Frequency and Phase Shifts Section 7.4 Pre–Calculus AB Pre–AP/Dual, Revised ©2014 viet.dang@humble.k12.tx.us 4/7/2019 12:30 PM 7.3: Amplitude and Vertical Shifts

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 4/7/2019 12:30 PM 7.3: Amplitude and Vertical Shifts

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 4/7/2019 12:30 PM 7.3: Amplitude and Vertical Shifts

7.3: Amplitude and Vertical Shifts Review of Graphs 4/7/2019 12:30 PM 7.3: Amplitude and Vertical Shifts

Trig Tables for Main Trig Functions y = csc (x) C Und 1 –1 y = sec (x) Und C 1 –1 y = cot (x) C Und 1 –1 4/7/2019 12:30 PM 7.3: Amplitude and Vertical Shifts

7.3: Amplitude and Vertical Shifts Example 1 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 4/7/2019 12:30 PM 7.3: Amplitude and Vertical Shifts

7.3: Amplitude and Vertical Shifts Example 2 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 4/7/2019 12:30 PM 7.3: Amplitude and Vertical Shifts

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 4/7/2019 12:30 PM 7.3: Amplitude and Vertical Shifts

7.3: Amplitude and Vertical Shifts Assignment Worksheet 4/7/2019 12:30 PM 7.3: Amplitude and Vertical Shifts