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Algebra 2

Prentice Hall, Basia Kennedy, Dan Ramirez

Chapter 13

Periodic Functions And Trigonometry

Educators


Problem 1

Identify one cycle in two different ways. Then determine the period of the function.

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Problem 2

Identify one cycle in two different ways. Then determine the period of the function.

Amrita B.
Numerade Educator

Problem 3

Identify one cycle in two different ways. Then determine the period of the function.

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Numerade Educator

Problem 4

Determine whether each function is or is not periodic. If it is, find the period.

Amrita B.
Numerade Educator

Problem 5

Determine whether each function is or is not periodic. If it is, find the period.

Vikash R.
Numerade Educator

Problem 6

Determine whether each function is or is not periodic. If it is, find the period.

Amrita B.
Numerade Educator

Problem 7

Determine whether each function is or is not periodic. If it is, find the period.

Vikash R.
Numerade Educator

Problem 8

Determine whether each function is or is not periodic. If it is, find the period.

Amrita B.
Numerade Educator

Problem 9

Determine whether each function is or is not periodic. If it is, find the period.

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Numerade Educator

Problem 10

Find the amplitude of each periodic function.

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Problem 11

Find the amplitude of each periodic function.

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Problem 12

Find the amplitude of each periodic function.

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Problem 13

Find the amplitude of each periodic function.

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Problem 14

Sketch the graph of a sound wave with the given period and amplitude.
period $0.02,$ amplitude 4

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Problem 15

Sketch the graph of a sound wave with the given period and amplitude.
period $0.005,$ amplitude 9

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Problem 16

Sketch the graph of a sound wave with the given period and amplitude.
Complete each statement with $x$ or $y .$
a. You use I-values to compute the amplitude of a function.
b. You use I-values to compute the period of a function.

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Problem 17

Writing Could you use a periodic function to represent each situation described below? Explain.
the average monthly temperature in your community, recorded every month for three years

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Problem 18

Writing Could you use a periodic function to represent each situation described below? Explain.
the population in your community, recorded every year for the last 50 years

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Problem 19

Writing Could you use a periodic function to represent each situation described below? Explain.
the number of cars per hour that pass through an intersection near where you live, recorded for two consecutive work days

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Problem 20

Health Use the graph below for Exercises 20 and $21 .$
A person's pulse rate is the number of times his or her heart beats in one minute. Each cycle in the graph represents one heartbeat. Find the pulse rate.

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Problem 21

Use the graph below for Exercises 20 and 21
An electrocardiogram (EKG or ECG) measures the electrical activity of a person's heart in millivolts over time.
a. What is the period of the EKG shown above?
b. What is the amplitude of the EKG?

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Problem 22

Use the graph below for Exercises 20 and 21
Open-Ended Sketch a graph of a periodic function that has a period of 3 and an amplitude of $2 .$

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Problem 23

Find the maximum, minimum, and period of each periodic function. Then copy the graph and sketch two more cycles.

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Problem 24

Find the maximum, minimum, and period of each periodic function. Then copy the graph and sketch two more cycles.

Amrita B.
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Problem 25

Find the maximum, minimum, and period of each periodic function. Then copy the graph and sketch two more cycles.

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Problem 26

Language Arts Functions that repeat over time common in everyay life. The English language has many words that stand for common periods of time. State the period of time from which each term derives.
annual

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Problem 27

Language Arts Functions that repeat over time common in everyay life. The English language has many words that stand for common periods of time. State the period of time from which each term derives.
biweekly

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Problem 28

Language Arts Functions that repeat over time common in everyay life. The English language has many words that stand for common periods of time. State the period of time from which each term derives.
quarterly

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Problem 29

Language Arts Functions that repeat over time common in everyay life. The English language has many words that stand for common periods of time. State the period of time from which each term derives.
hourly

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Problem 30

Language Arts Functions that repeat over time common in everyay life. The English language has many words that stand for common periods of time. State the period of time from which each term derives.
circadian

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Problem 31

Suppose $f$ is a periodic function. The period of $f$ is 5 and $f(1)=2 .$ Find $f(6),$ $f(11),$ and $f(-4) .$

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Problem 32

Suppose $g$ is a periodic function. The period of $g$ is $24, g(3)=67,$ and $g(8)=70$ Find each function value.

