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 = +- 0.27 sin 4theta Warm up

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 left If c < 0, shifts to the right

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

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

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

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.

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

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

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

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