6.4 Graphs of 6.4 Graphs of Sine and CosineSine and Cosine

90º

270º

180º

1

2

-2

-1

360º

-270º

-90º

Label your graph paper...Label your graph paper...

1

2

-2

-1

2

3

2

22

3

2

In radians...In radians...

Graph Graph y = siny = sin

90º-90º 270º-270º-1

-2

sin

0°

45°

90°

135°

180°

225°

270°

315°

360°

0

0

0

1

-1

0.707

0.707

-0.707

-0.707

180º 360º

2

2

2

2

2

2

2

2

2

1

y = sin x

90º-90º 270º-270º

1

-1

2

-2

MaximumMaximum

MinimumMinimum

intercept intercept

intercept

y = sin x

90º-90º 270º-270º

1

-1

2

-2 Period: 360°

Period: the least amount of space (degrees or radians) the function takes to complete one cycle.

y = sin x

90º-90º 270º-270º

1

-1

2

-2

Amplitude = 1

Amplitude: half the distance between the maximum and minimum

In other words, how high does it go from its axis?

Graph Graph y = cosy = cos

1

-1

2

-2

cos 0 1

1

-1

0

0

0.707

-0.707

-0.707

0.707

2

2

2

2

2

2

2

2

2

3

2

22

3

2

4

3

4

5

4

7

4

2

2

3

2

y = cos x

1

-1

2

-2

2--2

MaximumMaximum

MinimumMinimum

intercept intercept

y = cos x

1

-1

2

-2

2--2

Period: 2

Period: the least amount of space (degrees or radians) the function takes to complete one cycle.

y = cos x

1

-1

2

-2

2--2

Amplitude = 1

How high does it go from its axis?

1

-1

2

-2

y = cos xy = sin x

Try it on your calculator!

2

3

2

2

3

2

y= sin and y = cos are the mother functions.

Changing the equations changes the appearance of the graphs

We are going to talk about the AMPLITUDE, TRANSLATIONS, and PERIOD of relative equations

Mother Function relative function change?

y1 = sin x y2 = - sin x reflection over x-axis

y1 = sin x

y1 = sin x

y2 = 4 sin x

y2 = sin x1

2

amplitude = 4

amplitude = 1

2

generalization?

y = a sin x amplitude = a

Mother Function relative function change?

y1 = sin x y2 = sin (x - 45)

y2 = sin (x + 60)

horizontal translation, 45 degrees to the right.

horizontal translation, 60 degrees to the left.

y1 = sin x

generalization?y = sin (bx - c)

y = sin (bx – (- c))

is the horizontal translationto the right

to the left

y2 = sin (2x + 60)y1 = sin xhorizontal translation, 30 degrees to the left.

y2 = sin (3x - 270)y1 = sin xhorizontal translation, 90 degrees to the right.

c

b

Mother Function relative function change?

y1 = cos x y2 = 2 + cos x vertical translation, 2 units up.

y1 = cos x y2 = -3 + cos xvertical translation, 3 units down.

generalization?

y = d + cos x ‘d’ is the vertical translation

when d is positive, the graph moves up.

when d is negative, the graph moves down.

Mother Function relative function change?

y1 = sin x

y1 = sin x

y2 = sin 2x

y2 = sin x1

2

Period = 180or

Period = 7204or

generalization?

y = sin bx Period = 360b

or 2b

y = d + a sin (bx - c)

y = d + a cos (bx - c)a is the amplitude

is the horizontal translation

d is the vertical translation

period = 360b

Summary:Summary:

c

b

or 2b

Analyze the graph of

amplitude =

vertical translation:

horizontal translation:

1

2

period = 2

1

2

3

none

c

b(to the right)

3

)3

sin(2

1 xy

Analyze the graph of

amplitude =

vertical translation:

horizontal translation:

period = 2

2

1

c

b

none

3

4

2

2 (to the left)

)4 θ2cos(3 y

Analyze the graph of y x 2 3 2cos

amplitude =

vertical translation:

horizontal translation:

period = 360

2

180

Up 2

3

none

y = -2 + 3 cos (2x - 90°)

amplitude = 360

2

period =

vertical translation:

horizontal translation:

x y

45°

225°

90°

135°

180°

180 4 = 45table goes in increments of 45

1

-2

-5-2

1

1) horiz. tells you where to start

2) add the period to find out where to finish

3) divide period by 4 to find increments

4) plot points and graph45 + 180 = 225

Graph and AnalyzeGraph and Analyze

c

b

90

2 45

(to the right)

3

= 180°

down 2

high

low

high

mid

mid

The most important thing to remember about graphing is determining the starting point and the stopping point on the t-table.

You must know how to analyze the equation before you can graph it.

y = 1 + 3 sin (2 + )

amplitude =period =

vertical translation:

horizontal translation:

x y

0

1

4

1

-2

11) horiz. tells you where to start

2) add the period to find out where to finish

3) divide period by 4 to find increments

4) plot points and graph

Ex #6cEx #6c GraphGraph

c

b

2

c

b

2

2

=

2

2

4

4

3

up 1

2 2

4table goes in increments of

4

mid

mid

mid

high

low