Exercises 7.2 The General Sinusoidal Function

1.

[latex]f(x)=\sin x[/latex] and [latex]g(x)=\sin \left(x-\dfrac{\pi}{3}\right)[/latex]

1. 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]

2. Sketch the graphs of [latex]f[/latex] and [latex]g[/latex] on the same axes.
   
3. What is the horizontal shift from [latex]f[/latex] to [latex]g{?}[/latex]
4. 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]
5. 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]

1. 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]

2. Sketch the graphs of [latex]f[/latex] and [latex]g[/latex] on the same axes.
   
3. What is the horizontal shift from [latex]f[/latex] to [latex]g{?}[/latex]
4. 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]
5. 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]

1. 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]

2. Sketch the graphs of [latex]f[/latex] and [latex]g[/latex] on the same axes.
   
3. What is the horizontal shift from [latex]f[/latex] to [latex]g{?}[/latex]
4. Solve [latex]\tan \left(x+\dfrac{\pi}{4}\right)=1{,}[/latex] for [latex]-\pi \le x \le \pi{.}[/latex]
5. 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]

1. 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]

2. Sketch the graphs of [latex]f[/latex] and [latex]g[/latex] on the same axes.
   
3. What is the horizontal shift from [latex]f[/latex] to [latex]g{?}[/latex]
4. Solve [latex]\tan \left(x-\dfrac{\pi}{2}\right)=1{,}[/latex] for [latex]-\pi \le x \le \pi{.}[/latex]
5. 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]

1. 1. What are the amplitude and the horizontal shift?

Fill in the table of values.

1. Sketch the graph.
   
2. Solve [latex]-2\cos\left(x+\dfrac{\pi}{6}\right)=1{,}[/latex] for [latex]0 \le x \le 2\pi[/latex]
3. 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]

1. 1. What are the amplitude and the horizontal shift?

Fill in the table of values.

1. Sketch the graph.
   
2. Solve [latex]-6\sin\left(x-\dfrac{2\pi}{3}\right)=1{,}[/latex] for [latex]0 \le x \le 2\pi[/latex]
3. 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]

1. Find a formula for [latex]f(x)[/latex] as a shift of the sine function.
2. 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]

1. Find a formula for [latex]f(x)[/latex] as a shift of the sine function.
2. 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]

1. Find a formula for [latex]f(x)[/latex] as a shift of the tangent function.
2. 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]

1. Find a formula for [latex]f(x)[/latex] as a shift of the tangent function.
2. 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]

1. What are the period and the horizontal shift? (Hint: Factor out 2 from [latex]2x-\dfrac{\pi}{3}{.}[/latex])
2. 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]

3. Sketch the graph.
   
4. Solve [latex]\cos\left(2x-\dfrac{\pi}{3}\right)=1{,}[/latex] for [latex]0 \le x \le 2\pi[/latex]
5. 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]

1. What are the period and the horizontal shift? (Hint: Factor out 3 from [latex]3x+\dfrac{\pi}{2}{.}[/latex])
2. 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]

3. Sketch the graph.
   
4. Solve [latex]\sin\left(3x+\dfrac{\pi}{2}\right)=1{,}[/latex] for [latex]0 \le x \le 2\pi[/latex]
5. 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]

1. What are the period and the horizontal shift? (Hint: Factor out [latex]\pi[/latex] from [latex]\pi x+\dfrac{\pi}{3}{.}[/latex])
2. 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]

3. Sketch the graph.
   
4. Solve [latex]\sin\left(\pi x+\dfrac{\pi}{3}\right)=1{,}[/latex] for [latex]-2 \le x \le 2[/latex]
5. 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]

1. What are the period and the horizontal shift? (Hint: Factor out [latex]\pi[/latex] from [latex]\pi x-\dfrac{\pi}{3}{.}[/latex])
2. 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]

3. Sketch the graph.
   
4. Solve [latex]\cos\left(\pi x-\dfrac{\pi}{3}\right)=1{,}[/latex] for [latex]-2 \le x \le 2[/latex]
5. 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]

1. What are the midline, period, horizontal shift, and amplitude?
2. 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]

3. Sketch the graph.
   
4. Solve for [latex]0 \le x \le 2\pi{:}[/latex]
   [latex]3\sin\left(\dfrac{x}{2}-\dfrac{\pi}{6}\right)+4=1[/latex]
5. 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]

1. What are the midline, period, horizontal shift, and amplitude?
2. 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]

3. Sketch the graph.
   
4. Solve for [latex]0 \le x \le 6\pi{:}[/latex]
   [latex]2\cos\left(\dfrac{x}{3}-\dfrac{\pi}{4}\right)-1=1[/latex]
5. 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]

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.

1. What are the midline, period, and amplitude?
2. Write a formula for the average daily high temperature [latex]T(m){,}[/latex] where [latex]m[/latex] is the number of months since January.
3. 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.

1. What are the period, midline, and amplitude of [latex]V(t){?}[/latex]
2. Write a formula for [latex]V(t){.}[/latex]
3. 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.

1. What are the midline, period, and amplitude?
2. Graph [latex]h(t)[/latex] for two periods, labeling the points that correspond to high tide and low tide.
3. 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.

1. What are the midline, period, and amplitude?
2. 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.
3. When does the toy reach its maximum height the second time?

31.

32.

33.

34.