# Lesson 16

Features of Trigonometric Graphs (Part 2)

### Problem 1

A wheel rotates three times per second in a counterclockwise direction. The center of the wheel does not move.

What angle does the point $$P$$ rotate through in one second?

A:

$$\frac{2\pi}{3}$$ radians

B:

$$2\pi$$ radians

C:

$$3\pi$$ radians

D:

$$6\pi$$ radians

### Problem 2

A bicycle wheel is spinning in place. The vertical position of a point on the wheel, in inches, is described by the function $$f(t) = 13.5\sin(5 \boldcdot 2\pi t) + 20$$. Time $$t$$ is measured in seconds.

1. What is the meaning of 13.5 in this context?
2. What is the meaning of 5 in this context?
3. What is the meaning of 20 in this context?

### Problem 3

Each expression describes the vertical position, in feet off the ground, of a carriage on a Ferris wheel after $$t$$ minutes. Which function describes the largest Ferris wheel?

A:

$$100 \sin\left(\frac{2\pi t}{20}\right) + 110$$

B:

$$100 \sin\left(\frac{2\pi t}{30}\right) + 110$$

C:

$$200 \sin\left(\frac{2\pi t}{30}\right) + 210$$

D:

$$250 \sin\left(\frac{ 2\pi t}{20}\right) + 260$$

### Problem 4

Which trigonometric function has period 5?

A:

$$f(x) = \sin\left(\frac{1}{5}x\right)$$

B:

$$f(x) = \sin(5x)$$

C:

$$f(x) = \sin\left(\frac{5}{2\pi}x\right)$$

D:

$$f(x) = \sin\left(\frac{2\pi}{5} x\right)$$

### Problem 5

1. What is the period of the function $$f$$ given by $$f(t) = \cos(4\pi t)$$? Explain how you know.
2. Sketch a graph of $$f$$.

### Problem 6

Here is a graph of $$y=\cos(x)$$.

1. Sketch a graph of $$\cos(2x)$$.
2. How do the two graphs compare?

### Solution

(From Unit 6, Lesson 15.)

### Problem 7

Here is a table that shows the values of functions $$f$$, $$g$$, and $$h$$ for some values of $$x$$

$$x$$ $$f(x)$$ $$g(x)=f(a x)$$ $$h(x)=f(b x)$$
0 -125 -125 -125
3 -8 -64 -42.875
6 1 -27 -8
9 64 -8 -0.125
12 343 -1 1
15 1000 0 15.625
18 2197 1 64
21 4096 8 166.375
1. Use the table to determine the value of $$a$$ in the equation $$g(x)=f(ax)$$.
2. Use the table to determine the value of $$b$$ in the equation $$h(x)=f(bx)$$.