4.2: Periodic Functions

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Period, Midline and Amplitude

All sine and cosine graphs have the characteristic ”wave” shape we’ve seen in previous examples. But we can alter the size and frequency of the waves by changing the formula for the function. In the next example we consider three variations of the sine function.

Example 4.42

Make a table of values and sketch a graph for each of the functions. How does each differ from the graph of $y = \sin \theta$?

a $y=3 \sin \theta$
b $y=3+\sin \theta$
c $y=\sin 3 \theta$

Answer

a We make a table with multiples of $45^{\circ}$.

$\theta$	$0^{\circ}$	$45^{\circ}$	$90^{\circ}$	$135^{\circ}$	$180^{\circ}$	$225^{\circ}$	$270^{\circ}$	$315^{\circ}$	$360^{\circ}$
$y=3\sin \theta$	0	2.1	3	2.1	0	-2.1	-3	-2.1	0

We plot the points, and connect them with a sine-shaped wave. Compare the graph, shown at right, to the graph of $y=\sin \theta$. The graph is like a sine graph, except that it oscillates between a maximum value of 3 and a minimum value of -3. The amplitude of this function is 3.

$Screen Shot 2022-11-21 at 11.43.08 PM.png$

b Again, we make a table of values with multiples of $45^{\circ}$ and plot the points.

$\theta$	$0^{\circ}$	$45^{\circ}$	$90^{\circ}$	$135^{\circ}$	$180^{\circ}$	$225^{\circ}$	$270^{\circ}$	$315^{\circ}$	$360^{\circ}$
$y=3+\sin \theta$	3	3.7	4	3.7	3	2.3	2	2.1	3

This graph has the same amplitude as $y=\sin \theta$, but the entire graph is shifted up by 3 units, as shown at right. The midline of this function is the line $y=3$.

c This time we'll make a table with multiples of $15^{\circ}$.

$\theta$	$15^{\circ}$	$30^{\circ}$	$45^{\circ}$	$60^{\circ}$	$75^{\circ}$	$90^{\circ}$	$105^{\circ}$	$120^{\circ}$	$135^{\circ}$	$150^{\circ}$	$165^{\circ}$	$180^{\circ}$
$y=\sin 3 \theta$	0.7	1	0.7	0	-0.7	-1	-0.7	0	0.7	1	0.7	0

You can continue the table for $\theta$ between $180^{\circ}$ and $360^{\circ}$, and plot the points to find the graph shown at right. The graph has the same amplitude and midline as $y=\sin \theta$, but it completes three cycles from $0^{\circ}$ to $360^{\circ}$ instead of one cycle. The period of this graph is one-third of $360^{\circ}$, or $120^{\circ}$.

$Screen Shot 2022-11-21 at 11.50.19 PM.png$

The graphs in the previous example illustrate a general rule about sine and cosine graphs.

Amplitude, Period, and Midline.

1. The graph of

$y=A \cos \theta \quad \text { or } \quad y=A \sin \theta$

has amplitude $|A|$.

2. The graph of

$y=\cos B \theta \quad \text { or } \quad y=\sin B \theta$

has period $\frac{360^{\circ}}{|B|}$.

3. The graph of

$y=k+\cos \theta \quad \text { or } \quad y=k+\sin \theta$

has midline $y=k$.

Checkpoint 4.43

Sketch a graph for each of the following functions. Describe how each is different from the graph of $y=\cos \theta$.

a $f(\theta)=2+\cos \theta$
b $g(\theta)=\cos 2 \theta$
c $h(\theta)=2 \cos \theta$

Answer

a $Screen Shot 2022-12-22 at 11.11.36 PM.png$

The midline is $y=2$.

b $Screen Shot 2022-12-22 at 11.11.49 PM.png$

The period is $180^{\circ}$.

c $Screen Shot 2022-12-22 at 11.11.58 PM.png$

The amplitude is 2.

The quantities $A, B$, and $k$ in the equations above are called parameters, and their values for a particular function give us information about its graph.

