copyright © 2010 pearson education, inc. rational functions and models ♦identify a rational...

Copyright © 2010 Pearson Education, Inc.

Rational Functions Rational Functions and Modelsand Models

♦ Identify a rational function and state its domainIdentify a rational function and state its domain♦ Identify asymptotesIdentify asymptotes♦ Interpret asymptotesInterpret asymptotes♦ Graph a rational function by using Graph a rational function by using

transformationstransformations♦ Graph a rational function by handGraph a rational function by hand

4.64.6

Slide 4.6 - 2Copyright © 2010 Pearson Education, Inc.

Rational Function

A function f represented by

where p(x) and q(x) are polynomials and

q(x) ≠ 0, is a rational function.

f x p x

q x ,

Slide 4.6 - 3Copyright © 2010 Pearson Education, Inc.

Rational Function

The domain of a rational function includes all real numbers except the zeros of the denominator q(x).

The graph of a rational function is continuous except at x-values whereq(x) = 0.

Slide 4.6 - 4Copyright © 2010 Pearson Education, Inc.

Example 1For each rational function, determine any horizontal or vertical asymptotes.

a)

b)

c)

f (x)

2x 1

x2 1

g(x)

1

x

h(x)

x3 2x2 1

x2 3x 2

Slide 4.6 - 5Copyright © 2010 Pearson Education, Inc.

Example 1Solution

a)

b)

f (x)

2x 1

x2 1

g(x)

1

x

Is a rational function - both numerator and denominator are polynomials; domain is all real numbers; x2 + 1 ≠ 0

Is NOT a rational function Denominator is not a polynomial; domain is

{x | x > 0}

Slide 4.6 - 6Copyright © 2010 Pearson Education, Inc.

Example 1Solution continued

c) h(x)

x3 2x2 1

x2 3x 2

Is a rational function - both numerator and

denominator are polynomials; domain is

{x | x ≠1, x ≠ 2} because (x – 1)(x – 2) = 0

when x = 1 and x = 2.

Slide 4.6 - 7Copyright © 2010 Pearson Education, Inc.

Vertical Asymptotes

The line x = k is a vertical asymptote of the graph of f if f(x) ∞ or f(x) –∞ as x approaches k from

either the left or the right.

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Horizontal Asymptotes

The line y = b is a horizontal asymptote of the graph of f if

f(x) b as x

approaches

either ∞ or –∞.

Slide 4.6 - 9Copyright © 2010 Pearson Education, Inc.

Finding Vertical & Horizontal Asymptotes

Let f be a rational function given by

written in lowest terms.

Vertical AsymptoteTo find a vertical asymptote, set the denominator, q(x), equal to 0 and solve. If k is a zero of q(x), then x = k is a vertical asymptote. Caution: If k is a zero of both q(x) and p(x), then f(x) is not written in lowest terms, and x – k is a common factor.

f x p x

q x ,

Slide 4.6 - 10Copyright © 2010 Pearson Education, Inc.

Finding Vertical & Horizontal Asymptotes

Horizontal Asymptote

(a) If the degree of the numerator is less than the degree of the denominator, then y = 0 (the x-axis) is a horizontal asymptote.

(b) If the degree of the numerator equals the degree of the denominator, then y = a/b is a horizontal asymptote, where a is the leading coefficient of the numerator and b is the leading coefficient of the denominator.

Slide 4.6 - 11Copyright © 2010 Pearson Education, Inc.

Finding Vertical & Horizontal Asymptotes

Horizontal Asymptote

(c) If the degree of the numerator is greater than the degree of the denominator, then there are no horizontal asymptotes.

Slide 4.6 - 12Copyright © 2010 Pearson Education, Inc.

Example 4For each rational function, determine any horizontal or vertical asymptotes.Solutiong(x) is a translation of f(x) left one unit and down 2 units.The vertical asymptote is x = 1The horizontal asymptote isy = 2g(x) = f(x + 1) 2

2

1( )f x

x

2

1( ) 2.

( 1)g x

x

Slide 4.6 - 13Copyright © 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

Slide 4.6 - 14Copyright © 2010 Pearson Education, Inc.

ExampleSolution

a)

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

Slide 4.6 - 15Copyright © 2010 Pearson Education, Inc.

ExampleSolution continued

a)

Here’s a graph of f(x).

f (x)

6x 1

3x 3

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ExampleSolution continued

b)

Degree of numerator is one less than the degree of the denominator so the x-axis, or y = 0, is a horizontal asymptote.When x = ±2, the denominator, x2 – 4, equals 0 and the numerator, x + 1 does not equal 0, so the vertical asymptotes arex = 2 and x = 2.

g(x)

x 1

x2 4

Slide 4.6 - 17Copyright © 2010 Pearson Education, Inc.

ExampleSolution continued

b)

Here’s a graph of g(x).

g(x)

x 1

x2 4

Slide 4.6 - 18Copyright © 2010 Pearson Education, Inc.

ExampleSolution

c)

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

Slide 4.6 - 19Copyright © 2010 Pearson Education, Inc.

