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.Warm up

2cos3y

4cos3

2 y

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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.

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 leftIf c < 0, shifts to the right

State the phase shift for each function. Then graph the function.

1. y = cos(θ – π)

2. y = sin(4θ + π)

4

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

Vertical Shift

hckAy )sin( hckAy )cos(

State the vertical shift and the equation for the midline of the function. Then graph the function.

y = 3sinθ + 2

Vertical Shift

Steps for graphing it all

1. Determine the vertical shift and graph the midline.

2. Determine the amplitude. Dash lines for the max and min.

3. Determine the period and draw a dashed graph of the sine or cosine curve.

4. Determine the phase shift and translate your dashed graph to draw the final graph.

State the amplitude, period, phase shift, and vertical shift for:

Then graph the function.

Practice

1)4

cos(2 y

A=2Period=8πPhase shift= -4πVertical shift = -1

Write the equation of a cosine function with amplitude 5, period 2π, phase shift –π/8, and vertical shift -2.

1. Find the amplitude, “A”2. Find the vertical translation, “h”:3. Find “k”: Solve 2π/k = Period4. Solve for “c” phase shift = -c/k

Example

2)8

cos(5 y

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

Compound functions

Make a table of each individual function. then add or multiply. y = cos x + x

To graph compound functions

x Cos x X + cos x

0 1 0+1

π/2 0 π/2 +0

π -1 π-1

3π/2 0 3π/2+0

2π 1 2π+1