Lesson 16

Features of Trigonometric Graphs (Part 2)

  • Let's explore a trigonometric function modeling a situation.

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?

circle divided into 12 equivalent sections. point p on circumference.
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 \(f\).
horizontal axis, t, scale 0 to 1, by the fraction 1 over 8's. vertical axis, y, scale -1 to 1, by 1's. 

Problem 6

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

graph of y = cosine x. amplitude = 1. period = 2 pi.
  1. Sketch a graph of \(\cos(2x)\).
    horizontal axis, x, scale -2 pi to 2 pi, by the fraction pi over 2's. vertical axis -1 to 1's, by 1's. 
  2. How do the two graphs compare?
(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 \(h(x)=f(bx)\).
(From Unit 5, Lesson 9.)