Name: ________________________ Period: ___________________ Date: __________ ID: A
1
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 0.05 seconds, 4.5 C 0.025 seconds, 9B 0.05 seconds, 9 D 0.025 seconds, 4.5
Name: ________________________ ID: A
2
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
Name: ________________________ ID: A
3
Use the graph to find the value of y = sin θ θ θ θ for the value of θθθθ.
3. 14
π radians
A –0.7 B 0.7 C –0.9 D 0.9
4. Find the period of the graph shown below.
A 2π B23
π C12
π D 4π
5. A particular sound wave can be graphed using the function y = −3 sinx . Find the amplitude and period of the function.A amplitude = 3, period = 2π C amplitude = 2π, period = 3
B amplitude = 12
π, period = –3 D amplitude = –3, period = 12
Name: ________________________ ID: A
4
6. Sketch one cycle of y = 4 sin 4θ.A C
B D
Name: ________________________ ID: A
5
Write a cosine function for the graph.
7.
A y = −2 cos x + 3π4
Ê
Ë
ÁÁÁÁÁ
ˆ
¯
˜˜˜
− 2 C y = −2 cos x − 3π4
Ê
Ë
ÁÁÁÁÁ
ˆ
¯
˜˜˜
− 2
B y = 2 cos x − 3π4
Ê
Ë
ÁÁÁÁÁ
ˆ
¯
˜˜˜
− 2 D y = 2cos x + 3π4
Ê
Ë
ÁÁÁÁÁÁ
ˆ
¯
˜˜˜
− 2
8. Use the graph of y = tanx to find the value of y = tan 18
π . Round to the nearest tenth if necessary. If the tangent
is undefined at that point, write undefined.
A 0 B 1 C 0.4 D 2.4
Name: ________________________ ID: A
6
9. Sketch the graph of the tangent curve y = tan 2x.A C
B D
10. Identify the period for y = tan3π4
θ and tell where two asymptotes occur for the function.
A period = 3π4
; two asymptotes at x = –3π8
and x = 3π8
B period = 83
; two asymptotes at x = −43
and x = 43
C period = 43
; two asymptotes at x = 23
and x = 2
D period = 43
; two asymptotes at x = 0 and x = 43
Name: ________________________ ID: A
7
Graph the function in the interval from 0 to 2ππππ.
11. y = 4 cos 12
θ
A C
B D
Name: ________________________ ID: A
8
12. y = 2 cos x − π6
Ê
Ë
ÁÁÁÁÁ
ˆ
¯
˜˜˜
+ 2
A C
B D
Name: ________________________ ID: A
9
13. y = sec x
A C
B D
14. Write an equation for the translation 3 units down of y = sin x.A y = sin(x + 3) C y = sinx + 3B y = sin(x − 3) D y = sinx − 3
Name: ________________________ ID: A
10
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.
A y = 25 sin π3
xÊ
Ë
ÁÁÁÁÁ
ˆ
¯
˜˜˜
+ 20 C y = 20 sin π3
x + 3π2
Ê
Ë
ÁÁÁÁÁ
ˆ
¯
˜˜˜
+ 25
B y = −25 sin π3
xÊ
Ë
ÁÁÁÁÁ
ˆ
¯
˜˜˜
D y = sin π3
x − 3π2
Ê
Ë
ÁÁÁÁÁ
ˆ
¯
˜˜˜
+ 25
16. Find the amplitude of the periodic function.
A 3.5 B 3 C 7 D 1.75
Name: ________________________ ID: A
11
Short Answer
17. Use the graph of the sine function y = 2 sinx 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?
19. A particular sound wave is modeled in the laboratory with the function y = 3 sin 4 x − π2
Ê
Ë
ÁÁÁÁÁ
ˆ
¯
˜˜˜
+ 2.
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 = 32
sin 14
θ. Sketch a graph of this function.
Name: ________________________ ID: A
12
22. Sketch one cycle of y = −2sin18
x + πÊ
Ë
ÁÁÁÁÁÁ
ˆ
¯
˜˜˜and label. Then answer the questions below.
a. What is the range?b. What is the amplitude?c. What is the period?
23. Sketch one cycle of y = −2csc18
x + πÊ
Ë
ÁÁÁÁÁÁ
ˆ
¯
˜˜˜ and label. Then answer the questions below.
a. What are all the vertical asymptotes?b. What is the range?
