# Lesson 8

Rising and Falling

Let’s study graphs that repeat.

### Problem 1

A fan blade spins counterclockwise once per second.

Which of these graphs best depicts the height, \(h\), of \(P\) after \(s\) seconds? The fan blades are 1 foot long and the height is measured in feet from the center of the fan blades.

### Problem 2

Which situations are modeled accurately by a periodic function? Select **all** that apply.

the distance from the earth to the sun as a function of time

the vertical height of a point on a rotating wheel as a function of time

the area of a sheet of paper as a function of the number of times it is folded in half

the number of centimeters in \(x\) inches

the height of a swinging pendulum as a function of time

the height of a ball tossed in the air as a function of time

### Problem 3

Here is the graph of a function for some values of \(x\).

- Can you extend the graph to the whole plane so that the function \(f\) is periodic? Explain your reasoning.
- Can you extend the graph to the whole plane so that the function \(f\) is not periodic? Explain your reasoning.

### Problem 4

- Can a non-constant linear function be periodic? Explain your reasoning.
- Can a quadratic function be periodic? Explain your reasoning.

### Problem 5

Do \((7,1)\) and \((\text-5,5)\) lie on the same circle centered at \((0,0)\)? Explain how you know.

### Problem 6

The measure of angle \(\theta\) is between 0 and \(2\pi\) radians. Which statements *must* be true of \(\sin(\theta)\) and \(\cos(\theta)\)? Select **all** that apply.

\(\cos^2(\theta) + \sin^2(\theta) = 1\)

If \(\sin(\theta) = 0\), then \(\cos(\theta) = 1\).

If \(\sin(\theta) = 1\), then \(\cos(\theta) = 0\).

\(\cos(\theta) + \sin(\theta) = 1\).

The point \((\cos(\theta),\sin(\theta))\) lies on the unit circle.

### Problem 7

The center of a clock is the origin \((0,0)\) in a coordinate system. The hour hand is 4 units long. What are the coordinates of the end of the hour hand at:

- 3:00
- 8:00
- 11:00