Precalculus Sine and Cosine Graphs
Transcript of Precalculus Sine and Cosine Graphs
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Sine and Cosine Graphs
Reading and Drawing
Sine and Cosine Graphs
Some slides in this presentation contain animation. Slides will be
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before moving to the next one.
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This is the graph for y = sin x.
This is the graph for y = cos x.
π
π
π
π
π
22
3
20
22
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π
π
π
π
π
22
3
2
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y = sin x
y = cos x
One complete period is
highlighted on each of
these graphs.
or both y ! sin x and y ! cos x" the period is 2π. #rom the beginning of
a cycle to the end of that cycle" the distance along the x$axis is %&.'
π
π
π
π
π
22
3
20
22
32
π
π
π
π
π
22
3
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y = sin x
y = cos x
(mplitude deals with the
height of the graphs.
or both y ! sin x and y ! cos x" the amplitude is 1. )ach of these
graphs extends * unit above the x$axis and * unit below the x$axis.
*
$*
π
π
π
π
π
22
3
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22
32
π
π
π
π
π
22
3
2
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*
$*
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( sine graph has a phase shift if the -ey point
is shifted to the left or to the right.
ππ
22
3
2022
3
2
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1
-1
or y ! cos x" there is no phase shift.
The y$intercept is located at the point #+"*'.
,e will call that the key point.
π
π
π
π
π
22
3
20
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( cosine graph has a phase shift if the -ey point
is shifted to the left or to the right.
ππ 22
3
20
22
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y = a sin b (x - c )
or a sine graph which has no vertical shift" the euation for the
graph can be written as
or a cosine graph which has no vertical shift" the euation for the
graph can be written as
y = a cos b (x - c )
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Consider this cosine graph. The height of this graph is %" so a = 2.
The euation for this graph can be written as y ! 2 cos x.
ππ
πππ
−π−π
−π− %%
0
%+
%%
0%
2
1
-1
-2
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1f a sine graph is 2flipped3 over the x$axis" the value of a will be negative.
or the graph above" a = -3.
(n euation for this graph is y ! -3 sin x.
ππ
πππ
−π−π
−π− %%
0
%+
%%
0%
3
2
1
-1
-2
-3
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1f a cosine graph is 2flipped3 over the x$axis" the value of a will be negative.
or the graph above" a = -1.
(n euation for this graph is y ! -1 cos x or 4ust y ! - cos x.
π
π
π
π
π
22
3
20
22
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*
$*
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y ! a sin b #x $ c ' y ! a cos b #x $ c '
!b" affects the pe#io of the sine or cosine graph.
or sine and cosine graphs" the pe#io can $e ete#mine $y
%$
2pe#io
π
=
Conversely" when you already -now the period of a sine or cosine
graph" b can $e ete#mine $y
%pe#io
2$
π
=
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3
&
3
2
30
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2
3
& ππ
π
π
π
2
1
-1
-2
The period for this graph is .3
&π
( )%
0
0
5
%
period
%b =
π
π=
π=
6otice that a =2 on this graph since the graph extends % units above
the x$axis.
.x%
0sin%y =
2
3b =Since and a = 2" the sine euation for this graph is
7se the period to calculate b.
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( sine graph has a phase shift
if its -ey point has shifted to theleft or to the right.
( cosine graph has a phase shift
if its -ey point has shifted to the
left or to the right.
ππ
πππ
−π−π
−π− %%
0
%+
%%
0%
ππ
πππ
−π−π
−π− %%
0
%+
%%
0%
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y ! a sin b #x $ c ' y ! a cos b #x $ c '
!c " inicates the phase shift of the sine graph or of the
cosine graph. The x$coordinate of the -ey point is c.
2
'
22
3
2022
3 π
ππ
2π
This sine graph moved
units to the right. 2c ”, the phase
shift" is .
2
(n euation for this graph can be written as %2
xsiny
π
*
$*
y ! sin x
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22
3
20
22
32
2
'
This cosine graph above moved units to the left.
2c ”, the phase shift" is .
2
2
(n euation for this graph can be written as
%2
xcosyo# 2
xcosy
π
π
*
$*
y ! cos x
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Graphs whose euations can be written as a sine function can also be
written as a cosine function.
Given the graph above" it is possible to write an euation for the
graph. ,e will loo- at how to write both a sine euation that describes
this graph and a cosine euation that describes the graph.
