1. 7.3: Amplitude and Vertical Shifts

2. 7.3: Amplitude and Vertical Shifts
Quiz
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7.3: Amplitude and Vertical Shifts

3. 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

4. Transformation of Graph Equation
Equation: y = A trig function B(x – c) + d
A is the vertical stretch/compression or Amplitude for Sine and Cosine
B is the period
C is the horizontal shift or phase shift
D is the vertical shift
Frequency is defined as the number of cycles per second
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7.3: Amplitude and Vertical Shifts

5. 7.3: Amplitude and Vertical Shifts
Steps
Identify A to D
Identify the Period by using the equation, π 𝑩 for Tangent and Cotangent and 𝟐π 𝑩 for the others
Identify the Anchor Points (spacing) by 𝑷𝒆𝒓𝒊𝒐𝒅 𝟒
Identify “C”, the trig function, and the table for each trig function
Put “C” as the vertical shift and follow the table
For Sine and Tangent, use the horizontal shift when the first y = 0
For Cosine and Secant, use the horizontal shift when the first y = 1
For Cosecant and Cotangent, use the horizontal shift when the first y = undefined
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7.3: Amplitude and Vertical Shifts

6. 7.3: Amplitude and Vertical Shifts
Example 1
Graph 𝒚=𝟒𝐬𝐢𝐧 𝟏 𝟐 𝒙+𝟏 in one period and identify amplitude, period, vertical shift, and phase shift
y = sin (x) C 1 –1
y = 4sin (x) C 4 –4
y = 4sin (x) + 1 C 1 5 –3
Amplitude
Period
Phase Shift
Spacing (A.P.)
Vertical Shift
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7.3: Amplitude and Vertical Shifts

7. 7.3: Amplitude and Vertical Shifts
Example 1
Graph 𝒚=𝟒𝐬𝐢𝐧 𝟏 𝟐 𝒙+𝟏 in one period and identify amplitude, period, vertical shift, and phase shift
Amplitude
Period
Phase Shift
Spacing (A.P.)
Vertical Shift
y = 4sin (x) + 1 1 π 5 2π 3π –3 4π
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7.3: Amplitude and Vertical Shifts

8. 7.3: Amplitude and Vertical Shifts
Example 2
Graph 𝒚=−𝟐𝐭𝐚𝐧 𝒙+ 𝝅 𝟐 −𝟏 in one period and identify amplitude, period, vertical shift, and phase shift
y = –2tan (x) –1 –π
Und
–3π/4 1
–π/2 –1
–π/4 –3
y = –2tan (x) –π
Und
–3π/4 2
–π/2
–π/4 –2
y = tan (x)
Und –1 C 1
Amplitude
Period
Phase Shift
Spacing (A.P.)
Vertical Shift
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7.3: Amplitude and Vertical Shifts

9. 7.3: Amplitude and Vertical Shifts
Your Turn
Graph 𝒚=−𝟏+𝟐𝐬𝐢𝐧(𝟒𝒙−𝝅) in one period and identify amplitude, period, vertical shift, and phase shift
y = 2sin (x) –1
π/4 –1
3π/8 1
π/2
5π/8 –3
3π/4
Amplitude
Period
Phase Shift
Spacing (A.P.)
Vertical Shift
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7.3: Amplitude and Vertical Shifts

10. 7.3: Amplitude and Vertical Shifts
Example 3
Graph 𝒚= 𝐜𝐬𝐜 𝟏 𝟐 𝒙+ 𝝅 𝟐 in one period and identify amplitude, period, vertical shift, and phase shift
y = csc (x)
Und C 1 –1
Amplitude
Period
Phase Shift
Spacing (A.P.)
Vertical Shift
–π/2
π/2
3π/2
5π/2
7π/2
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7.3: Amplitude and Vertical Shifts

11. 7.3: Amplitude and Vertical Shifts
Example 4
Graph 𝒚=𝟑 𝐜𝐬𝐜 𝝅 𝟐 𝒙+𝟏 −𝟐 in one period and identify amplitude, period, vertical shift, and phase shift
y = csc (x)
Und C 1 –1
y = 3csc (x) C
Und 3 –3
y = 3csc (x) – 2 C
Und 1 –5
Amplitude
Period
Phase Shift
Spacing (A.P.)
Vertical Shift –2 –1 1 2
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7.3: Amplitude and Vertical Shifts

12. 7.3: Amplitude and Vertical Shifts
Example 4
Graph 𝒚=𝟑 𝐜𝐬𝐜 𝝅 𝟐 𝒙+𝟏 −𝟐 in one period and identify amplitude, period, vertical shift, and phase shift
y = 3csc (x) – 2
Und C 1 –5
Amplitude
Period
Phase Shift
Spacing (A.P.)
Vertical Shift –2 –1 1 2
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7.3: Amplitude and Vertical Shifts

13. 7.3: Amplitude and Vertical Shifts
Example 5
Graph 𝒚=− 𝐜𝐬𝐜 𝟐𝒙+ 𝝅 𝟒 in one period and identify amplitude, period, vertical shift, and phase shift
Amplitude
Period
Phase Shift
Spacing (A.P.)
Vertical Shift
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7.3: Amplitude and Vertical Shifts

14. 7.3: Amplitude and Vertical Shifts
Your Turn
Graph 𝒚= 𝐬𝐞𝐜 𝟐 𝒙+𝝅 in one period and identify amplitude, period, vertical shift, and phase shift
y = sec (x)
–5π/4
Und –π 1
–3π/4
–π/2 –1
–π/4
Amplitude
Period
Phase Shift
Spacing
Vertical Shift
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7.3: Amplitude and Vertical Shifts

15. 7.3: Amplitude and Vertical Shifts
Assignment
Worksheet
4/21/ :30 AM
7.3: Amplitude and Vertical Shifts