1. Writing Trig Functions
Section 7.4
Pre-Calculus AB Pre-AP/Dual, Revised ©2014
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7.4A: Frequency and Phase Shifts

2. Transformation Graph Equation
Equation: y = A trig function B(x – h) + k
A is the amplitude
B is the period
h is the horizontal shift or phase shift
k is the vertical shift
Frequency is defined as the number of cycles per second
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7.4A: Frequency and Phase Shifts

3. Steps for Transformation Graph Equation
Identify the amplitude (a) (|above point| + |below point|/2)
Identify the period (b)
Counting the number of hills from 0 to 2π OR
Find the length of the curve to get the period
Identify the horizontal shift (h) for sine and cosine and move over the parent function
Identify the vertical shift (k) by drawing a horizontal line cutting the graph in half
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7.4A: Frequency and Phase Shifts

4. 7.4A: Frequency and Phase Shifts
Amplitude
Amplitude (a) the maximum or minimum vertical distance between the graph and the x-axis. Amplitude is always positive.
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7.4A: Frequency and Phase Shifts

5. 7.4A: Frequency and Phase Shifts
Period
The period of a function is the x-interval needed for the function to complete one cycle. For b > 0, the period of y = a sin bx or y = a cos bx is 𝟐𝝅 𝒃 . If 0 < b < 1, the graph of the function is stretched horizontally.
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7.4A: Frequency and Phase Shifts

6. 7.4A: Frequency and Phase Shifts
Example 1
Write the equation of the graph below using a sine function
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7.4A: Frequency and Phase Shifts

7. 7.4A: Frequency and Phase Shifts
Example 1A
Write another equation of the graph below using a cosine function.
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7.4A: Frequency and Phase Shifts

8. 7.4A: Frequency and Phase Shifts
Example 1A
Write another equation of the graph below using a cosine function.
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7.4A: Frequency and Phase Shifts

9. 7.4A: Frequency and Phase Shifts
Example 1A
Write another equation of the graph below using a cosine function.
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7.4A: Frequency and Phase Shifts

10. 7.4A: Frequency and Phase Shifts
Example 2
Write the equation of the graph below using a sine and cosine function
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7.4A: Frequency and Phase Shifts

11. 7.4A: Frequency and Phase Shifts
Your Turn
Write the equation of the graph below using a sine function
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7.4A: Frequency and Phase Shifts

12. 7.4A: Frequency and Phase Shifts
Example 3
Write the equation of the graph below using a cosine function
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7.4A: Frequency and Phase Shifts

13. 7.4A: Frequency and Phase Shifts
Example 4
Write the equation of the graph below using a sine and cosine function
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7.4A: Frequency and Phase Shifts

14. 7.4A: Frequency and Phase Shifts
Your Turn
Write the equation of the graph below using a cosine function
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7.4A: Frequency and Phase Shifts