precalculus sine and cosine graphs

8/20/2019 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

more meaningful if you allow each slide to finish its presentation

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

32

π

π

π

π

π

22

3

2

022

32


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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

20

22

32


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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

20

22

32

π

π

π

π

π

22

3

2

022

32

*

$*


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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

22

32


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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

32


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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

32

*

$*


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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

33

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

-

-

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

-

-

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

-

-

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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precalculus sine and cosine graphs

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