# Lesson 8

Rising and Falling

### 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.

### Solution

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### Problem 2

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

A:

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

B:

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

C:

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

D:

the number of centimeters in $$x$$ inches

E:

the height of a swinging pendulum as a function of time

F:

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

### Solution

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### Problem 3

Here is the graph of a function for some values of $$x$$.

1. Can you extend the graph to the whole plane so that the function $$f$$ is periodic? Explain your reasoning.
2. Can you extend the graph to the whole plane so that the function $$f$$ is not periodic? Explain your reasoning.

### Solution

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### Problem 4

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

### Solution

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### Problem 5

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

### Solution

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(From Unit 6, Lesson 1.)

### 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.

A:

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

B:

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

C:

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

D:

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

E:

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

### Solution

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(From Unit 6, Lesson 5.)

### 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:

1. 3:00
2. 8:00
3. 11:00

### Solution

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(From Unit 6, Lesson 7.)