The sine function will be written as y ! a sin b #x / c '.
The cosine function will be written as y ! a cos b #x / c '.
3
&
3
2
333
2
3
& π
π
π
π
π
2
1
-1
-2
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y ! a sin b #x / c '
or the sine function" the values for a" b" and c must be determined.
The height of the graph is 5" so a = &.
The period of the graph is %2
3
3
&
22%3
&
=π
π
== period b %23=b
The -ey point has shifted to " so the phase shift is3
π
%3
π
%3
π
c
3
&
3
2
333
2
3
& π
π
π
π
π
2
1
-1
-2
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y ! a sin b #x / c '
a = &2
3$ =
3c π
π
π
32
3sin&
32
3sin& x y or x y
3
&
3
2
333
2
3
& π
π
π
π
π
2
1
-1
-2
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This is an euation for the graph written as a sine function.
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3
&
3
2
333
2
3
& π
π
π
π
π
2
1
-1
-2
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y ! a cos b #x / c '
To write the euation as cosine function" the values for a" b" and c
must be determined. 1nterestingly" a and b are the same for cosine as
they were for sine. Only c is different.
The height of the graph is 5" so a = &.
The period of the graph is %2
3
3
&
22%3
&
=π
π
== period b 2
3$ =
The -ey point has not shifted" so there is no phase shift. That means
that c ! +.
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a = &2
3$ = 0c =
x2
3cos&yo# 0x
2
3cos&y
=
y ! a cos b #x / c '
3
&
3
2
333
2
3
& π
π
π
π
π
2
1
-1
-2
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This is an euation for the graph written as a cosine function.
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1t is important to be able to draw a sine graph when you are given the
corresponding euation. Consider the euation
8egin by loo-ing at a" b" and c .
. x siny
8 2 2
. x siny
8 2 2
22 π
=
c ba
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The amplitude is %. 9aximums will be at %.
9inimums will be at $%.
The negative sign means that the graph has 2flipped3 about the x $axis.
π
(2sin2 x y
%a%a =−=
2
-2
2
-2
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1n order to correctly label the x$intercepts" maximums" and minimums on
the graph" you will need to divide the period into 5 eual parts or
inc#ements. (n increment" : of the period" is the distance between an x$intercept and
a maximum or minimum.
One increment
The increment is : of the period. Since the period for
is π " the increment is %&
o# &
1 π
π
π
2sin2 x y
(
π
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To label the graph" begin at the phase shift. (dd one increment at a time
to label x$intercepts" maximums" and minimums.
8 x 2 sin2 y
2
-2
(
3π
48
48
3 48
5
(
1)
(
1'
(
13
(
11
(
* π
(
'π
(
)π
(
0
(
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,hat does the graph for the euation loo- li-e;x2
1cos'y
π
c ba2
1'
9aximums will be at
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The phase shift is
That means that the -ey point
shifts from the origin to
πc
%
%
7se to calculate the period of the graph.
π== &
21
22
b
period
One complete period is highlighted here.
%2
1cos' π x y
2
1=b
'
-'
π
'
-'
π
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Remember that the increment #: of the period' is the distance between
an x$intercept and a maximum or minimum.
Since the period for is 5π " the increment is π .
Don=t forget that x$intercepts" maximums" and minimums can be labeled
by beginning at the phase shift and adding one increment at a time.
x y 2
1cos'
'&3202
-π + π
This is the graph for
%21cos' π x y
0 + π π + π
'
-'
π
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Sometimes a sine or cosine graph may be shifted up or down. This is
called a vertical shift.
y = a sin b (x - c ) +d %
The euation for a sine graph with a vertical shift can be written as
The euation for a cosine graph with a vertical shift can be written as
y = a cos b (x - c ) +d %
1n both of these euations" d represents the vertical shift.
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( good strategy for graphing a sine or cosine function that has a
vertical shift>
?Graph the function without the vertical shift
? Shift the graph up or down d units.
Consider the graph for
The euation is in the form y = a cos b (x - c ) +d %
2d3 euals 0" so the vertical shift is 0.
The graph ofwas drawn in the previous example.
%32
1cos' x y
x y 2
1
cos'
x y 2
1cos'
'&3202
'
-'
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A,> Bage %5
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