1. State the amplitude and period for each function
State the amplitude and period for each function. Then graph each function. 1. 2.
3. Write an equation of the sine function with amplitude 0.27 and period π/2.
1. 3, pi
2. 2/3, 8pi
3. y = sin 4theta
Warm up

2. Lesson 6-5 Translations of Sine and Cosine Functions
Objective: Find the phase shift and the vertical translation for sine & cosine functions.
Write the equations of sine & cosine functions given the amplitude period, phase shift and vertical translation.
Graph compound functions.

3. Phase Shift
A horizontal translation or shift of a trigonometric function
y = Asin(kθ + c) or y = Acos(kθ +c)
The phase shift is -c/k, where k > 0
If c > 0, shifts to the left
If c < 0, shifts to the right

4. State the phase shift for each function. Then graph the function.
y = cos(θ – π)
y = sin(4θ + π)

5. Vertical Shift
The graph shifts vertically based on the “h” in the equations:
if h<0 the midline moves down
if h>0 the midline moves up
The midline is y = h

6. Vertical Shift
State the vertical shift and the equation for the midline of the function. Then graph the function.
y = 3sinθ + 2

7. Steps for graphing it all
Determine the vertical shift and graph the midline.
Determine the amplitude. Dash lines for the max and min.
Determine the period and draw a dashed graph of the sine or cosine curve.
Determine the phase shift and translate your dashed graph to draw the final graph.

8. Practice
State the amplitude, period, phase shift, and vertical shift for:
Then graph the function.
A=2
Period=8π
Phase shift= -4π
Vertical shift = -1

9. Example
Write the equation of a cosine function with amplitude 5, period 2π, phase shift –π/8, and vertical shift -2.
Find the amplitude, “A”
Find the vertical translation, “h”:
Find “k”: Solve 2π/k = Period
Solve for “c” phase shift = -c/k

10. Compound functions
Compound functions are made up of sums or products of trig functions and other functions.
ex: y = sin x + cos x
y= cos x + x

11. To graph compound functions
Make a table of each individual function. then add or multiply. y = cos x + x x
Cos x
X + cos x 1
0+1
π/2
π/2 +0 π -1
π-1
3π/2
3π/2+0 2π
2π+1