Example 4.44

State the period, midline, and amplitude of the graph of $y=-3+4 \sin 3 \theta$, and graph the function.

Answer

$Screen Shot 2022-12-22 at 11.15.20 PM.png$

For this function, $A=4, B=3$, and $k=-3$. Its amplitude is 4, its period is $\frac{360^{\circ}}{3}=120^{\circ}$, and its midline is $y=-3$. The graph is shown at right.

Checkpoint 4.45

State the period, midline, and amplitude of the graph of $y=1-3 \sin 2 \theta$, and graph the function.

Answer

$Screen Shot 2022-12-22 at 11.18.20 PM.png$

Amplitude 3, period $180^{\circ}$, midline $y=1$

Sinusoidal Functions

Many interesting functions have graphs shaped like sines or cosines, even though they may not be functions of angles. These functions are called sinusoidal.

Example 4.46

Imagine a grandfather clock. As the minute hand sweeps around, the height of its tip changes with time. Which of the graphs shown below best represents the height of the tip of the minute hand as a function of time?

$Screen Shot 2022-12-22 at 11.21.20 PM.png$

Answer

Figure (a) is not the graph of a function at all: It does not pass the vertical line test. That is, some values of $t$, such as $t=0$, correspond to more than one value of $h$, which is not possible in the graph of a function.

Figure (b) shows the height of the minute hand varying between a maximum and minimum value. The height decreases at a constant rate (the graph is straight and the slope is constant) until the minimum is reached, and then increases at a constant rate.

But notice that during the 10 minutes from 12:10 to 12:20 the height of the minute hand decreases about half the diameter of the clock, while from 12:20 to 12:30 the height decreases only about a quarter of the diameter of the clock, as shown below.

$Screen Shot 2022-12-22 at 11.24.14 PM.png$

Thus the height of the minute hand does not decrease at a constant rate. Figure (c) is the best choice. The graph is curved because the slopes are not constant. The graph is steep when the height is changing rapidly, and the graph is nearly horizontal when the height is changing slowly. The height changes slowly near the hour and the half-hour, and more rapidly near the quarter-hours.

Checkpoint 4.47

As the moon revolves around the earth, the percent of the disk that we see varies sinusoidally with a period of approximately 30 days. There are eight phases, starting with the new moon, when the moon’s disk is dark, followed by waxing crescent, first quarter, waxing gibbous, full moon, waning gibbous, last quarter, and waning crescent. Which graph best represents the phases of the moon?

$Screen Shot 2022-12-22 at 11.26.39 PM.png$

Answer: (b)

Example 4.48

The table shows the number of hours of daylight in Glasgow, Scotland on the first of each month.

Month	Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
Daylight Hours	7.1	8.7	10.7	13.1	15.3	17.2	17.5	16.7	13.8	11.5	9.2	7.5

a Sketch a sinusoidal graph of daylight hours as a function of time, with $t=1$ in January.

b Estimate the period, amplitude, and midline of the graph.

Answer

a. Plot the data points and fit a sinusoidal curve by eye, as shown below.

$Screen Shot 2022-12-22 at 11.33.57 PM.png$

b. The period of the graph is 12 months. The midline is approximately $y = 12.25$, and the amplitude is approximately 5.25.

Checkpoint 4.49

The figure shows the number of daylight hours in Jacksonville, Florida, in Anchorage, Alaska, at the Arctic Circle, and at the Equator.

$Screen Shot 2022-12-22 at 11.35.29 PM.png$

a Which graph corresponds to each location?

b What are the maximum and minimum number of daylight hours in Jacksonville?

c For how long are there 24 hours of daylight per day at the Arctic Circle?

Answer

a From top to bottom in January: Equator, Jacksonville, Anchorage, Arctic Circle

b 14 hours and 10 hours

c Four months

Other Periodic Functions

There are other periodic functions besides sinusoidal functions. Any function that repeats a pattern at intervals of fixed length is periodic.

Periodic Function

The function $y = f(x)$ is periodic if there is a smallest value of $p$ such that

$f(x+p) = f(x)$

for all $x$. The constant $p$ is called the period of the function.

