What is periodic ?
A function is periodic if it has a pattern of y-values that repeats at regular intervals.
What is cycle ?
One complete pattern is called cycle. A cycle may begin at any point on the graph.
What is period ?
The horizontal length of one cycle is called period of the function.
Amplitude :
Amplitude is half the difference between the maximum value (peak) and the minimum value (trough) in a cycle.
y = (Max + Min)/2
Determine whether the graph represents a periodic function. If so, identify the period and amplitude.
Problem 1 :
Solution:
It has a pattern, so it is a periodic function.
The horizontal length of the pattern is 2 units.
So, period is 2.
Amplitude = (max - min)/2
= (1 - 0)/2
= 1/2
= 0.5
Problem 2 :
Solution:
It has a pattern, so it is a periodic function.
The horizontal length of the pattern is π/2.
So, period is π/2.
Amplitude = (max - min)/2
= (1.5 - (-1.5))/2
= (1.5+1.5)/2
= 3/2
= 1.5
Problem 3 :
Solution :
The graph doesn't have pattern. So, it is not a periodic function.
Problem 4 :
Solution:
It has a pattern, but y-values is not repeating for each cycle. So, it is not a periodic function.
Identify the amplitude and period of the graph of the function.
Problem 5 :
Solution:
The horizontal length of one pattern = 3π.
So, the period = 3π.
Maximum = 1, minimum = -1
Amplitude = (Max - Min)/2
= (1 - (-1))/2
= (1 + 1)/2
= 2/2
= 1
So, amplitude = 1
Problem 6 :
Solution:
The horizontal length of one pattern = 1
So, the period = 1
Maximum = 0.5, minimum = -0.5
Amplitude = (Max - Min)/2
= (0.5 - (-0.5))/2
= (0.5 + 0.5)/2
= 1/2
= 0.5
So, amplitude = 0.5
Problem 7 :
Solution:
The horizontal length of one pattern = π
So, the period = π
Maximum = 4, minimum = -4
Amplitude = (Max - Min)/2
= (4 - (-4))/2
= (4 + 4)/2
= 8/2
= 4
So, amplitude = 4
Problem 8 :
Solution:
The horizontal length of one pattern = 8π
So, the period = 8π
Maximum = 3, minimum = -3
Amplitude = (Max - Min)/2
= (3 - (-3))/2
= (3 + 3)/2
= 6/2
= 3
So, amplitude = 3
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