2.7 Rational Functions and Their Graphs Graphing Rational Functions.

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2.7 Rational Functions and Their Graphs Graphing Rational Functions

Transcript of 2.7 Rational Functions and Their Graphs Graphing Rational Functions.

2.7 Rational Functions and

Their Graphs

Graphing Rational Functions

Graphing Rational Functions

1) Find the y-intercept: Substitute 0 in for x2) Find the x-intercepts: Set the numerator equal to 0 and

solve.3) Find the Vertical Asymptote: Set the denominator

equal to zero and solve.4) Find the Horizontal or Slant Asymptote: Use the

numerator’s degree (m) and the denominator’s degree (n) to determine which method to use.

5) Find Extra Points: Find points to the left and right of each vertical asymptote.

6) Graph: Graph steps 1-6.

( )( )

( )

m

n

p x ax bf x

q x cx d

Degree Comparison Alternate Asymptote

nm

nm

dcx

bax

xq

xpxf

n

m

)(

)()(

0y

Slant Asymptote:Long Division

Horizontal Asymptote:

( )

( )

p xy

q x

nm ay

c

Horizontal Asymptote:

1int . :

4y

Graphing Rational Functions (m=n)

2

21.

1

2

4

x xf x

x

2

2

2 0 0 1 - intercept :) 0

0 4fb y

1int . : ,1

2x

2

2

2 1 - intercepts : 0

4)

x x

xc x

0 2 1 1x x

2 1 1

2 2

x x

x x

a

b

2y

Graphing Rational Functions (m=n)

2 Vertical Asymptote : 4 0)d x 2,2x

2 Horizontal Asymptote :

1)e y

2 2 0x x

2

21.

1

2

4

x xf x

x

2 1 1

2 2

x x

x x

c

Graphing Rational Functions (m=n)

x y3 43 1

)f

2

2

2 3 3 13

3 4f

18 3 13

9 4f

203 4

5f

2

2

2 3 3 13

3 4f

18 3 13

9 4f

143 2.8

5f

2

21.

1

2

4

x xf x

x

2 1 1

2 2

x x

x x

d

1int . :

2y

Graphing Rational Functions (m<n)

2

3

5 62.

xf x

x x

2

0 3 - intercept : 0

0)

5 0 6y fb

int . : 3x

2

3 - intercepts) : 0

5 6

xx

xc

x

0 3x

3

3 2

x

x x

a

b

Graphing Rational Functions (m<n)

2 Vertical Asym) ptote : 5 6 0d x x 2,3x

Horizontal Asymptote) : 0e y

3 2 0x x

2

3

5 62.

xf x

x x

3

3 2

x

x x

d

c

Graphing Rational Functions (m<n)

x y

41

42

2.5 22

)f

2

3

5 62.

xf x

x x

3

3 2

x

x x

4 3.5

e

1int . :

3y

Graphing Rational Functions (m>n)

2

3 2

.1

3

x xf x

x

20 2 0 1

- intercept :) 00 3

y fb

int . :1x

2 2 1 - intercepts : 0

3)

x xxc

x

20 1x

21

3

x

x

b

a

Graphing Rational Functions (m>n)

Vertical Asymptot 3) e : 0xd 3x

Slant Asymp :) totee

2

3 2

.1

3

x xf x

x

2

1

3

x

x

23 2 1x x x x

2 3x x1x

1

3x

1y x d

c

Graphing Rational Functions (m>n)

x y

4 9

)f

94 9

1f

24 1

44 3

f

2

3 2

.1

3

x xf x

x

2

1

3

x

x

e

int . : 0y

Special Rational Functions

2

2

44.

2 1

xf x

x

2

2

4 0 - intercept : 0

2 1)

0y fb

int . : 0x

2

2

4 - intercept : 0

2 1) s

xx

xc

20 4x

b

a

Special Rational Functions

2 Vertical Asymptote : 2 1 0)d x

none

4 Horizontal Asymptote :

2)e y

2

2

44.

2 1

xf x

x

2y

1

2x i

d

c

Special Rational Functions

x y

1 0.8

41

5f

2

2

4 11

4 1 1f

2

2

44.

2 1

xf x

x

41

5f

2

2

4 11

4 1 1f

1 0.8

e

int . : 3y

Special Rational Functions

20 0 6

- intercept) : 00 2

y fb

int . : 3x - intercepts : 0 3) xc x

2

56

. 2

x xf x

x

3 2

2

x x

x

3x

b

a

Special Rational Functions

Vertical Asymptot 2) e : 0xd 2x

none

2

56

. 2

x xf x

x

3 2

2

x x

x

3x

d

c

Horizontal Asymptote:

Special Rational Functions

)f

2

56

. 2

x xf x

x

3 2

2

x x

x

3x

e