex] |
[latex]1[/latex] |
[latex]-2[/latex] |
[latex]2[/latex] |
| [latex]3\pi[/latex] |
[latex]\pi[/latex] |
[latex]0[/latex] |
[latex]0[/latex] |
[latex]4[/latex] |
| [latex]\frac{9\pi}{2}[/latex] |
[latex]\frac{3\pi}{2}[/latex] |
[latex]-1[/latex] |
[latex]2[/latex] |
[latex]6[/latex] |
| [latex]6\pi[/latex] |
[latex]2\pi[/latex] |
[latex]0[/latex] |
[latex]0[/latex] |
[latex]4[/latex] |
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.
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.
Solution
We would like a function of the form [latex]y=k+a\sin Bt{,}[/latex] so we must find the values of the parameters [latex]A,~B[/latex] and [latex]k{.}[/latex]
- The midline of the graph is [latex]y=\dfrac{70+110}{2}=90{,}[/latex] and the amplitude is [latex]110-90=20{,}[/latex] so [latex]A=20[/latex] and [latex]k=90{.}[/latex]
- The graph repeats 60 times per minute, so the period is [latex]\dfrac{1}{60}[/latex] minute, and [latex]B=\dfrac{2\pi}{\dfrac{1}{60}}=120\pi{.}[/latex]
Thus,
[latex]y=90+20\sin 120 \pi t[/latex]
The graph of the function is shown below.
(In Section 7.2, we’ll consider sinusoidal functions that start at other positions on the cycle.)
In Exercise 5, note the starting point of the graph and choose the most appropriate sinusoidal function to model the function.
Checkpoint 7.11.
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?
Solution
- [latex]\displaystyle y=-35\cos 100\pi t[/latex]
- 50 cycles per second
The transformations of shifting and stretching can be applied to the tangent function as well. The graph of [latex]y=\tan x[/latex] does not have an amplitude, but we can see any vertical stretch by comparing the function values at the guide points.
- Graph [latex]y=1+3\tan 2x{.}[/latex]
- Describe the transformations of the graph compared to [latex]y=\tan x{.}[/latex]
Solution
- Recall that the period of the tangent function is [latex]\pi{.}[/latex] We make a table of values for one cycle of the function, choosing multiples of [latex]\dfrac{\pi}{4}[/latex] as the inputs for the tangent function. Then we plot the guide points and sketch a tangent function through them. The graph is shown below.
| [latex]x[/latex] |
[latex]2x[/latex] |
[latex]\tan 2x[/latex] |
[latex]1+3\tan 2x[/latex] |
| [latex]0[/latex] |
[latex]{0}[/latex] |
[latex]0[/latex] |
[latex]1[/latex] |
| [latex]\dfrac{\pi}{8}[/latex] |
[latex]{\dfrac{\pi}{4}}[/latex] |
[latex]1[/latex] |
[latex]4[/latex] |
| [latex]\dfrac{\pi}{4}[/latex] |
[latex]{\dfrac{\pi}{2}}[/latex] |
[latex]---[/latex] |
[latex]---[/latex] |
| [latex]\dfrac{3\pi}{8}[/latex] |
[latex]{\dfrac{3\pi}{4}}[/latex] |
[latex]-1[/latex] |
[latex]-2[/latex] |
| [latex]\dfrac{\pi}{2}[/latex] |
[latex]{\pi}[/latex] |
[latex]0[/latex] |
[latex]1[/latex] |
- Writing the formula as[latex]y=A\tan Bx+k=3\tan 2x+1[/latex]we see that the graph is stretched vertically by a factor of [latex]A=3{.}[/latex] The midline is [latex]y=1{,}[/latex] so the graph is shifted up by 1 unit. Finally, the coefficient [latex]B=2[/latex] compresses the graph horizontally by a factor of 2, so the period of the graph is [latex]\dfrac{\pi}{2}{,}[/latex] and there are four cycles between [latex]0[/latex] and [latex]2\pi[/latex]
Checkpoint 7.13.
- Identify the midline and period of the tangent graph shown below.
- Find an equation of the form[latex]y=A\tan Bx + k[/latex]for the graph.
Solution
- Midline: [latex]y=-2{,}[/latex] period [latex]=2[/latex]
- [latex]\displaystyle y=-3+2\tan \pi x[/latex]
Vocabulary
- Transformation
- Amplitude
- Period
- Midline
Concepts
- Changes to the amplitude, period, and midline of the basic sine and cosine graphs are called 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.
-
Amplitude, Period, and Midline of Sinusoidal Functions.
- The graph of
[latex]y=A\cos x ~~~~{or}~~~~ y=A\sin x[/latex]
has amplitude [latex]\lvert A \rvert{.}[/latex]
- The graph of
[latex]y=\cos Bx ~~~~{or}~~~~ y=\sin Bx[/latex]
has period [latex]\dfrac{2\pi}{B}{.}[/latex]
- The graph of
[latex]y=k+\cos x ~~~~{or}~~~~ y=k+\sin x[/latex]
has midline [latex]y=k{.}[/latex]
- 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 [latex]0[/latex] to [latex]2\pi[/latex] by multiples of [latex]\dfrac{\pi}{4}{.}[/latex]
- Count from [latex]0[/latex] to [latex]2\pi[/latex] by multiples of [latex]\dfrac{\pi}{6}{.}[/latex]
- Transformation. The maximum value of a certain sinusoidal function is [latex]M{,}[/latex] and its minimum value is [latex]m{.}[/latex] What is the midline of the function? What is its amplitude?
- [latex]f(x)=k+A\tan x{,}[/latex] and [latex]f(0)=4,~~f(\dfrac{\pi}{4})=6{.}[/latex] What are the values of [latex]k[/latex] and [latex]A{?}[/latex]