Example 4.50

Which of the functions shown below are periodic? If the function is periodic, give its period.

$Screen Shot 2022-12-22 at 11.44.40 PM.png$

Answer

a This graph is periodic with period 360.

b This graph is not periodic.

c This graph is periodic with period 8.

Checkpoint 4.51

Which of the functions shown below are periodic? If the function is periodic, give its period.

$Screen Shot 2022-12-23 at 12.04.51 AM.png$

Answer

a Period: 2

b Period: 3

c Not periodic

Example 4.52

A patient receives regular doses of medication to maintain a certain level of the drug in his body. After each dose, the patient’s body eliminates a certain percent of the medication before the next dose is administered. The graph shows the amount of the drug, in milliliters, in the patient’s body as a function of time in hours.

$Screen Shot 2022-12-23 at 12.06.40 AM.png$

a How much of the medication is administered with each dose?

b How often is the medication administered?

c What percent of the drug is eliminated from the body between doses?

Answer

a The medication level increase from 30 ml to 50 ml at each cycle of the graph, so 20 ml of medication are administered at each dose.

b The medication level peaks sharply evey four hours, when each new dose is administered.

c The medication level declines by $20$ ml between doses, or $\frac{20}{50} = 0.6$, or $60%$.

Checkpoint 4.53

You are sitting on your front porch late one evening, and you see a light coming down the road tracing out the path shown below, with distances in inches. You realize that you are seeing a bicycle light, fixed to the front wheel of the bike.

$Screen Shot 2022-12-23 at 12.16.54 AM.png$

a Approximately what is the period of the graph?

b How far above the ground is the light?

c What is the diameter of the bicycle wheel?

Answer

a 75 in

b 4 in

c 24 in

Review the following skills you will need for this section.

Algebra Refresher 4.5

Sketch the graph. Give the equations of any vertical or horizontal asymptotes.

1. $f(x) = \dfrac{1}{x-2}$

2. $g(x) = \dfrac{1}{x+3}$

3. $h(x) = \dfrac{1}{x^2-1}$

4. $H(x) = \dfrac{-1}{(x+1)^2}$

5. $G(x) = \dfrac{2x+1}{x}$

6. $F(x) = \dfrac{x}{x-1}$

Algebra Refresher Answers

1. $Screen Shot 2022-12-23 at 12.22.26 AM.png$

$x=2, \quad y=0$

2. $Screen Shot 2022-12-23 at 12.22.36 AM.png$

$x=-3, \quad y=0$

3. $Screen Shot 2022-12-23 at 12.22.46 AM.png$

$x=-1, \quad x=1 \quad y=0$

4. $Screen Shot 2022-12-23 at 12.23.01 AM.png$

$x=-1, \quad y=0$

5. $Screen Shot 2022-12-23 at 12.23.06 AM.png$

$x=0, \quad y=2$

6. $Screen Shot 2022-12-23 at 12.23.25 AM.png$

$x=1, \quad y=1$

Section 4.3 Summary

Vocabulary

• Sinusoidal

• Period

• Periodic

• Parameter

Concepts

Amplitude, Period, and Midline.

1. The graph of

$y=A \cos \theta \text { or } y=A \sin \theta$

has amplitude $|A|$.

2. The graph of

$y=\cos B \theta \text { or } y=\sin B \theta$

has period $\frac{360^{\circ}}{|B|}$.

3. The graph of

$y=k+\cos \theta \text { or } y=k+\sin \theta$

has midline $y=k$.

2 The graph of $y=k+A \sin B \theta$ has amplitude $A$, period $\frac{360^{\circ}}{B}$, and midline $y=k$. The same is true for the graph of $y=k+A \cos B \theta$.

3 Functions that have graphs shaped like sines or cosines are called sinusoidal.

Periodic Function.

4 The function $y=f(x)$ is periodic if there is a smallest value of $p$ such that

$f(x+p)=f(x)$

for all $x$. The constant $p$ is called the period of the function.

