Name: ________________________ Period: ___________________ Date: __________
ID: A
Precalculus ~ Review Sheet 4.4-4.5
Multiple Choice
1. The screen below shows the graph of a sound recorded on an oscilloscope. What is the period and the amplitude?
(Each unit on the t-axis equals 0.01 seconds.)
A
B
0.05 seconds, 4.5
0.05 seconds, 9
C
D
0.025 seconds, 9
0.025 seconds, 4.5
1
Name: ________________________
ID: A
2. A sound wave has a period of 0.02 seconds and an amplitude of 3 units. Sketch a graph of the sound wave.
A
B
C
D
2
Name: ________________________
ID: A
Use the graph to find the value of y = sin θ for the value of θ.
3.
1
π radians
4
–0.7
B 0.7
4. Find the period of the graph shown below.
A
C
–0.9
D
0.9
2
1
π
C
π
D 4π
3
2
5. A particular sound wave can be graphed using the function y = −3 sin x . Find the amplitude and period of the
function.
A amplitude = 3, period = 2π
C amplitude = 2π, period = 3
1
1
B amplitude = π, period = –3
D amplitude = –3, period =
2
2
A
2π
B
3
Name: ________________________
ID: A
6. Sketch one cycle of y = 4 sin 4θ.
A
C
B
D
4
Name: ________________________
ID: A
Write a cosine function for the graph.
7.
ÊÁ
3π ˆ˜˜˜
y = −2 cos ÁÁÁÁ x −
−2
4 ˜˜¯
Ë
ÁÊ
3π ˜ˆ˜˜
−2
y = 2cos ÁÁÁÁ x +
B
D
4 ˜˜¯
Ë
1
8. Use the graph of y = tan x to find the value of y = tan π . Round to the nearest tenth if necessary. If the tangent
8
is undefined at that point, write undefined.
A
A
ÊÁ
3π ˆ˜˜˜
y = −2 cos ÁÁÁÁ x +
−2
4 ˜˜¯
Ë
ÊÁ
3π ˆ˜˜˜
y = 2 cos ÁÁÁÁ x −
−2
4 ˜˜¯
Ë
0
B
1
C
C
0.4
5
D
2.4
Name: ________________________
ID: A
9. Sketch the graph of the tangent curve y = tan 2x.
A
C
B
D
3π
θ and tell where two asymptotes occur for the function.
4
3π
3π
3π
; two asymptotes at x = –
and x =
4
8
8
8
4
4
; two asymptotes at x = − and x =
3
3
3
4
2
; two asymptotes at x = and x = 2
3
3
4
4
; two asymptotes at x = 0 and x =
3
3
10. Identify the period for y = tan
A
period =
B
period =
C
period =
D
period =
6
Name: ________________________
ID: A
Graph the function in the interval from 0 to 2π.
11.
y = 4 cos
1
θ
2
A
C
B
D
7
Name: ________________________
ID: A
ÁÊ
π ˜ˆ
12. y = 2 cos ÁÁÁÁ x − ˜˜˜˜ + 2
6¯
Ë
A
C
B
D
8
Name: ________________________
ID: A
13. y = sec x
A
C
B
D
14. Write an equation for the translation 3 units down of y = sin x.
A
C
y = sin (x + 3)
y = sin x + 3
B
D
y = sin (x − 3)
y = sin x − 3
9
Name: ________________________
ID: A
15. The graph below shows height as a function of time for a ride on a Ferris wheel. Find a sine equation for the
graph.
ÊÁ π ˆ˜
y = 25 sin ÁÁÁÁ x ˜˜˜˜ + 20
Ë3 ¯
ÊÁ π ˆ˜
B
y = −25 sin ÁÁÁÁ x ˜˜˜˜
Ë3 ¯
16. Find the amplitude of the periodic function.
A
A
3.5
B
3
C
D
C
ÊÁ π
3π ˆ˜˜˜
y = 20 sinÁÁÁÁ x +
+ 25
2 ˜˜¯
Ë3
ÊÁ π
3π ˆ˜˜˜
y = sinÁÁÁÁ x −
+ 25
2 ˜˜¯
Ë3
7
10
D
1.75
Name: ________________________
ID: A
Short Answer
17. Use the graph of the sine function y = 2 sin x shown below.
a. How many cycles occur in the graph?
b. Find the period of the graph.
c. Find the amplitude of the graph.
18. Consider the graph of the cosine function shown below.
a. Find the period and amplitude of the cosine function.
b. At what values of x for 0 ≤ θ ≤ 2π do the maximum value(s), minimum values(s), and zeros occur?
ÁÊ
π ˜ˆ
19. A particular sound wave is modeled in the laboratory with the function y = 3 sin 4 ÁÁÁÁ x − ˜˜˜˜ + 2 .
2¯
Ë
a. Find the amplitude of the function.
b. Find the period of the function.
c. Describe any phase shift in the graph.
d. Describe any vertical shift in the graph.
20. a. Sketch at least one cycle of a sine curve with amplitude 1 and period of 4π. Assume a > 0.
b. Write an equation for the graph.
21. An electromagnetic wave is modeled with the function y =
11
3
1
sin θ. Sketch a graph of this function.
2
4
Name: ________________________
ID: A
ÁÊ 1
˜ˆ
22. Sketch one cycle of y = −2sin ÁÁÁÁ x + π ˜˜˜˜ and label. Then answer the questions below.
Ë8
¯
a. What is the range?
b. What is the amplitude?
c. What is the period?
