Exercises 7.2 The General Sinusoidal Function
Skills
- Graph trigonometric functions using a table of values #1–6, 11–16
- Find a formula for a transformation of a trigonometric function #7–10, 17–26
- Solve trigonometric equations graphically #1–6, 11–16
- Model periodic phenomena with trigonometric functions #27–30
- Fit a circular function to data #31–34
Problems: #6, 8, 12, 14, 18, 30, 32
Homework 7.2
1.
[latex]f(x)=\sin x[/latex] and [latex]g(x)=\sin \left(x-\dfrac{\pi}{3}\right)[/latex]
- Fill in the table of values.
| [latex]x[/latex] |
[latex]-\pi[/latex] |
[latex]\dfrac{-5\pi}{6}[/latex] |
[latex]\dfrac{-2\pi}{3}[/latex] |
[latex]\dfrac{-\pi}{2}[/latex] |
[latex]\dfrac{-\pi}{3}[/latex] |
[latex]\dfrac{-\pi}{6}[/latex] |
[latex]0[/latex] |
[latex]\dfrac{\pi}{6}[/latex] |
[latex]\dfrac{\pi}{3}[/latex] |
[latex]\dfrac{\pi}{2}[/latex] |
[latex]\dfrac{2\pi}{3}[/latex] |
[latex]\dfrac{5\pi}{6}[/latex] |
[latex]\pi[/latex] |
| [latex]f(x)[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
| [latex]g(x)[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
- Sketch the graphs of [latex]f[/latex] and [latex]g[/latex] on the same axes.
- What is the horizontal shift from [latex]f[/latex] to [latex]g{?}[/latex]
- Find all values of [latex]x[/latex] for which [latex]\sin \left(x-\dfrac{\pi}{3}\right)=1{,}[/latex] for [latex]-\pi \le x \le \pi{.}[/latex]
- Find all values of [latex]x[/latex] for which [latex]\sin \left(x-\dfrac{\pi}{3}\right)=0{,}[/latex] for [latex]-\pi \le x \le \pi{.}[/latex]
2.
[latex]f(x)=\cos x[/latex] and [latex]g(x)=\cos \left(x+\dfrac{\pi}{3}\right)[/latex]
- Fill in the table of values.
| [latex]x[/latex] |
[latex]-\pi[/latex] |
[latex]\dfrac{-5\pi}{6}[/latex] |
[latex]\dfrac{-2\pi}{3}[/latex] |
[latex]\dfrac{-\pi}{2}[/latex] |
[latex]\dfrac{-\pi}{3}[/latex] |
[latex]\dfrac{-\pi}{6}[/latex] |
[latex]0[/latex] |
[latex]\dfrac{\pi}{6}[/latex] |
[latex]\dfrac{\pi}{3}[/latex] |
[latex]\dfrac{\pi}{2}[/latex] |
[latex]\dfrac{2\pi}{3}[/latex] |
[latex]\dfrac{5\pi}{6}[/latex] |
[latex]\pi[/latex] |
| [latex]f(x)[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
| [latex]g(x)[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
- Sketch the graphs of [latex]f[/latex] and [latex]g[/latex] on the same axes.
- What is the horizontal shift from [latex]f[/latex] to [latex]g{?}[/latex]
- Find all values of [latex]x[/latex] for which [latex]\cos \left(x+\dfrac{\pi}{3}\right)=1{,}[/latex] for [latex]-\pi \le x \le \pi{.}[/latex]
- Find all values of [latex]x[/latex] for which [latex]\cos \left(x+\dfrac{\pi}{3}\right)=0{,}[/latex] for [latex]-\pi \le x \le \pi{.}[/latex]
3.
[latex]f(x)=\tan x[/latex] and [latex]g(x)=\tan \left(x+\dfrac{\pi}{4}\right)[/latex]
- Fill in the table of values.
| [latex]x[/latex] |
[latex]-\pi[/latex] |
[latex]\dfrac{-3\pi}{4}[/latex] |
[latex]\dfrac{-\pi}{2}[/latex] |
[latex]\dfrac{-\pi}{4}[/latex] |
[latex]0[/latex] |
[latex]\dfrac{\pi}{4}[/latex] |
[latex]\dfrac{\pi}{2}[/latex] |
[latex]\dfrac{3\pi}{4}[/latex] |
[latex]\pi[/latex] |
| [latex]f(x)[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
| [latex]g(x)[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
- Sketch the graphs of [latex]f[/latex] and [latex]g[/latex] on the same axes.
