1. Translation Images of Circular Functions
Lesson 4.8

2. 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)

3. 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

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

5. 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

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

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

8. Homework
Pages 282 – – 9 15 – 16