ÊÁ 1
ˆ˜
23. Sketch one cycle of y = −2csc ÁÁÁ x + π ˜˜˜˜ and label. Then answer the questions below.
ÁË 8
¯
a. What are all the vertical asymptotes?
b. What is the range?
ÁÊ 1 ˜ˆ
24. Graph one cycle of y = cos ÁÁÁÁ π x ˜˜˜˜ − 3 and label. Then answer the questions below.
Ë2 ¯
a. What is the y-intercept?
b. What is the period?
c. What is the amplitude?
d. What are all the minimum points?
ÊÁ 1 ˆ˜
25. Graph one cycle of y = sec ÁÁÁ π x ˜˜˜˜ − 3 and label. Then answer the questions below.
ÁË 2 ¯
a. What is the domain?
b. What is the amplitude?
c. What kind of function is it?
ÊÁ 1 ˆ˜
26. Graph one cycle of y = tan ÁÁÁ x ˜˜˜˜ and label. Then answer the questions below.
ÁË 4 ¯
a. What is the period?
b. What are all the x-intercepts?
c. What are all the asymptotes?
ÁÊ 1 ˜ˆ
27. Graph one cycle of y = cot ÁÁÁÁ x ˜˜˜˜ and label. Then answer the questions below.
Ë4 ¯
a. What is the domain?
b. What are the extrema?
12
Name: ________________________
ID: A
Determine whether the function shown below is or is not periodic. If it is, find the period.
28.
29.
30. Find the period, range, and amplitude of the cosine function y =
13
3
t
cos .
2
2
Name: ________________________
ID: A
31. Write the equation for the sine function shown below.
Essay
32. A sound wave is modeled with cosine function having an equation of y =
1
2π
cos
θ.
4
3
a. Find the period of the cosine function. Explain your method.
b. Find the amplitude of the cosine function. Explain your method.
Other
33. Suppose that you record the average number of cars per month that pass through an intersection in a particular city
every month for 10 years. Then you graph the data of the month where x is a number corresponding to the month
(January = 1, February = 2, etc.), and y is the number of cars. Is this function periodic? Explain your reasoning.
34. Sketch the graphs of y = sin x , y = sin 2x , and y = 2 sin x . Describe how the graphs are alike and how they are
different.
14
ID: A
Precalculus ~ Review Sheet 4.4-4.5
Answer Section
MULTIPLE CHOICE
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
A
A
B
B
A
C
B
C
A
C
A
D
A
D
C
A
SHORT ANSWER
17. a. 1 cycle
b. 2π
c. 2
18. a. The period is π and the amplitude is 4.
b. maximum values: 0 : π : 2π
1
3
minimum values: π : π
2
2
1
3
5
7
zeros: π : π : π : π
4
4
4
4
19. a. 3
1
b. π
2
c. phase shift is
π
units to the right
2
d. vertical shift is 2 units up
1
ID: A
20. a.
1
b. y = sin θ
2
21.
22.
a.
b.
c.
ÈÍ
˘
ÍÎ −2,2 ˙˙˚
2
16π
2
ID: A
23.
a.
b.
x = 8π n and n ∈ Ω
ÊÁ −∞,−2 ˘˙˙ ∪ ÈÍÍ 2,∞ ˆ˜
˙˚ ÍÎ
Ë
¯
24.
a. ÊÁË 0,−2 ˆ˜¯
b. 4
c. 1
d. ÊÁË 2 + 4n,−4 ˆ˜¯ and n ∈ Ω
3
ID: A
25.
ÔÏ
Ô¸
a. ÌÔ x: x ∈ ò except x ≠ 1 + 2n and n ∈ Ω ˝Ô
Ó
˛
b. none
c. even
26.
a.
b.
4π
ÊÁ 2π n,0 ˆ˜ and n ∈ Ω
Ë
¯
c. x = 2π + 4π n and n ∈ Ω
4
ID: A
27.
28.
29.
30.
31.
ÏÔ
¸Ô
a. ÌÔ x: x ∈ ò except x ≠ 2π + 4π n and n ∈ Ω ˝Ô
Ó
˛
b. none
not periodic
periodic; about 6
3
3
3
period = 4π , range: − ≤ y ≤ , amplitude =
2
2
2
y = sin 3θ
5
ID: A
ESSAY
32.
[4]
a.
The period of function y = a cos b θ , with b > 0, is
2π
. For the equation
b
1
2π
2π
cos
θ,b=
.
4
3
3
2π
2π
=
Substitute.
2π
b
3
3
= 2π ⋅
Multiply by the reciprocal to divide.
2π
y =
=3
Simplify.
The period of the function is 3.
b.
[3]
[2]
[1]
The amplitude for a function of the form y = a cos b θ , with b > 0,
1
2π
1
is | a |. For the equation y = cos
θ,a= .
4
3
4
1
The amplitude of the function is .
4
one incorrect computation OR an error in reasoning
two computational errors OR poor explanation
one or more answers missing OR no explanation
OTHER
33. Answers may vary. Sample: It will not be a periodic function. The number of cars passing through an
intersection at least partially depends upon the population. Population growth is normally exponential and not
periodic.
6
ID: A
34.
Answers may vary. Sample:
All three graphs pass through (0º, 0), (180º, 0) and (360º, 0). The graphs of y = sin x and y = sin 2x have
amplitude 1 while the graph of y = 2 sin x has amplitude 2. The graphs of y = sin x and y = 2 sin x have a
period of 360º, while the graph of y = sin 2x has a period of 180º.
Note: We will be using radians instead of degrees to explain this answer :-)
7