slide 3.5- 1 copyright © 2007 pearson education, inc. publishing as pearson addison-wesley

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

OBJECTIVES

Rational Functions

Learn the definition of a rational function.

Learn to find vertical asymptotes (if any).

Learn to find horizontal asymptotes (if any).

Learn to graph rational functions.

Learn to graph rational functions with oblique asymptotes.

Learn to graph a revenue curve.

SECTION 3.5

1

2

3

4

5

6


RATIONAL FUNCTION

A function f that can be expressed in the form

where the numerator N(x) and the denominator D(x) are polynomials and D(x) is not the zero polynomial, is called a rational function. The domain of f consists of all real numbers for which D(x) ≠ 0.

f x N x D x ,


EXAMPLE 1 Finding the Domain of a Rational Function

Find the domain of each rational function.

a. f x 3x2 12

x 1

Solution

a. The domain of f (x) is the set of all real numbers for which x – 1 ≠ 0; that is, x ≠ 1 .

b. g x x

x2 6x 8

c. h x x2 4

x 2

In interval notation: ,1 U 1,



Solution continued

b. Find the values of x for which the denominator x2 – 6x + 8 = 0, then exclude those values from the domain.

x 2 x 4 0

x 2 0 or x 4 0

x 2 or x 4

In interval notation: ,2 U 2, 4 U 4,

The domain of g (x) is the set of all real numbers such that x ≠ 2 and x ≠ 4 .



Solution continued

c. The domain of h(x) is the set of all real numbers for which x – 2 ≠ 0; that is, x ≠ 2 .

In interval notation: ,2 U 2,

The domain of g (x) is the set of all real numbers such that x ≠ 2.


VERTICAL ASYMPTOTES

The line with equation x = a is called a vertical asymptote of the graph of a function f if

f x as x a or x a .


VERTICAL ASYMPTOTES


LOCATING VERTICAL ASYMPTOTES OF RATIONAL FUNCTIONS

If

where the N(x) and D(x) do not have a common factor and a is a real zero of D(x), then the line with equation x = a is a vertical asymptote of the graph of f.

f x N x D x is a rational function,


EXAMPLE 2 Finding Vertical Asymptotes

Find all vertical asymptotes of the graph of each rational function.

a. f x 1

x 1

Solution

a. No common factors, zero of the denominator is x = 1. The line with equation x = 1 is a vertical asymptote of f (x).

b. g x 1

x2 9

c. h x 1

x2 1


EXAMPLE 2 Finding Vertical Asymptotes

Solution continued

b. No common factors. Factoring x2 – 9 = (x + 3)(x – 3), we see the zeros of the denominator are x = –3 and x = 3. The lines with equations x = 3 and x = –3 are the two vertical asymptotes of f (x).

c. The denominator x2 + 1 has no real zeros. Hence, the graph of the rational function h (x) has no vertical asymptotes.


EXAMPLE 3 Rational Function Whose Graph Has a Hole

Find all vertical asymptotes of the graph of each rational function.

a. h x x2 4

x 2b. g x x 2

x2 4

The graph is the line with equation y = x + 2, with a gap (hole) corresponding to x = 2.

Solution

a. h x x2 4

x 2

x 2 x 2 x 2

x 2, x 2



Solution continued



Solution continued

The graph has a hole at x = –2. However, the graph of g(x) also has a vertical asymptote at x = 2.

b. g x x 2

x2 4

x 2

x 2 x 2

1

x 2, x 2



Solution continued


HORIZONTAL ASYMPTOTES

The line with equation y = k is called a horizontal asymptote of the graph of a function f if

f x k as x or x .


RULES FOR LOCATINGHORIZONTAL ASYMPTOTES

Let f be a rational function given by

where N(x) and D(x) have no common factors. Then whether the graph of f has one horizontal asymptote or no horizontal asymptote is found by comparing the degree of the numerator, n, with that of the denominator, m:

an xn an 1x

n 1 ... a2 x2 a1x a0

bm xm bm 1xm 1 ... b2 x2 b1x b0

,an 0,bn 0

f x N x D x


1. If n < m, then the x-axis (y = 0) is the horizontal asymptote.

3. If n > m, then the graph of f has no horizontal asymptote.

y an

bm

2. If n = m, then the line with equation

is the horizontal asymptote, where an and bm are the leading coefficients of N(x) and D(x), respectively.

RULES FOR LOCATINGHORIZONTAL ASYMPTOTES


EXAMPLE 4 Finding the Horizontal Asymptote

Find the horizontal asymptotes (if any) of the graph of each rational function.

a. f x 5x 2

1 3x

Solution

a. Numerator and denominator have degree 1.

b. g x 2x

x2 1

c. h x 3x2 1

x 2

is the horizontal asymptote.y 5

3

5

3


EXAMPLE 4 Finding the Horizontal Asymptote

Solution continued

degree of denominator > degree of numerator y = 0 (the x-axis) is the horizontal asymptote

b. g x 2x

x2 1

c. h x 3x2 1

x 2

degree of numerator > degree of denominator the graph has no horizontal asymptote


PROCEDURE FOR GRAPHINGA RATIONAL FUNCTION

1. Find the intercepts. The x-intercepts are found by solving the equation N(x) = 0. The y-intercept is f (0).

2. Find the vertical asymptotes (if any). Solve D(x) = 0. This step gives the vertical asymptotes of the graph. Sketch the vertical asymptotes.