ExampleSolution

c)

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

h(x)

x2 1

x 1

Slide 4.6 - 20Copyright © 2010 Pearson Education, Inc.

Slant, or Oblique, Asymptotes

A third type of asymptote, which is neither vertical nor horizontal, occurs when the numerator of a rational function has degree one more than the degree of the denominator.

Slide 4.6 - 21Copyright © 2010 Pearson Education, Inc.

Slant, or Oblique, Asymptotes

The line y = x + 1 is a slant asymptote, or oblique asymptote of the graph of f.

f (x)

x2 2

x 1x 1

3

x 1

Slide 4.6 - 22Copyright © 2010 Pearson Education, Inc.

Graphs and Transformations of Rational Functions

Graphs of rational functions can vary greatly in complexity.

We begin by graphing and then use

transformations to graph other rational

functions.

y

1

x

Slide 4.6 - 23Copyright © 2010 Pearson Education, Inc.

Example 5Sketch a graph of and identify any asymptotes.

Solution

y

1

x

Vertical asymptote:x = 0Horizontal asymptote:y = 0

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Example 6Use the graph of to sketch a

graph of

Include all asymptotesin your graph. Writeg(x) in terms of f(x).

f x 1

x2

g x 1

x 2 2.

Slide 4.6 - 25Copyright © 2010 Pearson Education, Inc.

Example 6Solutiong(x) is a translation of f(x) left 2 units and then a reflection acrossthe x-axis. Vertical asymptote: x = –2Horizontal asymptote: y = 0

g(x) = –f(x + 2)

Slide 4.6 - 26Copyright © 2010 Pearson Education, Inc.

Example 7

Let

a) Use a calculator to graph f. Find the domain of f.

b) Identify any vertical or horizontal asymptotes.

c) Sketch a graph of f that includes the asymptotes.

f (x)

2x2 1

x2 4.

Slide 4.6 - 27Copyright © 2010 Pearson Education, Inc.

Example 7Solution

a) Here’s the calculator display using “Dot Mode.” The function is undefined when x2 – 4 = 0, or when x = ±2.

The domain of f isD = {x|x ≠ 2, x ≠ –2}.

Slide 4.6 - 28Copyright © 2010 Pearson Education, Inc.

Example 7Solution

b) When x = ±2, the denominator x2 – 4

= 0 (the numerator does not), so the vertical asymptotes are x = ±2.

Degree of numerator = degree of denominator, ratio of leading coefficients is 2/1 = 2, so the horizontal asymptote is y = 2.

f (x)

2x2 1

x2 4

Slide 4.6 - 29Copyright © 2010 Pearson Education, Inc.

Example 7Solution

c) Here’s another version of the graph.

Slide 4.6 - 30Copyright © 2010 Pearson Education, Inc.

Graphing Rational Functions by Hand

Let define a rational function

in lowest terms. To sketch its graph, follow these steps.

STEP 1: Find all vertical asymptotes.

STEP 2: Find all horizontal or oblique asymptotes.

STEP 3: Find the y-intercept, if possible, by evaluating f(0).

f (x)

p x q x

Slide 4.6 - 31Copyright © 2010 Pearson Education, Inc.

Graphing Rational Functions by Hand

STEP 4: Find the x-intercepts, if any, by solving f(x) = 0. (These will be the zeros of the numerator p(x).)

STEP 5: Determine whether the graph will intersect its nonvertical asymptote y = b by solving f(x) = b, where b is the y-value of the horizontal asymptote, or by solving f(x) = mx + b, where y = mx + b is the equation of the oblique asymptote.

Slide 4.6 - 32Copyright © 2010 Pearson Education, Inc.

Graphing Rational Functions by Hand

STEP 6: Plot selected points as necessary. Choose an x-value in each interval of the domain determined by the vertical asymptotes and x-intercepts.

STEP 7: Complete the sketch.

Slide 4.6 - 33Copyright © 2010 Pearson Education, Inc.

Example 8

Graph

Solution

STEP 1: Vertical asymptote: x = 3

STEP 2: Horizontal asymptote: y = 2

STEP 3: f(0) = , y-intercept is

f x 2x 1

x 3.

1

3

1

3

Slide 4.6 - 34Copyright © 2010 Pearson Education, Inc.

Example 8

Solution continued

STEP 4: Solve f(x) = 0

The x-intercept is

f x 2x 1

x 3

1

2

2x 1

x 30

2x 10

x 1

2

Slide 4.6 - 35Copyright © 2010 Pearson Education, Inc.

Example 8

Solution continued

STEP 5: Graph does not intersect its horizontal asymptote, sincef(x) = 2 has no solution.

STEP 6: The points

are on the graph.

STEP 7: Complete the sketch (next slide)

f x 2x 1

x 3

4,1 , 1,

3

2

, 6,13

3

Slide 4.6 - 36Copyright © 2010 Pearson Education, Inc.

Example 8

Solution continued

STEP 7

f x 2x 1

x 3