Key Information• Starting Last Unit Today

– Graphing– Factoring– Solving Equations– Common Denominators– Domain and Range (Interval Notation)

• Factoring will be critical next week

• Retest is this Thursday PM and Friday AM

• Must have test corrections done by then

• Factoring Quiz Tuesday. Only 2 left!!

Warm-upWhat is the degree of each of these?

4 23 3 7 1x x x 6 4 24 7 8x x x

8 10x

12

4

6

1

0

Section 8-2 & 8-3

Graphing Rational Functions

Objectives

• I can determine vertical asymptotes of a rational function and graph them

• I can determine horizontal asymptotes of a rational function and graph them

• I can find x and y intercepts to help graph

• I can graph rational functions using a calculator

Rational Functions

• A rational function is any ratio of two polynomials, where denominator cannot be ZERO!

• Examples:

1)(

x

xxf

103

1)(

2

xx

xxf

Asymptotes

• Asymptotes are the boundary lines that a rational function approaches, but never crosses.

• We draw these as Dashed Lines on our graphs.

• There are two types of asymptotes we will study in Alg-2: – Vertical– Horizontal

Vertical Asymptotes

• Vertical Asymptotes exist where the denominator would be zero.

• They are graphed as Vertical Dashed Lines

• There can be more than one!

• To find them, set the denominator equal to zero and solve for “x”

Example #1

• Find the vertical asymptotes for the following function:

1)(

x

xxf

•Set the denominator equal to zero

•x – 1 = 0, so x = 1

•This graph has a vertical asymptote at x = 1

1 2 63 4 5 7 8 9 10

4

3

2

7

5

6

8

9

x-axis

y-axis

0

1-2-6 -3-4-5-7-8-910

-4

-3

-2

-1

-7

-5

-6

-8

-9

0

-1

Vertical Asymptote at

X = 1

Other Examples:

• Find the vertical asymptotes for the following functions:

3

3)(

x

xg

)5)(2(

1)(

xx

xxg

3: xVA

5;2: xxVA

Horizontal Asymptotes

• Horizontal Asymptotes are also Dashed Lines drawn horizontally to represent another boundary.

• To find the horizontal asymptote you compare the degree of the numerator with the degree of the denominator

• See next slide:

Horizontal Asymptote

Given the Rational Function:

Compare DEGREE of Numerator to Denominator

If N < D , then y = 0 is the HA

If N = D, then the HA is

If N > D, then the graph has NO HA

Numerator( )

Denominatorf x

N

D

LCy

LC

Example #1

• Find the horizontal asymptote for the following function:

1)(

x

xxf

•Since the degree of numerator is equal to degree of denominator (m = n)

•Then HA: y = 1/1 = 1

•This graph has a horizontal asymptote at y = 1

1 2 63 4 5 7 8 9 10

4

3

2

7

5

6

8

9

x-axis

y-axis

0

1-2-6 -3-4-5-7-8-910

-4

-3

-2

-1

-7

-5

-6

-8

-9

0

-1

Horizontal Asymptote at

y = 1

Other Examples:

• Find the horizontal asymptote for the following functions:

3

3)(

x

xg

13

13)(

2

2

xx

xxg

5

1)(

3

x

xxg

0: yHA

3: yHA

NoneHA :

Intercepts

• x-intercepts (there can be more than one)

• Set Numerator = 0 and solve for “x”

• y-intercept (at most ONE y-intercept)

• Let all x’s =0 and solve

Graphing Rational Expressions• Factor rational expression and reduce• Find VA (Denominator = 0)• Find HA (Compare degrees)• Find x-intercept(s) (Numerator = 0)• Find y-intercept (All x’s = 0)• Next type the function into the graphing calculator and

look up ordered pairs from the data table to graph the function.

• Remember that the graph will never cross the VA

Calculator

ALWAYS use parenthesis!

(Numerator) (Denominator)y

Key Data to Graph

)1)(3(

32)(

xx

xxf

1,3: xxVA

0: yHA

3int : ( ,0)

2x

)1,0(:int y

Graph: f(x) =

• Vertical asymptote:• x – 2 = 0 so at x = 2• Dashed line at x = 2• m = 0, n = 1 so m<n• HA at y = 0• No x-int• y-int = (0, -1)• Put into graphing calc.• Pick ordered pairs

2

2

x

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

-7

-6-5

-4

-3

-2-1

12

3

4

56

7

f(x) = )3)(2(

6

xx

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

-7

-6-5

-4

-3

-2-1

12

3

4

56

7Vertical Asymptotes:

x – 2 = 0 and x + 3 = 0

x = 2, x = -3

m = 0, n = 2 m < n

HA at y = 0

No x-int

y-int (0, -1)

Graph on right

Calculator

(6) (( 2)( 3))y x x

6( )

( 2)( 3)f x

x x

Homework

• WS 12-1

• Must know how to factor for next week!!!

• Factoring Quiz Tuesday