Chapter 4:Graphing &

Inverse Functions

Chapter 4:Graphing &

Inverse Functions

Sections 4.2, 4.3, & 4.5

Transformations

Sections 4.2, 4.3, & 4.5

Transformations

What happens when the

function has a coefficient?

f (x) = sin x

f (x) = sin x

f (x) = 2sin x

f (x) = sin x

f (x) = 3sin x

f (x) = sin x

f (x) = 4sin x

f (x) = ½sin x

f (x) = sin x

f (x) = ¼sin x

f (x) = sin x

A indicates the ???????????the amplitude is A times larger than that of the basic sine curve (amp = 1)

A indicates the amplitude

f (x) = A sin x

What about

cosin

e?

f (x) = cos x

f (x) = cos x f (x) = cos x

f (x) = cos x

f (x) = 3cos x

f (x) = 2cos x

f (x) = cos x

f (x) = ¼cos x

f (x) = ½cos x

the amplitude is A times larger than that of the basic sine curve

the amplitude is A times larger than that of the basic sine or cosine curve

f (x) = A sin x or f (x) = A cos x A indicates the amplitude

Amplitude is the distance

from the midline…so

always positive.

|A| indicates the amplitudethe amplitude is |A| times larger than that of the basic sine or cosine curve

What about

negative

coeffi

cients?

f (x) = sin x

f (x) = -sin x

f (x) = cos x

f (x) = -cos x

f (x) = A sin x or

f (x) = A cos x

If A is negativethe graph is reflected

across the x-axis.

f (x) = A sin x or

f (x) = A cos x

Domain: Range:Amplitude:Period:

all A y A

A2

f (x) = cos x

f (x) = -3cos x

What happens when the ANGLE has a coefficient?

f (x) = sin x

f (x) = sin 2x

f (x) = sin 2x f (x) = sin x

f (x) = sin x

f (x) = sin 4x

f (x) = sin x

f (x) = sin ½x

f (x) = sin x

f (x) = sin ¼x

indicates the ?????????indicates the periodB indicates

the ?????????

f (x) = sin Bx

the period of the function is the period of the basic curve divided by B (period = 2p)

2pB

___

f (x) = cos x

f (x) = cos ½x

f (x) = cos 2x

indicates the period

f (x) = sin Bx orf (x) = cos Bx

the period of the function is the period of the basic curve divided by B (period = 2p)

2pB

___

Period for tangent and

cotangent will be based on

its period of π.

f (x) = sin Bx

What about

negative

coeffi

cients

of t

he

angle?

f (x) = sin x

f (x) = sin -x

SINE odd functionf(-x) = - f(x)

origin symmetry

Also graph of: f (x) = -sin x !

f (x) = cos x

f (x) = cos -x

COSINE even function

f(-x) = f(x)y-axis

symmetry

Also graph of: f (x) = cos x !

f (x) = sin Bx or

f (x) = cos Bx

If B is negativethe graph is reflected

across the y-axis.

f(x)=sin Bx or f(x)=cos Bx

Domain: Range:Amplitude:

Period:

all 1 1y

12

B

B < 0 means y-axis reflection

What happens when a constant is

added to the function?

f(x)= sin x

f(x)= sin x + 1

f(x)= sin x

f(x)= sin x - 2

f(x)= cos x

f(x)= cos x – 3

f (x) = sin x + Dor

f (x) = cos x + D

D indicates displacement.The displacement is a vertical translation (shift) upward for D > 0 and downward for D < 0

What happens when a constant is

added to the angle?

f(x)=sin x

f(x)=sin(x+ ) 4

f(x)=cos x

f(x)=cos(x- ) 3

2

f(x)=cos x

f(x)=cos(2x- )2

= cos[2(x- )]

Phase shift is NOT p!

Phase shift is NOT p!

Coefficients affect the phase shift!

f(x)=sin x

f(x)=sin( x+ )6

3

= sin[ (x+ )]

1

2

1

2

Alternate method:

10

2 6x

1

2 6x

12 2

2 6x

(negative means left)

3x

f(x)=sin x

f(x)=sin( x+ )6

3

= sin[ (x+ )]

1

2

1

2

indicates phase shift.

= sin(Bx+C)

= cos(Bx+C)The phase shift is a horizontal translation left for C > 0 and right for C < 0

= sin[B(x+C)]

= cos[B(x+C)]B

___-C

or f (x)

f (x)

-C

Period:

B < 0 reflect y

f (x)= A sin [B(x + C )] + D

f (x)= A cos [B(x + C )] + D

Phase shift:C > 0 left C < 0 right

-C

Amplitude: AA < 0 reflect x

___2pB

Displacement: DD > 0 up D < 0 down

f (x)= 3sin(2x)+1

f (x)= 2sin(x - π) -2Looks like sine reflected also…

2sin 2f x x

=-1cos2(x - π/2)

+3

f (x)=-cos(2x - π) +3

f (x)= 2 sin [½(x + 0)] - 2f(x)=2sin(½ x)-2

What is the equation?

f (x)= A sin [B(x + C )] + Df (x)= A sin [B(x + C )] - 2f (x)= 2 sin [B(x + C )] - 2f (x)= 2 sin [½(x + C )] - 2

Could also be written with shifted

cosine: 1

2cos 22

f x x

MODE

On a TI-84 calculator:

Function (Func Par Pol Seq)Radian (Radian Degree)

y = y = 2 sin ( (1/2) x ) - 2

WINDOW

Xmin = -4πXmax = 4πXscl = π/2

Ymin = -4Ymax = 4Yscl = 1

SCALE determines the location of the

tic marks.

On a TI-Nspire calculator:5 Settings2: Document SettingsAngle: Radian

y = 2 sin ( (1/2) x ) - 2

XMin: -4πXMax: 4πXScale: π/2

YMin: -4YMax: 4YScale: 1

B Graph

4: Window/Zoom1: Window Settings

3: Graph Entry1: Function

The End