- What is the horizontal shift from [latex]f[/latex] to [latex]g{?}[/latex]
- Solve [latex]\tan \left(x+\dfrac{\pi}{4}\right)=1{,}[/latex] for [latex]-\pi \le x \le \pi{.}[/latex]
- Solve [latex]\tan \left(x+\dfrac{\pi}{4}\right)=0{,}[/latex] for [latex]-\pi \le x \le \pi{.}[/latex]
4.
[latex]f(x)=\tan x[/latex] and [latex]g(x)=\tan \left(x-\dfrac{\pi}{2}\right)[/latex]
- Fill in the table of values.
| [latex]x[/latex] |
[latex]-\pi[/latex] |
[latex]\dfrac{-3\pi}{4}[/latex] |
[latex]\dfrac{-\pi}{2}[/latex] |
[latex]\dfrac{-\pi}{4}[/latex] |
[latex]0[/latex] |
[latex]\dfrac{\pi}{4}[/latex] |
[latex]\dfrac{\pi}{2}[/latex] |
[latex]\dfrac{3\pi}{4}[/latex] |
[latex]\pi[/latex] |
| [latex]f(x)[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
| [latex]g(x)[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
- Sketch the graphs of [latex]f[/latex] and [latex]g[/latex] on the same axes.
- What is the horizontal shift from [latex]f[/latex] to [latex]g{?}[/latex]
- Solve [latex]\tan \left(x-\dfrac{\pi}{2}\right)=1{,}[/latex] for [latex]-\pi \le x \le \pi{.}[/latex]
- Solve [latex]\tan \left(x-\dfrac{\pi}{2}\right)=0{,}[/latex] for [latex]-\pi \le x \le \pi{.}[/latex]
5.
[latex]y = -2\cos\left(x+ \dfrac{\pi}{6}\right)[/latex]
-
- What are the amplitude and the horizontal shift?
Fill in the table of values.
| [latex]x[/latex] |
[latex]x+\dfrac{\pi}{6}[/latex] |
[latex]\cos\left(x+\dfrac{\pi}{6}\right)[/latex] |
[latex]-2\cos\left(x+\dfrac{\pi}{6}\right)[/latex] |
| [latex]\hphantom{0000}[/latex] |
[latex]-\pi[/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]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] |
- Sketch the graph.
- Solve [latex]-2\cos\left(x+\dfrac{\pi}{6}\right)=1{,}[/latex] for [latex]0 \le x \le 2\pi[/latex]
- Solve [latex]-2\cos\left(x+\dfrac{\pi}{6}\right)=0{,}[/latex] for [latex]0 \le x \le 2\pi[/latex]
6.
[latex]y = -6\sin\left(x- \dfrac{2\pi}{3}\right)[/latex]
-
- What are the amplitude and the horizontal shift?
Fill in the table of values.
| [latex]x[/latex] |
[latex]x-\dfrac{2\pi}{3}[/latex] |
[latex]\sin\left(x-\dfrac{2\pi}{3}\right)[/latex] |
[latex]-6\sin\left(x-\dfrac{2\pi}{3}\right)[/latex] |
| [latex]\hphantom{0000}[/latex] |
[latex]-\pi[/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]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] |
- Sketch the graph.
- Solve [latex]-6\sin\left(x-\dfrac{2\pi}{3}\right)=1{,}[/latex] for [latex]0 \le x \le 2\pi[/latex]
- Solve [latex]-6\sin\left(x-\dfrac{2\pi}{3}\right)=0{,}[/latex] for [latex]0 \le x \le 2\pi[/latex]
7.
The figure shows the graph of [latex]y = f(x){.}[/latex]
- Find a formula for [latex]f(x)[/latex] as a shift of the sine function.
- Find a formula for [latex]f(x)[/latex] as a shift of the cosine function.
8.
The figure shows the graph of [latex]y = f(x){.}[/latex]
- Find a formula for [latex]f(x)[/latex] as a shift of the sine function.
- Find a formula for [latex]f(x)[/latex] as a shift of the cosine function.
9.
The figure shows the graph of [latex]y = f(x){.}[/latex]
- Find a formula for [latex]f(x)[/latex] as a shift of the tangent function.
- Find another formula for [latex]f(x)[/latex] as a different shift of the tangent function.