24. Graph one cycle of y = cos12
πxÊ
Ë
ÁÁÁÁÁÁ
ˆ
¯
˜˜˜
− 3 and label. Then answer the questions below.
a. What is the y-intercept?b. What is the period?c. What is the amplitude?d. What are all the minimum points?
25. Graph one cycle of y = sec12
πxÊ
Ë
ÁÁÁÁÁÁ
ˆ
¯
˜˜˜
− 3 and label. Then answer the questions below.
a. What is the domain?b. What is the amplitude?c. What kind of function is it?
26. Graph one cycle of y = tan14
xÊ
Ë
ÁÁÁÁÁÁ
ˆ
¯
˜˜˜ and label. Then answer the questions below.
a. What is the period?b. What are all the x-intercepts?c. What are all the asymptotes?
27. Graph one cycle of y = cot14
xÊ
Ë
ÁÁÁÁÁÁ
ˆ
¯
˜˜˜ and label. Then answer the questions below.
a. What is the domain?b. What are the extrema?
Name: ________________________ ID: A
13
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 = 32
cos t2
.
Name: ________________________ ID: A
14
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 = 14
cos2π3
θ .
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 = sinx , y = sin 2x, and y = 2 sinx . Describe how the graphs are alike and how they are different.
ID: A
1
Precalculus ~ Review Sheet 4.4-4.5Answer Section
MULTIPLE CHOICE
1. A 2. A 3. B 4. B 5. A 6. C 7. B 8. C 9. A 10. C 11. A 12. D 13. A 14. D 15. C 16. A
SHORT ANSWER
17. a. 1 cycleb. 2πc. 2
18. a. The period is π and the amplitude is 4.b. maximum values: 0 : π : 2π
minimum values: 12
π : 32
π
zeros: 14
π : 34
π : 54
π : 74
π
19. a. 3
b. 12
π
c. phase shift is π2
units to the right
d. vertical shift is 2 units up
ID: A
2
20. a.
b. y = sin 12
θ
21.
22. a. −2,2È
ÎÍÍÍ
˘˚˙˙
b. 2c. 16π
ID: A
3
23. a. x = 8πn andn ∈ Ω
b. −∞,−2ÊËÁÁ
˘
˚˙˙ ∪ 2,∞È
ÎÍÍÍÍ
ˆ¯˜
24. a. 0,−2Ê
ËÁÁ ˆ
¯˜
b. 4c. 1d. 2+ 4n,−4Ê
ËÁÁ ˆ
¯˜ andn ∈ Ω
ID: A
4
25.
a. x: x ∈ ò exceptx ≠ 1+ 2n andn ∈ ΩÏÌÓ
ÔÔÔÔ
¸˝˛
ÔÔÔÔ
b. nonec. even
26. a. 4πb. 2πn,0Ê
ËÁÁ ˆ
¯˜ andn ∈ Ω
c. x = 2π + 4πn andn ∈ Ω
ID: A
5
27.
a. x: x ∈ ò exceptx ≠ 2π + 4πn andn ∈ ΩÏÌÓ
ÔÔÔÔ
¸˝˛
ÔÔÔÔ
b. none 28. not periodic 29. periodic; about 6
30. period = 4π, range:−32
≤ y ≤ 32
, amplitude= 32
31. y = sin 3θ
ID: A
6
ESSAY
32.
[4] a. The period of function y = a cosbθ , with b > 0, is 2πb
. For the equation
y = 14
cos2π3
θ , b = 2π3
.
2πb
= 2π2π3
Substitute.
= 2π ⋅ 32π Multiply by the reciprocal to divide.
= 3 Simplify.
The period of the function is 3.
b. The amplitude for a function of the form y = a cosbθ , with b > 0,
is | a |. For the equation y = 14
cos2π3
θ , a = 14
.
The amplitude of the function is 14
.
[3] one incorrect computation OR an error in reasoning[2] two computational errors OR poor explanation[1] 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.
ID: A
7
34.
Answers may vary. Sample:All three graphs pass through (0º, 0), (180º, 0) and (360º, 0). The graphs of y = sinx and y = sin 2x have amplitude 1 while the graph of y = 2 sinx has amplitude 2. The graphs of y = sinx and y = 2 sinx 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 :-)