Study Questions

1 Sketch two examples of a function with period 8: one that is sinusoidal, and one that is not.

2 Delbert says that $\tan \left(\theta+180^{\circ}\right)=\tan \theta$ for any value of $\theta$. Is he correct? Explain why or why not.

Each statement in Questions 3–6 is false. Write a corrected statement.

3 The period of a sine or cosine function is the distance between horizontal intercepts.

4 The amplitude is the vertical distance between the maximum and minimum values.

5 The midline of the cosine graph is the vertical line $\theta=180^{\circ}$.

6 The cosine graph looks just like the sine graph except flipped upside down.

Skills

1 Graph periodic functions #1–6, 11–14, 27–28

2 Write equations for sinusoidal functions #7–12

3 Graph sinusoidal functions #13–18, 25–32

4 Find amplitude, period and midline #19–32

5 Fit a sinusoidal function to data or to a description #33–42

6 Find coordinates of points on a sinusoidal graph #43–48

7 Identify periodic functions and give their periods #49–52

8 Sketch graphs to model sinusoidal functions #53–66

9 Analyze periodic graphs #67–74

Homework 4.3

1. An ant is walking clockwise around the face of a sundial in the garden. The sundial is a circle with a 12-inch diameter, and the ant makes one circuit of the sundial in 24 seconds.

a Sketch the sundial and a coordinate system with its origin at the center of the dial. Draw the $y$-axis to align with the gnomon (pointer) of the sundial.

b Suppose we start timing the ant when it is at the tip of the gnomon. Complete the table showing the ant’s location as an angle in standard position, and its $y$-coordinate at that time.

$t$	0	2	4	6	8	10	12	14	16	18	20	22	24
$\theta$	$90^{\circ}$	$60^{\circ}$
$y=f(t)$	$6$	$3\sqrt{3}$

c Sketch a graph of the ant's $y$-coordinate as a function of time.

$Screen Shot 2022-12-26 at 10.07.03 PM.png$

d If the ant makes a second trip around the sundial, what will the graph look like from $t = 24$ to $t = 48$? Explain the statement $f(t + 24) = f(t)$.

2. Repeat Problem 1, but make a table and graph of the ant’s $x$-coordinate as a function of time.

$t$	0	2	4	6	8	10	12	14	16	18	20	22	24
$\theta$	$90^{\circ}$	$60^{\circ}$
$x=g(t)$	0	3

$Screen Shot 2022-12-26 at 10.21.11 PM.png$

3. Delbert is standing at the point (5, 0) on the square shown. If he walks around the square in the counter-clockwise direction, his $y$-coordinate is a function of the distance, $d$, he has walked.

$Screen Shot 2022-12-26 at 10.24.17 PM.png$

a What is the $y$-coordinate of Delbert’s position when he has walked 2 units? 5 units? 8 units? (Start filling in the table in part (c).)

b After Delbert reaches the upper right corner of the square, he will turn left and start walking along the top of the square. What is his $y$-coordinate from $d = 5$ to $d = 15$?

c Fill in the rest of the table with the $y$-coordinates of Delbert’s position when he has walked a distance $d$.

$d$	0	2	5	8	10	12	15	18	20	22	25	28	30	32	35	38	40
$y$

d Plot a graph of Delbert's coordinate as a function of the distance he has walked.

$Screen Shot 2022-12-26 at 10.28.02 PM.png$

4. Repeat Problem 3, except consider Delbert’s $x$-coordinate as a function of the distance he has walked.

$d$	0	2	5	8	10	12	15	18	20	22	25	28	30	32	35	38	40
$x$

$Screen Shot 2022-12-26 at 10.30.22 PM.png$

a Refer to Problem 3. Suppose you know where Delbert is on the square at some given time. Can you predict where he will be after he has walked another 40 units?

b Let $y=f(d)$. What does your answer say about $f(d)$ and $f(d+40)$ for any positive value of $d$?

c How does the graph of $f$ for $0 \leq d \leq 40$ compare with the graph for $40 \leq d \leq 80$?

d Describe how the graph of $f$ would continue for $0 \leq d \leq 400$.

