Translation Images of Circular Functions Lesson 4.8

Definitions Phase Shift: shifts the graph to the right and to the left. (connected to the x coordinate) y = sin (x + 2π) Shifts the graph to the LEFT 2π Vertical Translations: shifts the graph up and down y + 2 = sin (x) y = -2 + sin (x)

Graph y = cos (x – π/2) What is the phase shift of this function from y = cos x? π/2 to the right What is the shift of this function from y = sin x? 0; they are the same function

Theorem For all real numbers x, cos (x – π/2) = sin x (remember the Complements Theorem: Cos(π/2 – x) = sin x)

Compare the graphs of y = sin x and Example 2 Compare the graphs of y = sin x and y = sin (x + 5π/6) + 2 vertical translation: 2 up phase shift: 5π/6 to the left

Vertical Translation: Example 3 Compare the graphs of y = tan x and y = -1 + tan x. Phase Shift: nothing Vertical Translation: 1 down

Example 4 Phase shift: π/2 to the left Vertical Translation: 3 Amplitude: 2 Period: 2π/6 = π/3

Homework Pages 282 – 284 3 – 9 15 – 16