Transcript Lesson 6-5 Translations of Sine and Cosine Functions

State the amplitude and period for each
function. Then graph each function.
 1.
y  3 cos 2


2. y  2 cos 
3

4
3. Write an equation of the sine function
with amplitude 0.27 and period π/2.
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.
1.
y = cos(θ – π)
2.
y = sin(4θ + π)



4
Vertical Shift

The graph shifts vertically based on the “h” in
the equations:
y  A sin(k  c)  h
y  A cos(k  c)  h
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
1.
2.
3.
4.
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:
y  2 cos(   )  1
4
 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.

1.
2.
3.
4.
y  5 cos(  )  2
8
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
0
1
0+1
π/2
0
π/2 +0
π
-1
π-1
3π/2
0
3π/2+0
2π
1
2π+1