14.1 reteaching

Name Date Class
LESSON
14-1
Reteach
Graphs of Sine and Cosine
Transformations of the sine and cosine functions change the amplitude and/or the period of
the graph.
The amplitude is half the difference
For y asinbx or y acosbx:
between the greatest and least
• the amplitude is |a|,
values of the function.
• the period is ___ 2
FbF .
One full cycle appears in each period.
Use the graph of f x sinx to sketch the graph of g x 0.5sin2x.
Step 1 Compare g x 0.5sin2x to y asinbx.
Find a to identify the amplitude.
a 0.5 and F0.5F 0.5, so the amplitude is 0.5.
Step 2 Find b to identify the period.
b 2, and ___ 2
FbF ___ 2 , so the period is .
F2F
Step 3 Graph f x sinx.
Step 4 Graph g x 0.5sin2x on the same plane as f x .
The maximum value
of g x is 0.5 and the
minimum value is 0.5.
One full cycle appears in
the interval from 0 to .
The amplitude is 1. The maximum and
minimum values of f x are 1 and 1.
The period is 2. One full cycle appears in
the interval from 0 to 2. Two full cycles
appear in the interval from 2 to 2.
The x-intercepts are at multiples of .
The amplitude is 0.5. The maximum and
minimum values of g x are 0.5 and 0.5.
The period is . One full cycle appears in the
interval from 0 to . Two full cycles appear in
the interval from 0 to 2 and from 2 to 0.
The x-intercepts are at multiples of __ 2 .
Complete to graph h x 0.5cos2x.
1. Find the amplitude of h x . a 0.5
2. Find the period of h x . ___ 2
FbF
3. What are the maximum and
minimum values of h x ? 0.5, 0.5
4. How many full cycles appear
in the interval from 0 to ?
1 cycle
5. Sketch the graph of f x cosx. Then graph h x 0.5cos2x.
Copyright © by Holt, Rinehart and Winston.
6 Holt Algebra 2
All rights reserved.

Name Date Class
LESSON
14-1
Reteach
Graphs of Sine and Cosine (continued)
A phase shift is a horizontal translation. Sine and cosine can be translated horizontally
by y sin x h and y cos x h .
A phase shift or horizontal translation of h units moves the graph left h units for h 0 or
right h units for h 0.
Use the graph of f x cosx to sketch the graph of g x cos x __
2 .
Step 1 Compare g x cos x __
2 to y a cosbx.
Find the amplitude and period.
The amplitude and
a 1 and F1F 1, so the amplitude is 1.
period of g are the
b 1, and ___ 2
FbF ___ 2
same as for y cosx.
2, so the period is 2.
F1F
Step 2 Find h to identify the phase shift.
x h x __ Because h 0, the shift is to the right.
__
, so h
2 2 .
The phase shift is __ radians to the right.
Intercepts occur
2
at integer
Step 3 Identify the first two positive x-intercepts.
The x-intercepts of f x cosx occur at __ and
3
2 ___
2 .
multiples of .
The x-intercepts of g x occur at __
3
, or , and ___ , or 2.
2 __ 2
Step 4 Identify the maximum and minimum values.
The maxima and minima of f x cosx occur
at 0 and .
The maxima and minima of g x occur at
3
, or ___
2 .
0 __ 2 , or __ 2 , and __ 2
Step 5 Graph f x cosx and g x cos x __
2 .
2 __ 2
Use the graph of f x sinx to sketch the graph of g x sin x __
2 .
6. Identify h. What is the phase shift?
h __ 2 ; phase shift: __ radians to the right
2
7. Identify the x-intercepts from 0 to 2.
__
2 , ___ 3
2
8. Identify the maxima and minima from 0 to 2.
Maximum of 1 at ; minimum of –1 at 0 and 2
9. Sketch the graphs of f x and g x .
Copyright © by Holt, Rinehart and Winston.
7 Holt Algebra 2
All rights reserved.