3. Find the horizontal asymptotes (if any). Use the rules found in an earlier slide.


4. Test for symmetry. If f (–x) = f (x), then f is symmetric with respect to the y-axis. If f (–x) = – f (x), then f is symmetric with respect to the origin.

5. Find the sign of f (x). Use the sign graphs and test numbers associated with the zeros of N(x) and D(x), to determine where the graph of f is above the x-axis and where it is below the x-axis.

6. Sketch the graph. Plot the points and asymptotes found in steps 1-5 and symmetry to sketch the graph of f.


EXAMPLE 5 Graphing a Rational Function

Sketch the graph of f x x

x2 4.

Step 2 Find the vertical asymptotes (if any).

f 0 0

02 40 y-intercept is 0

Set f x 0,x

x2 40 or x 0 x-intercept is 0

SolutionStep 1 Find the intercepts.

Solve x2 4 0, x 2 vertical asymptotes are x = 2 and x = –2.



degree of denominator > degree of numeratory = 0 (the x-axis) is the horizontal asymptote

Solution continued

Step 3 Find the horizontal asymptotes (if any).

Step 4 Test for symmetry.

f x x

x 2 4

x

x2 4 f x

Symmetric with respect to the origin



The three zeros 0, –2 and 2 of the numerator and denominator divide the x-axis into four intervals

Solution continued

Step 5 Find the sign of f in the intervals determined by the zeros of the numerator and denominator.

f x x

x2 4

x

x 2 x 2

, 2 , 2,0 , 0,2 , 2,



Solution continued



Solution continued

Step 6 Sketch the graph.

f x x

x2 4



Sketch the graph of f x x2 2

x 2 x 1 .

Step 2 Solve (x + 2)(x – 1) = 0; x = –2, x = 1

f 0 02 2

0 2 0 1 1 y-intercept is –1

SolutionStep 1 Since x2 + 2 > 0, no x-intercepts

vertical asymptotes are x = –2 and x = 1



y = 1 is the horizontal asymptote

Solution continued

Step 3 degree of den = degree of num

Step 4 Symmetry. None

Step 5 The zeros of the denominator –2 and 1 yield the following figure:



Solution continued



Solution continued


f x x2 2

x 2 x 1



Sketch a graph of f x x2

x2 1.

Step 2 Because x2 +1 > 0 for all x, the domain is the set of all real numbers. Since there are no zeros for the denominator, there are no vertical asymptotes.

Solution

Step 1 Since f (0) = 0 and setting f (x) = 0, we have 0. x-intercept and y-intercept are 0.



y = 1 is the horizontal asymptote

Solution continued

Step 3 degree of den = degree of num

Step 4 Symmetry.

f x x 2

x 2 1

x2

x2 1 f x

Symmetric with respect to the y-axis

Step 5 The graph is always above the x-axis, except at x = 0.



Solution continued


f x x2

x2 1


OBLIQUE ASUMPTOTES

Suppose f x N x D x ,

is greater than the degree of D(x). Then

and the degree of N(x)

f x N x D x Q x R x

D x .

Thus, as x , f x Q x 0 Q x .That is the graph of f approaches the graph of the oblique asymptote defined by Q(x).


EXAMPLE 8 Graphing a Rational Function with an Oblique Asymptote

Sketch the graph of f x x2 4

x 1.

Step 2 Solve x + 1 = 0; x = –1; domain is set of all real numbers except –1.

f 0 0 4

0 1 4 y-intercept is –4.

SolutionStep 1 Solve x2 – 4 = 0, x-intercepts: –2, 2

Vertical asymptote is x = –1.



y = x – 1 is an oblique asymptote.

Solution continued

Step 3 degree of num > degree of den

Step 4 Symmetry. None

f x x2 4

x 1x 1

3

x 1

Step 5 Sign of f in the intervals determined by the zeros of the numerator and denominator: –2, 2, and –1.



Solution continued



Solution continued


f x x2 4

x 1


EXAMPLE 9 Graphing a Revenue Curve

The revenue curve for an economy of a country is given by

R x x 100 x x 10

,

a. Find and interpret R(10), R(20), R(30), R(40), R(50), and R(60).

b. Sketch the graph of y = R(x) for 0 ≤ x ≤ 100.c. Use a graphing calculator to estimate the tax

rate that yields the maximum revenue.

where x is the tax rate in percent and R(x) is the tax revenue in billions of dollars.



a. R 10 10 100 10 10 10

45 billion dollars

Solution

If income is taxed at a rate of 10%, total revenue for the government will be 45 billion dollars.

R 20 53.3 billion dollars


R 40 48 billion dollars





Solution continued

Here is the graph ofy = R(x)for0 ≤ x ≤ 100.



Solution continued

c. From the calculator graph of

by using the ZOOM and TRACE features, you can see that the tax rate of about 23% produces he maximum tax revenue of about 53.67 billion dollars for the government.

Y 100x x2

x 10,

slide 3.5- 1 copyright © 2007 pearson education, inc. publishing as pearson addison-wesley

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