Section 7.2 More on Graphs of Sine and Cosine: Phase Shift
OBJECTIVE 1:
Sketching Graphs of the Form y = sin(x − C) and y = cos(x − C)
The third factor that can affect the graph of a sine or cosine curve is known as phase shift. In general the
C
number
is known as the phase shift.
€
€
B
2π 2π
C
=
= 2π , and phase shift = = C .
B
1
1
€
π
3π
The x-coordinates of the quarter points are C, C + , C + π , C +
, C + 2π .
2
2
In Objective 1, A = 1 and B=1, so amplitude = 1, period is P =
€
€
7.2.2 Determine the amplitude, range, period, and phase shift, then sketch the graph of the function
_______________.
€
OBJECTIVE 2:
Sketching Graphs of the Form y = Asin(Bx − C) and y = Acos(Bx − C)
Steps for Sketching Functions of the Form and
Step 1: If
y = Asin(Bx − C) and y = Acos(Bx − C)
, rewrite the function in an equivalent€form such that
€the odd property of the sine function or
. Use
the even property of the cosine function. It is often helpful to factor out B when
such that
€
€  
 
C 
C 
y = Asin(Bx − C) = Asin B x −  or y = Acos(Bx − C) = Acos B x − 
B 
B 
 
 
In the factored form, the amplitude, period, and phase shift are more apparent.
Step 2: The amplitude is
€
. The range is
Step 3: The period is
.
Step 4: The phase shift is
.
[− A, A ] .
€
€
Step 5: The x-coordinate of the first quarter point is
one complete cycle is
and adding
. The x-coordinate of the last quarter point is
. Subdivide this interval into 4 equal subintervals of length
. An interval for
by starting with
to the x-coordinate of each successive quarter point.
Step 6: Multiply the y-coordinates of the quarter points of
the y-coordinates of the corresponding quarter points for
or
by A to determine
y = Asin(Bx − C) and y = Acos(Bx − C)
Step 7: Connect the quarter points to obtain one complete cycle.
€
€
In the following exercises, determine the amplitude, range, period, and phase shift and then sketch the
graph.
7.2.7
7.2.10
OBJECTIVE 3:
Sketching Graphs of the Form y = Asin(Bx − C) + D and y = Acos(Bx − C) + D
Steps
for
Sketching
Functions
of
the
Form
and
€
Step
1:
If
,
rewrite
the
function
in
an
equivalent
form
such
that
function
or
the
even
property
of
the
cosine
function.
It
is
often
helpful
to
factor
out
B
when
€
.
Use
the
odd
property
of
the
sine
such
that
or
.
In
the
factored
form,
the
amplitude,
period,
and
phase
shift
are
more
apparent.
Step
2:
The
amplitude
is
Step
3:
The
period
is
.
The
range
is
.
.
Step
4:
The
phase
shift
is
.
Step
5:
The
x‐coordinate
of
the
first
quarter
point
is
interval
for
one
complete
cycle
is
by
starting
with
and
adding
.
The
x‐coordinate
of
the
last
quarter
point
is
.
An
.
Subdivide
this
interval
into
4
equal
subintervals
of
length
to
the
x‐coordinate
of
each
successive
quarter
point.
Step
6:
Multiply
the
y‐coordinates
of
the
quarter
points
of
the
y‐coordinates
of
the
corresponding
quarter
points
for
or
by
A
then
add
D
to
determine
and
.
Step
7:
Connect
the
quarter
points
to
obtain
one
complete
cycle.
In the following exercises, determine the amplitude, range, period, and phase shift and then sketch the
graph.
7.2.17
7.2.20
7.2.23