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Problem 33

Motion You are sitting on a pier watching the waves when you notice a bottle in the water. The bottle bobs so that it is between 2.5 $\mathrm{ft}$ and 4.5 $\mathrm{ft}$ below the pier. You know you can reach 3 $\mathrm{ft}$ below the pier. Suppose the bottle reaches its highest point every 5 $\mathrm{s} .$
a. Sketch a graph of the bottle's distance below the pier for 15 s. Assume that at $t=0,$ the bottle is closest to the pier.
b. Find the period and the amplitude of the function.
c. Estimatien Use your graph to estimate the length of time the bottle is within reach during each cycle.

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Problem 34

calendar $\mathrm{A}$ day - the time Earth takes to rotate from one noon to the next- is a basic measure of time. A solar year is about 365.2422 days. We try to keep our calendar in step with the solar year.
a. If every calendar year has 365 days, by how many days would the calendar year and the solar year differ after 100 years?
b. If every fourth year has an extra "leap" day added, by how many days would the two systems differ after 100 years?
c. If every hundred years the "leap" day is omitted, by how many days would the two systems differ after 100 years?
d. Critical Thinking Why is it important for the difference between the calendar year and the solar year to be zero?

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Problem 35

A periodic function goes through 5 complete cycles in 4 min. What is the period of the function?
$\begin{array}{lllll}{\text { A. } \frac{1}{5} \min } & {\text { B. } \frac{1}{4} \min } & {\text { C. } 48 \mathrm{s}} & {\text { D. } 75 \mathrm{s}}\end{array}$

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Problem 36

The period of a periodic function is 8 s. How many cycles does it go through in 30 $\mathrm{s?}$
F. $\frac{4}{15}$ cycle
G. 3.75 cycles
H. 22 cycles
J. 240 cycles

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Problem 37

Which graph is NOT the graph of a periodic function?

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Problem 38

The amplitude of a periodic function is 2.5 and its minimum value is $0 .$ What is the function's maximum value?
$\mathrm{F} .-2.5$
G. 0
H. 2.5

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Problem 39

A periodic function completes $m$ cycles in $n$ seconds. What is the period of the function? Show your work.

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Problem 40

A machine begins recording two periodic functions at the same time. The first has a period of 6 s. The second has a period of 7 s. After 20 $\mathrm{s}$ , the machine begins recording a third periodic function, with a period of 8 $\mathrm{s}$ . How many seconds after the machine begins recording the third function are all three functions at the beginning of their periods Explain.

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Problem 41

Sketch a normal curve for a distribution that has mean 57 and standard deviation 12 . Label the $x$ -axis values at one, two, and three standard deviations from the mean.

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Problem 42

Find the next two terms in each sequence. Write a formula for the $n$ th term. Identify each formula as explicit or recursive.
$$
1,3,5,7,9, \dots
$$

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Problem 43

Find the next two terms in each sequence. Write a formula for the $n$ th term. Identify each formula as explicit or recursive.
$$
4,6,8,10,12, \dots
$$

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Problem 44

Find the next two terms in each sequence. Write a formula for the $n$ th term. Identify each formula as explicit or recursive.
$$
3,6,11,18,27, \dots
$$

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Problem 45

Write an equation for each conic section. Then sketch the graph.
parabola with vertex $(-3,2)$ and focus $(-3,7)$

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Problem 46

Write an equation for each conic section. Then sketch the graph.
$$
(5,-3), \text { one focus at }(5,0), \text { and one vertex at }(5,-1)
$$

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Numerade Educator

Problem 47

Write an equation for each conic section. Then sketch the graph.
$$
\begin{array}{l}{\text { ellipse with center }(-2,1), \text { vertices at }(-6,1) \text { and }(2,1), \text { and co-vertices at }} \\ {(-2,3) \text { and }(-2,-1)}\end{array}
$$

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