section 2.6 rational functions hand out rational functions sheet!

19
Section 2.6 Rational Functions Hand out Rational Functions Sheet!

Upload: clarissa-leonard

Post on 03-Jan-2016

236 views

Category:

Documents


3 download

TRANSCRIPT

Page 1: Section 2.6 Rational Functions Hand out Rational Functions Sheet!

Section 2.6 Rational Functions

Hand out Rational Functions Sheet!

Page 2: Section 2.6 Rational Functions Hand out Rational Functions Sheet!

Section 2.6 Rational Functions

Objective: To find asymptotes and domain

of rational functions.

Page 3: Section 2.6 Rational Functions Hand out Rational Functions Sheet!

The ratio of two polynomial functions is called a rational function.

f (x) x

x 1

Examples

1

( )3 8

xf x

x x

Denominator cannot be zero!

1x 3, 8 x

Page 4: Section 2.6 Rational Functions Hand out Rational Functions Sheet!

What is an asymptote?

An asymptote is a line that a graph approaches as it moves away from the origin.

•Asymptotes can be vertical or horizontal.

•Vertical asymptotes cannot ever be crossed, but horizontal ones can.

•Vertical asymptotes are determined by the zeros of the denominator.

•Horizontal asymptotes are determined by comparing the degrees of the numerator vs. denominator.

Page 5: Section 2.6 Rational Functions Hand out Rational Functions Sheet!

ASYMPTOTE RULES Vertical Asymptotes: located at zeros of q(x). 

Horizontal Asymptotes: (at most one) a. If degree of the numerator < degree of the denominator: y = 0 b. If degree of the numerator = degree of the denominator y = ratio of lead coefficients

c. If degree of the numerator > degree of the denominator no horizontal asymptote  Slant Asymptote: only if the degree of the numerator is exactly one more than the degree of denominator. Divide numerator by denominator. y = ax+b is the slant asymptote. **In order to get a good sketch of the graph, you must plot some points between and beyond each x-intercept and vertical asymptote.

Page 6: Section 2.6 Rational Functions Hand out Rational Functions Sheet!

Copyright © 2010 Pearson Education, Inc.

Example 4For each rational function, determine any

horizontal or vertical asymptotes.

a)

b)

c)

f (x)

6x 1

3x 3

g(x)

x 1

x2 4

h(x)

x2 1

x 1

Page 7: Section 2.6 Rational Functions Hand out Rational Functions Sheet!

Copyright © 2010 Pearson Education, Inc.

ExampleSolution

a)

HA: Degree of numerator and denominator are both 1. Since the ratio of the leading coefficients is 6/3, the horizontal asymptote is y = 2.

VA: When x = –1, the denominator, 3x + 3, equals 0 and the numerator, 6x – 1 does not equal 0, so the vertical asymptote is x = 1.

f (x)

6x 1

3x 3

Page 8: Section 2.6 Rational Functions Hand out Rational Functions Sheet!

Copyright © 2010 Pearson Education, Inc.

ExampleSolution continued

a)

Here’s a graph of f(x).

f (x)

6x 1

3x 3

Page 9: Section 2.6 Rational Functions Hand out Rational Functions Sheet!

Copyright © 2010 Pearson Education, Inc.

ExampleSolution continued

b)

HA: Degree of numerator is one less than the degree of the denominator so the x-axis, or y = 0, is a horizontal asymptote.

VA: When x = ±2, the denominator, x2 – 4, equals 0 and the numerator, x + 1 does not equal 0, so the vertical asymptotes are x = 2 and x = 2.

g(x)

x 1

x2 4

Page 10: Section 2.6 Rational Functions Hand out Rational Functions Sheet!

Copyright © 2010 Pearson Education, Inc.

ExampleSolution continued

b)

Here’s a graph of g(x).

g(x)

x 1

x2 4

Page 11: Section 2.6 Rational Functions Hand out Rational Functions Sheet!

Copyright © 2010 Pearson Education, Inc.

ExampleSolution

c)

HA: Degree of numerator is greater than the degree of the denominator so there are no horizontal asymptotes.

VA: When x = –1, both the numerator and denominator equal 0 so the expression is not in lowest terms: g(x) = x – 1, x ≠ –1. There are no vertical asymptotes.

h(x)

x2 1

x 1

Page 12: Section 2.6 Rational Functions Hand out Rational Functions Sheet!

Copyright © 2010 Pearson Education, Inc.

ExampleSolution

c)

Here’s the graph of h(x).A straight line with thepoint (–1, –2) missing.

Why isn’t there a vertical asymptote at x=-1?

h(x)

x2 1

x 1

Page 13: Section 2.6 Rational Functions Hand out Rational Functions Sheet!

Graph f (x) 2

x 4

x

y

Vertical asymptote at x = 4

Horizontal asymptote at y = 0

x intercept?

y intercept?

x f(x)

5

2

0 -1/2

2

-1

None1

0,2

Page 14: Section 2.6 Rational Functions Hand out Rational Functions Sheet!

x

y

Graph 2

6( )

6f x

x x

Vertical asymptote ?

Horizontal asymptote ?

X intercepts?

Y intercepts?

Domain?

x = 2 and x = -3

y = 0

(0, -1)

None

2, 3x

Evaluate at convenient values of x.

Page 15: Section 2.6 Rational Functions Hand out Rational Functions Sheet!

x

y

Graph f (x) x2 25

x 5

5x

Factor:

5 5

( )5

x xf x

x

Graph looks like ( ) 5

with a hole at 5.

f x x

x

5xDomain:

Page 16: Section 2.6 Rational Functions Hand out Rational Functions Sheet!

Analyze the function. Find the following:

Vertical asymptotes:

horizontal asymptotes:

x intercepts:

y intercepts:

Domain:

2 1

3

xy

x

x

y

Page 17: Section 2.6 Rational Functions Hand out Rational Functions Sheet!

Analyze the function. Find the following:

Vertical asymptotes:

horizontal asymptotes:

x intercepts (reduce 1st):

y intercepts:

Domain:

2

2 8

9 20

xy

x x

2( 4)

4 5

x

x x

2

5x

Page 18: Section 2.6 Rational Functions Hand out Rational Functions Sheet!

Analyze the graph of:

2

2

2 3( )

1

xf x

x

X intercepts?

Vertical asymptote ?

Horizontal asymptote ?

Domain?

Range?

Y intercepts?

Page 19: Section 2.6 Rational Functions Hand out Rational Functions Sheet!

Homework

• P.148

• 7-19 all, 31-34 all, 35, 39

• (table of values not required)