10.
The figure shows the graph of [latex]y = f(x){.}[/latex]
- Find a formula for [latex]f(x)[/latex] as a shift of the tangent function.
- Find another formula for [latex]f(x)[/latex] as a different shift of the tangent function.
11.
[latex]y = \cos\left(2x-\dfrac{\pi}{3}\right)[/latex]
- What are the period and the horizontal shift? (Hint: Factor out 2 from [latex]2x-\dfrac{\pi}{3}{.}[/latex])
- Fill in the table of values.
| [latex]x[/latex] |
[latex]2x[/latex] |
[latex]2x-\dfrac{\pi}{3}[/latex] |
[latex]\cos\left(2x-\dfrac{\pi}{3}\right)[/latex] |
| [latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/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] |
- Sketch the graph.
- Solve [latex]\cos\left(2x-\dfrac{\pi}{3}\right)=1{,}[/latex] for [latex]0 \le x \le 2\pi[/latex]
- Solve [latex]\cos\left(2x-\dfrac{\pi}{3}\right)=0{,}[/latex] for [latex]0 \le x \le 2\pi[/latex]
12.
[latex]y = \sin\left(3x+\dfrac{\pi}{2}\right)[/latex]
- What are the period and the horizontal shift? (Hint: Factor out 3 from [latex]3x+\dfrac{\pi}{2}{.}[/latex])
- Fill in the table of values.
| [latex]x[/latex] |
[latex]3x[/latex] |
[latex]3x+\dfrac{\pi}{2}[/latex] |
[latex]\sin\left(3x+\dfrac{\pi}{2}\right)[/latex] |
| [latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/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] |
- Sketch the graph.
- Solve [latex]\sin\left(3x+\dfrac{\pi}{2}\right)=1{,}[/latex] for [latex]0 \le x \le 2\pi[/latex]
- Solve [latex]\sin\left(3x+\dfrac{\pi}{2}\right)=0{,}[/latex] for [latex]0 \le x \le 2\pi[/latex]
13.
[latex]y = \sin\left(\pi x+\dfrac{\pi}{3}\right)[/latex]
- What are the period and the horizontal shift? (Hint: Factor out [latex]\pi[/latex] from [latex]\pi x+\dfrac{\pi}{3}{.}[/latex])
- Fill in the table of values.
| [latex]x[/latex] |
[latex]\pi x[/latex] |
[latex]\pi x+\dfrac{\pi}{3}[/latex] |
[latex]\sin\left(\pi x+\dfrac{\pi}{3}\right)[/latex] |
| [latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/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] |
- Sketch the graph.
- Solve [latex]\sin\left(\pi x+\dfrac{\pi}{3}\right)=1{,}[/latex] for [latex]-2 \le x \le 2[/latex]
- Solve [latex]\sin\left(\pi x+\dfrac{\pi}{3}\right)=0{,}[/latex] for [latex]-2 \le x \le 2[/latex]
14.
[latex]y = \cos\left(\pi x-\dfrac{\pi}{3}\right)[/latex]
- What are the period and the horizontal shift? (Hint: Factor out [latex]\pi[/latex] from [latex]\pi x-\dfrac{\pi}{3}{.}[/latex])
- Fill in the table of values.
| [latex]x[/latex] |
[latex]\pi x[/latex] |
[latex]\pi x-\dfrac{\pi}{3}[/latex] |
[latex]\cos\left(\pi x-\dfrac{\pi}{3}\right)[/latex] |
| [latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/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] |
- Sketch the graph.
- Solve [latex]\cos\left(\pi x-\dfrac{\pi}{3}\right)=1{,}[/latex] for [latex]-2 \le x \le 2[/latex]
- Solve [latex]\cos\left(\pi x-\dfrac{\pi}{3}\right)=0{,}[/latex] for [latex]-2 \le x \le 2[/latex]
15.
[latex]y = 3\sin\left(\dfrac{x}{2}-\dfrac{\pi}{6}\right)+4[/latex]
- What are the midline, period, horizontal shift, and amplitude?