6. Refer to Problem 4. Sketch a graph of Delbert's $x$-coordinate, $x=g(d)$, for $0 \leq d \leq 120$.

7. Write an equation for a sine function with amplitude 6.

8. Write an equation for a cosine function with amplitude $\frac{1}{2}$.

9. Write an equation for a cosine function with midline $-5$.

10. Write an equation for a sine function with midline 2.

11. Write an equation for a sine function with period $90^{\circ}$.

12. Write an equation for a cosine function with period $720^{\circ}$.

For Problems 13–18,

a Graph the function.

b State the amplitude, period and midline of the function.

13. $y = 3\cos \theta$

$Screen Shot 2022-12-26 at 10.37.49 PM.png$

14. $y=4\sin \theta$

$Screen Shot 2022-12-26 at 10.38.03 PM.png$

15. $y=3 + \sin \theta$

$Screen Shot 2022-12-26 at 10.38.10 PM.png$

16. $y = -2 + \cos \theta$

$Screen Shot 2022-12-26 at 10.38.20 PM.png$

17. $y = \cos 3\theta$

$Screen Shot 2022-12-26 at 10.38.30 PM.png$

18. $y = \sin 2\theta$

$Screen Shot 2022-12-26 at 10.38.40 PM.png$

For Problems 19-24, graph the function in the Trig window (ZOOM 7), but change Ymin to -10 and Ymax to 10. State the amplitude, period, and midline.

19. $y=3+4\cos \theta$

20. $y = -4+3\sin \theta$

21. $y = 5\sin 2\theta$

22. $y = 6\cos 4\theta$

23. $f(\theta) = -4+3\sin 3\theta$

24. $f(\theta) = 2 + 4\cos 3\theta$

For Problems 25-32,

a State the amplitude, period, and midline for the graph.

b Write an equation for the graph using sine or cosine.

25. $Screen Shot 2022-12-26 at 10.44.53 PM.png$

26. $Screen Shot 2022-12-26 at 10.45.02 PM.png$

27. $Screen Shot 2022-12-26 at 10.45.17 PM.png$

28. $Screen Shot 2022-12-26 at 10.45.25 PM.png$

29. $Screen Shot 2022-12-26 at 10.45.39 PM.png$

30. $Screen Shot 2022-12-26 at 10.45.54 PM.png$

31. $Screen Shot 2022-12-26 at 10.46.00 PM.png$

32. $Screen Shot 2022-12-26 at 10.46.07 PM.png$

For Problems 33–36, the table describes a sine or cosine function. Find an equation for the function.

33.

$\theta$	$0^{\circ}$	$45^{\circ}$	$90^{\circ}$	$135^{\circ}$	$180^{\circ}$	$225^{\circ}$	$270^{\circ}$	$315^{\circ}$	$360^{\circ}$
$f(\theta)$	7	5.56	2	-1.54	-3	-1.54	2	5.54	7

34.

$\theta$	$0^{\circ}$	$45^{\circ}$	$90^{\circ}$	$135^{\circ}$	$180^{\circ}$	$225^{\circ}$	$270^{\circ}$	$315^{\circ}$	$360^{\circ}$
$f(\theta)$	1	3.12	4	3.12	1	-1.12	-2	-1.12	1

35.

$\theta$	$0^{\circ}$	$45^{\circ}$	$90^{\circ}$	$135^{\circ}$	$180^{\circ}$	$225^{\circ}$	$270^{\circ}$	$315^{\circ}$	$360^{\circ}$
$f(\theta)$	0	-2.83	-4	-2.83	0	2.83	4	2.83	0

36.

$\theta$	$0^{\circ}$	$45^{\circ}$	$90^{\circ}$	$135^{\circ}$	$180^{\circ}$	$225^{\circ}$	$270^{\circ}$	$315^{\circ}$	$360^{\circ}$
$f(\theta)$	-9	-6.36	0	6.36	9	6.36	0	-6.36	-9

For Problems 37–42, write the equation of a sine or cosine function with the given properties.

