1. Frequency and Phase Shifts
Section 7.4
Pre-Calculus AB Pre-AP/Dual, Revised ©2014
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7.3: Amplitude and Vertical Shifts

2. Transformation Graph Equation
Equation: y = A trig function (bx – h) + k or book’s equation: y = A trig function (bx – c) + d
A is the amplitude
a: vertically stretches by a factor of a,
1/a: Vertically compresses by a factor of 1/a
–a: Reflects over the x-axis
B is the period or frequency
Period equation: 2π/b for sine and cosine, π/b for tangent
B: horizontally compresses by a factor of 1/b
1/B: horizontally stretches by a factor of b
–B: Reflects over the y-axis
k is the vertical shift
h is the horizontal shift or phase shift
Phase Shift equation: h/b or c/b
(x – h) shifts graph right
(x + h) shifts graph left
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7.3: Amplitude and Vertical Shifts

3. 7.3: Amplitude and Vertical Shifts
Example 1
Describe the transformation, y = 4 cost + 1
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7.3: Amplitude and Vertical Shifts

4. 7.3: Amplitude and Vertical Shifts
Example 2
Describe the transformation, y = –1/2 sint – 2
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7.3: Amplitude and Vertical Shifts

5. 7.3: Amplitude and Vertical Shifts
Your Turn
Describe the transformation, y = 5cost + 3
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7.3: Amplitude and Vertical Shifts

6. 7.3: Amplitude and Vertical Shifts
Example 3
Given 𝒚=−𝟒𝐬𝐢𝐧 𝟏 𝟐 𝒙+𝟏 +𝟑, identify amplitude, period, vertical shift, and phase shift for one period.
Amplitude
Period
Vertical Shift
Phase Shift
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7.3: Amplitude and Vertical Shifts

7. 7.3: Amplitude and Vertical Shifts
Example 4
Given 𝒚=𝐬𝐢𝐧𝟐 𝒙+𝝅 −𝟏 identify amplitude, period, vertical shift, and phase shift for one period.
Amplitude
Period
Vertical Shift
Phase Shift
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7.3: Amplitude and Vertical Shifts

8. 7.3: Amplitude and Vertical Shifts
Your Turn
Given 𝒚=𝟐𝐜𝐨𝐬 𝟑𝒙−𝟒 −𝟏, identify amplitude, period, vertical shift, and phase shift for one period.
Amplitude
Period
Vertical Shift
Phase Shift
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7.3: Amplitude and Vertical Shifts

9. 7.3: Amplitude and Vertical Shifts
Example 5
Given 𝒚=𝟐𝐭𝐚𝐧 𝟑𝒙− 𝝅 𝟐 −𝟏 identify amplitude, period, vertical shift, and phase shift for one period.
Amplitude
Period
Vertical Shift
Phase Shift
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7.3: Amplitude and Vertical Shifts

10. 7.3: Amplitude and Vertical Shifts
Example 6
Given 𝒚=𝟓𝐬𝐞𝐜 𝟑𝒙+ 𝝅 𝟕 , identify amplitude, period, vertical shift, and phase shift for one period.
Amplitude
Period
Vertical Shift
Phase Shift
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7.3: Amplitude and Vertical Shifts

11. 7.3: Amplitude and Vertical Shifts
Your Turn
Given 𝒚=𝟑𝐬𝐞𝐜 𝒙+ 𝝅 𝟒 , identify amplitude, period, vertical shift, and phase shift for one period.
Amplitude
Period
Vertical Shift
Phase Shift
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7.3: Amplitude and Vertical Shifts

12. 7.3: Amplitude and Vertical Shifts
Example 7
Given 𝒚=− 𝟏 𝟐 𝐭𝐚𝐧 −𝟑𝒙+ 𝝅 𝟔 , identify amplitude, period, vertical shift, and phase shift for one period.
Amplitude
Period
Vertical Shift
Phase Shift
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7.3: Amplitude and Vertical Shifts

13. 7.3: Amplitude and Vertical Shifts
Your Turn
Given 𝒚=−𝐜𝐨𝐭 𝟐𝒙+ 𝝅 𝟒 , identify amplitude, period, vertical shift, and phase shift for one period.
Amplitude
Period
Vertical Shift
Phase Shift
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7.3: Amplitude and Vertical Shifts

14. 7.3: Amplitude and Vertical Shifts
Example 8
Write an equation using y = sin x where a = 2, period is π, phase shift is π/2 and vertical shift is up 1.
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7.3: Amplitude and Vertical Shifts

15. 7.3: Amplitude and Vertical Shifts
Example 9
Write an equation using y = cos x where a = 3 reflected across the x-axis, period is 2π, phase shift is π/2 to the right and vertical shift is down 1.
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7.3: Amplitude and Vertical Shifts

16. 7.3: Amplitude and Vertical Shifts
Example 10
Write an equation using y = tan x where a = 3, period is 4π, and phase shift is π/3 to the right.
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7.3: Amplitude and Vertical Shifts

17. 7.3: Amplitude and Vertical Shifts
Your Turn
Write an equation using y = cos x where a = 4, period is 2, and phase shift is 2/π to the right and one unit up
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7.3: Amplitude and Vertical Shifts

18. Transformation Graph Equation
A is the amplitude
a: vertically stretches by a factor of a,
1/a: Vertically compresses by a factor of 1/a
–a: Reflects over the x-axis
B is the period or frequency
Period equation: 2π/b for sine and cosine, π/b for tangent
B: horizontally compresses by a factor of 1/b
1/B: horizontally stretches by a factor of b
–B: Reflects over the y-axis
k is the vertical shift
h is the horizontal shift or phase shift
Phase Shift equation: h/b or c/b
(x – h) shifts graph right
(x + h) shifts graph left
2/23/ :01 AM
7.3: Amplitude and Vertical Shifts

19. 7.3: Amplitude and Vertical Shifts
Example 11
Describe the transformation of 𝒚=−𝟐𝐜𝐨𝐬 𝟑𝒙−𝝅 −𝟏
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7.3: Amplitude and Vertical Shifts

20. 7.3: Amplitude and Vertical Shifts
Example 12
Describe the transformation of 𝒚= 𝟏 𝟒 𝐜𝐬𝐜 𝟐 𝟑 𝒙− 𝟑 𝟐 𝝅 +𝟑
2/23/ :01 AM
7.3: Amplitude and Vertical Shifts