# 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?

\(\frac{2\pi}{3}\) radians

\(2\pi\) radians

\(3\pi\) radians

\(6\pi\) 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.

- What is the meaning of 13.5 in this context?
- What is the meaning of 5 in this context?
- 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?

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

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

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

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

### Problem 4

Which trigonometric function has period 5?

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

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

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

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

### Problem 5

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

### Problem 6

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

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

### 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 |

- Use the table to determine the value of \(a\) in the equation \(g(x)=f(ax)\).
- Use the table to determine the value of \(b\) in the equation \(h(x)=f(bx)\).