This document contains information about periodic functions including definitions of key terms like period, amplitude, and cycle. It includes examples of analyzing periodic functions to determine whether they are periodic and calculating their period and amplitude. Specific examples determine if relations are functions, identify periods of periodic graphs, and find the period and amplitude of a graph showing alternating current electricity in the United States over time.

1. Top School in Gudgao
By:
School.edhole.com

2. 13.1 – Exploring
Periodic Data
School.edhole.com

3. Determine whether each relation is a function.
1. {(2, 4), (1, 3), (–3, –1), (4, 6)} 2. {(2, 6), (–3, 1), (–2, 2)}
3. {(x, y)| x = 3} 4. {(x, y)| y = 8}
5. {(x, y)| x = y2} 6. {(x, y)| x2 + y2 = 36}
7. {(a, b)| a = b3} 8. {(w, z)| w = z – 36}
Exploring Periodic Data
School.edhole.com

4. Solutions
1. {(2, 4), (1, 3), (–3, –1), (4, 6)}; yes, this is a function because each element of
the domain is paired with exactly one element in the range.
2. {(2, 6), (–3, 1), (–2, 2)}; yes, this is a function because each element of the
domain is paired with exactly one element in the range.
3. {(x, y)| x = 3}; no, this is not a function because it is a vertical line and fails
the vertical line test.
4. {(x, y)| y = 8}; yes, this is a function because it is a horizontal line and passes
the vertical line test.
Exploring Periodic Data
School.edhole.com

5. Solutions (continued)
5. {(x, y)| x = y2}; no, this is not a function because an element of the domain is
paired with more than one element in the range.
Example: 4 = 22 and 4 = (–2)2
6. {(x, y)| x2 + y2 = 36}; no, this is not a function because it is a circle and fails
the vertical line test.
7. {(a, b)| a = b3}; yes, this is a function because each element of the domain is
paired with exactly one element in the range.
8. {(w, z)| w = z – 36}; yes, this is a function because each element of the
domain is paired with exactly one element in the range.
Exploring Periodic Data
School.edhole.com

6. Periodic Function: a repeating pattern of y-values (outputs) at regular intervals.
Cycle: One complete pattern of the function. A cycle can occur at any point on
the graph of the function
Period: the horizontal length of one cycle of the function.
Amplitude: half of the distance between the minimum and maximum values of
the function.
Amplitude
Vocabulary
Period Period School.edhole.com

7. Analyze this periodic function. Identify one cycle in two different ways.
Then determine the period of the function.
Begin at any point on the graph.
Trace one complete pattern.
The beginning and ending x-values of each
cycle determine the period of the
function.
Each cycle is 7 units long. The period of the function is 7.
Exploring Periodic Data
School.edhole.com

8. Determine whether each function is or is not periodic. If it is, find the period.
a.
The pattern of y-values in one section repeats exactly in other sections.
The function is periodic.
Find points at the beginning and end of one cycle.
Subtract the x-values of the points: 2 – 0 = 2.
The pattern of the graph repeats every 2 units, so the period is 2.
Exploring Periodic Data
School.edhole.com

9. (continued)
b.
The pattern of y-values in one section repeats exactly in other sections.
The function is periodic.
Find points at the beginning and end of one cycle.
Subtract the x-values of the points: 3 – 0 = 3.
The pattern of the graph repeats every 3 units, so the period is 3.
Exploring Periodic Data
School.edhole.com

10. Find the amplitudes of the two functions in
Additional Example 2.
a. amplitude = (maximum value – minimum value) Use definition of
1
2
1
2
The amplitude of the function is 2.
Exploring Periodic Data
amplitude.
1
2
= [2 – (–2)] Substitute.
= (4) = 2 Subtract within parentheses and simplify.
School.edhole.com

11. (continued)
b. amplitude = (maximum value – minimum value) Use definition of
1
2
1
2
The amplitude of the function is 3.
Exploring Periodic Data
amplitude.
1
2
= [6 – 0] Substitute.
= (6) = 3 Subtract within parentheses and simplify.
School.edhole.com

12. The oscilloscope screen below shows the graph of the alternating
current electricity supplied to homes in the United States. Find the
period and amplitude.
1
360
1 unit on the t-axis = s
Exploring Periodic Data
School.edhole.com

13. 1
60
One cycle of the electric current occurs from 0 s to s.
The maximum value of the function is 120, and the minimum is –120.
1
60
1
60
period = – 0 Use the definitions.
= Simplify.
1
2
1
2
amplitude = [120 – (–120)]
= (240) = 120
1
60
The period of the electric current is s. The amplitude is 120 volts.
School.edhole.com