37. Midline $y=-4$, amplitude 6, period $120^{\circ}$
38. Midline $y=5$, amplitude $\frac{3}{2}$, period $180^{\circ}$
39. Maximum points at $\left(0^{\circ}, 5\right)$ and $\left(360^{\circ}, 5\right)$, minimum point at $\left(180^{\circ}, 1\right)$
40. Maximum point at $\left(90^{\circ}, 1\right)$, minimum point at $\left(270^{\circ},-3\right)$
41. Horizontal intercepts at $45^{\circ}$ and $135^{\circ}$, vertical intercept at $\left(0^{\circ}, 12\right)$
42. Horizontal intercepts at $30^{\circ}$ and $90^{\circ}$, vertical intercept at $\left(0^{\circ},-8\right)$

For Problems 43–48, give the coordinates of the points on the graph.

43. $f(\theta) = -3\cos \theta$

$Screen Shot 2022-12-26 at 11.10.58 PM.png$

44. $f(\theta) = 4\sin \theta$

$Screen Shot 2022-12-26 at 11.11.10 PM.png$

45. $f(\theta) = \sin 4\theta$

$Screen Shot 2022-12-26 at 11.11.21 PM.png$

46. $f(\theta) = -\cos 3\theta$

$Screen Shot 2022-12-26 at 11.11.30 PM.png$

47. $f(\theta) = -3 + \cos \theta$

$Screen Shot 2022-12-26 at 11.11.36 PM.png$

48. $f(\theta) = 1 + \sin \theta$

$Screen Shot 2022-12-26 at 11.11.50 PM.png$

Which of the graphs in Problems 49-52 are periodic? If the graph is periodic, give its period.

49. $Screen Shot 2022-12-26 at 11.18.19 PM.png$

50. $Screen Shot 2022-12-26 at 11.18.24 PM.png$

51. $Screen Shot 2022-12-26 at 11.18.30 PM.png$

52. $Screen Shot 2022-12-26 at 11.18.38 PM.png$

For Problems 53–56, sketch a periodic function that models the situation.

53. At a ski slope, the lift chairs take 5 minutes to travel from the bottom, at an elevation of 3000 feet, to the top, at elevation 4000 feet. The cable supporting the ski lift chairs is a loop turning on pulleys at a constant speed. At the top and bottom, the chairs are at a constant elevation for a few seconds to allow skiers to get on and off.

a Sketch a graph of $h(t)$, the height of one chair at time $t$. Show at least two complete trips.

b What is the period of $h(t)$?

54. The heater in Paul’s house doesn’t have a thermostat; it runs on a timer. It uses 300 watts when it is running. Paul sets the heater to run from 6 am to noon, and again from 4 pm to 10 pm in the evening.

a Sketch a graph of $P(t)$, the power drawn by the heater as a function of time. Show at least two days of heater use.

b What is the period of $P(t)$?

55. Francine adds water to her fish pond once a week to keep the depth at 30 centimeters. During the week the water evaporates at a constant rate of 0.5 centimeters per day.

a Sketch a graph of $D(t)$, the depth of the water as a function of time. Show at least two weeks.

b What is the period of $D(t)$?

56. Erin’s fox terrier, Casey, is very energetic and bounces excitedly at dinner time. Casey can jump 30 inches high, and each jump takes him 0.8 seconds.

a Sketch a graph of Casey’s height, $D(t)$, as a function of time. Show at least two jumps.

b What is the period of $h(t)$?

For Problems 57–62, sketch a sinusoidal function that models the situation.

57. Delbert’s bicycle wheel is 24 inches in diameter, and he has a light attached to the spokes 10 inches from the center of the wheel. It is dark, and he is cycling home slowly from work. The bicycle wheel makes one revolution every second.

a At $t = 0$, the light is at its highest point the bicycle wheel. Sketch a graph of the light’s height as a function of $t$.