- Fill in the table of values.
| [latex]x[/latex] |
[latex]\dfrac{x}{2}[/latex] |
[latex]\dfrac{x}{2}-\dfrac{\pi}{6}[/latex] |
[latex]\sin\left(\dfrac{x}{2}-\dfrac{\pi}{6}\right)[/latex] |
[latex]3\sin\left(\dfrac{x}{2}-\dfrac{\pi}{6}\right)+4[/latex] |
| [latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]0[/latex] |
[latex]\hphantom{0000}[/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]\hphantom{0000}[/latex] |
[latex]\pi[/latex] |
[latex]\hphantom{0000}[/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]\hphantom{0000}[/latex] |
[latex]2\pi[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
- Sketch the graph.
- Solve for [latex]0 \le x \le 2\pi{:}[/latex]
[latex]3\sin\left(\dfrac{x}{2}-\dfrac{\pi}{6}\right)+4=1[/latex]
- Solve for [latex]0 \le x \le 2\pi{:}[/latex]
[latex]3\sin\left(\dfrac{x}{2}-\dfrac{\pi}{6}\right)+4=4[/latex]
16.
[latex]y = 2\cos\left(\dfrac{x}{3}-\dfrac{\pi}{4}\right)-1[/latex]
- What are the midline, period, horizontal shift, and amplitude?
- Fill in the table of values.
| [latex]x[/latex] |
[latex]\dfrac{x}{3}[/latex] |
[latex]\dfrac{x}{3}-\dfrac{\pi}{4}[/latex] |
[latex]\cos\left(\dfrac{x}{3}-\dfrac{\pi}{4}\right)[/latex] |
[latex]2\cos\left(\dfrac{x}{3}-\dfrac{\pi}{4}\right)-1[/latex] |
| [latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]0[/latex] |
[latex]\hphantom{0000}[/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]\hphantom{0000}[/latex] |
[latex]\pi[/latex] |
[latex]\hphantom{0000}[/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]\hphantom{0000}[/latex] |
[latex]2\pi[/latex] |
[latex]\hphantom{0000}[/latex] |
[latex]\hphantom{0000}[/latex] |
- Sketch the graph.
- Solve for [latex]0 \le x \le 6\pi{:}[/latex]
[latex]2\cos\left(\dfrac{x}{3}-\dfrac{\pi}{4}\right)-1=1[/latex]
- Solve for [latex]0 \le x \le 6\pi{:}[/latex]
[latex]2\cos\left(\dfrac{x}{3}-\dfrac{\pi}{4}\right)-1=-1[/latex]
17.
Find a formula for a sinusoidal function that has an amplitude of 2, a period of 3, and is shifted 4 units to the left and 5 units upward compared with the sine function. Sketch the graph for [latex]0 \le x \le 3{.}[/latex]
18.
Find a formula for a sinusoidal function that has an amplitude of 3, a period of 24, and is shifted 2 units to the right and 4 units upward compared with the cosine function. Sketch the graph for [latex]0 \le x \le 24{.}[/latex]
19.
Find a formula for a sinusoidal function that has an amplitude of 5, a period of 360, its midline at [latex]y=12{,}[/latex] and passes through [latex](0,7){.}[/latex] Sketch the graph for [latex]0 \le x \le 360{.}[/latex]
20.
Find a formula for a sinusoidal function that has an amplitude of 50, a period of 30, its midline at [latex]y=50{,}[/latex] and passes through [latex](0,100){.}[/latex] Sketch the graph for [latex]0 \le x \le 30{.}[/latex]
Exercise Group.
For Problems 21–26, find a formula for the circular function whose graph is shown.
- Write the function in the form [latex]f(x)=A\sin (B(x-h)){.}[/latex]
- Write the function in the form [latex]f(x)=A\cos (B(x-h))[/latex]
21.
22.
23.
24.
25.
26.
27.
The average daily high temperature in Fairbanks, Alaska, can be approximated by a sinusoidal function with a period of 12 months. The low temperature of [latex]-1.6°[/latex] occurs in January, and the high temperature of [latex]72.3°[/latex] in July.
- What are the midline, period, and amplitude?
- Write a formula for the average daily high temperature [latex]T(m){,}[/latex] where [latex]m[/latex] is the number of months since January.
- Graph [latex]T(m)[/latex] for two periods, labeling the points that correspond to highest and lowest average temperature.
28.
Depending on its phase, the moon looks like a disk that is partially visible and partially in shadow. The visible fraction ranges from 0% to 100% and can be approximated by a sinusoidal function [latex]V(t){,}[/latex] where [latex]t[/latex] is the number of days since the last full moon. The time between successive full moons (a lunar month) is 29.5 days.
- What are the period, midline, and amplitude of [latex]V(t){?}[/latex]
- Write a formula for [latex]V(t){.}[/latex]
- Graph your function over two periods, labeling the points that correspond to full moon, half moon, and new moon.
29.
The tide in Yorktown is approximated by the function
[latex]h(t)=1.4-1.4\cos(0.51t),[/latex]
measured in feet above low tide, where [latex]t[/latex] is the number of hours since the last low tide.
- What are the midline, period, and amplitude?
- Graph [latex]h(t)[/latex] for two periods, labeling the points that correspond to high tide and low tide.
- If the last low tide occurred at 5:00 am, predict when the next high and low tides will occur.
30.
The height of a child’s toy suspended at the end of a spring is approximated by a sinusoidal function. The toy’s height ranges between 200 centimeters and 260 centimeters above the ground, and it completes one up-and-down cycle every 0.8 seconds.
- What are the midline, period, and amplitude?
- Let [latex]h(t)[/latex] be the height of the toy in centimeters, where [latex]t=0[/latex] seconds corresponds to a time when the object was at the midline and moving upward. Graph [latex]h(t)[/latex] for two periods, labeling the points that correspond to the high and low positions of the toy.
- When does the toy reach its maximum height the second time?
Exercise Group.
In Problems 31–34,
- Estimate the amplitude, period, and midline of a circular function that fits the data.
- Write a formula for the function.
31.
| [latex]t[/latex] |
[latex]0[/latex] |
[latex]0.25[/latex] |
[latex]0.5[/latex] |
[latex]0.75[/latex] |
[latex]1[/latex] |
[latex]1.25[/latex] |
[latex]1.5[/latex] |
[latex]1.75[/latex] |
[latex]2[/latex] |
[latex]2.25[/latex] |
[latex]2.5[/latex] |
| [latex]f(t)[/latex] |
[latex]5.2[/latex] |
[latex]4.26[/latex] |
[latex]2[/latex] |
[latex]-0.26[/latex] |
[latex]-1.2[/latex] |
[latex]-0.26[/latex] |
[latex]2[/latex] |
[latex]4.26[/latex] |
[latex]5.2[/latex] |
[latex]4.26[/latex] |
[latex]2[/latex] |
32.
| [latex]s[/latex] |
[latex]0[/latex] |
[latex]1[/latex] |
[latex]2[/latex] |
[latex]3[/latex] |
[latex]4[/latex] |
[latex]5[/latex] |
[latex]6[/latex] |
[latex]7[/latex] |
[latex]8[/latex] |
[latex]9[/latex] |
[latex]10[/latex] |
| [latex]g(s)[/latex] |
[latex]3[/latex] |
[latex]2.58[/latex] |
[latex]2.4[/latex] |
[latex]2.58[/latex] |
[latex]3[/latex] |
[latex]3.42[/latex] |
[latex]3.6[/latex] |
[latex]3.42[/latex] |
[latex]3[/latex] |
[latex]2.58[/latex] |
[latex]2.4[/latex] |
33.
| [latex]x[/latex] |
[latex]0[/latex] |
[latex]0.1[/latex] |
[latex]0.2[/latex] |
[latex]0.3[/latex] |
[latex]0.4[/latex] |
[latex]0.5[/latex] |
[latex]0.6[/latex] |
[latex]0.7[/latex] |
[latex]0.8[/latex] |
| [latex]H(x)[/latex] |
[latex]5[/latex] |
[latex]7.9[/latex] |
[latex]9.8[/latex] |
[latex]9.8[/latex] |
[latex]7.9[/latex] |
[latex]5[/latex] |
[latex]2.1[/latex] |
[latex]0.2[/latex] |
[latex]0.2[/latex] |
34.
| [latex]t[/latex] |
[latex]0[/latex] |
[latex]0.79[/latex] |
[latex]1.57[/latex] |
[latex]2.36[/latex] |
[latex]3.14[/latex] |
[latex]3.93[/latex] |
[latex]4.71[/latex] |
[latex]5.50[/latex] |
[latex]6.28[/latex] |
| [latex]V(t)[/latex] |
[latex]1[/latex] |
[latex]-0.17[/latex] |
[latex]-3[/latex] |
[latex]-5.8[/latex] |
[latex]-7[/latex] |
[latex]-5.3[/latex] |
[latex]-3[/latex] |
[latex]-0.17[/latex] |
[latex]1[/latex] |