3.5 - 1 Rational function A function of the form where p(x) and q(x) are polynomials, with q(x) ≠ 0, is called a rational function.

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Rational function

A function of the form

( )( ) ,

( )p x

xq x

f

where p(x) and q(x) are polynomials, with q(x) ≠ 0, is called a rational function.

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Rational Function

Some examples of rational functions are

2

2 2

1 1 3 3 6( ) , ( ) , and ( )

2 5 3 8 16x x x

x x xx x x x x

f f f

Since any values of x such that q(x) = 0 are excluded from the domain of a rational function, this type of function often has a discontinuous graph, that is, a graph that has one or more breaks in it.

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Domain: (– , 0) (0, ) Range: (– , 0) (0, )

RECIPROCAL FUNCTION 1

( )xx

f

x y

– 2 – ½ – 1 – 1

– ½ – 2

0 undefined

½ 2

1 1

2 ½

decreases on the intervals (–,0) and (0, ).

1( )x

xf

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Asymptotes

Let p(x) and q(x) define polynomials. For the rational function defined by written in lowest terms, and for real numbers a and b:

1. If (x) as x a, then the line is a vertical asymptote.2. If (x) b as x , then the line y = b is a horizontal asymptote.

( )( ) ,

( )p x

xq x

f

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Determining Asymptotes

To find the asymptotes of a rational function defined by a rational expression in lowest terms, use the following procedures.1. Vertical Asymptotes Find any vertical asymptotes by setting the denominator equal to 0 and solving for x. If a is a zero of the denominator, then the line x = a is a vertical asymptote.

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2. Other AsymptotesDetermine any other asymptotes. Consider three possibilities:(a) If the numerator has lower degree than the denominator, then there is a horizontal asymptote y = 0 (the x-axis).

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2. Other AsymptotesDetermine any other asymptotes. Consider three possibilities:

(b) If the numerator and denominator have the same degree, and the function is of the form

where an, bn ≠ 0,

then the horizontal asymptote has equation.n

n

ay

b

0

0

( ) ,n

nn

n

a x ax

b x b

f

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2. Other AsymptotesDetermine any other asymptotes. Consider three possibilities:

(c) If the numerator is of degree exactly one more than the denominator, then there will be an oblique (slanted) asymptote. To find it, divide the numerator by the denominator and disregard the remainder. Set the rest of the quotient equal to y to obtain the equation of the asymptote.

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Steps for Graphing Functions

A comprehensive graph of a rational function exhibits these features:1. all x- and y-intercepts;2. all asymptotes: vertical, horizontal, and/or oblique;3. the point at which the graph intersects its nonvertical asymptote (if there is any such point);4. the behavior of the function on each domain interval determined by the verticalasymptotes and x-intercepts.

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Graphing a Rational Function

Let define a function where p(x) and q(x) are polynomials and the rational expression is written in lowest terms. To sketch its graph, follow these steps.Step 1 Find any vertical asymptotes.Step 2 Find any horizontal or oblique asymptotes.Step 3 Find the y-intercept by evaluating (0).

( )( )

( )p x

xq x

f

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Step 4 Find the x-intercepts, if any, by solving (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 or y = mx + b by solving (x) = b or(x) = mx + b.

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

Step 7 Complete the sketch.

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Example 5 GRAPHING A RATIONAL FUNCTION WITH THE x-AXIS AS HORIZONTAL ASYMPTOTE

Interval Test Point

Value of (x)

Sign of (x) Graph Above or Below

x-Axis

(– , – 3) – 4 Negative Below

(– 3, – 1) – 2 Positive Above

(– 1, ½ ) 0 Negative Below

(½ , ) 2 Positive Above

15

15

13

13

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Example 7 GRAPHING A RATIONAL FUNCTION THAT INTERSECTS ITS HORIZONTAL ASYMPTOTE

Solution

Graph2

2

3 3 6( ) .

8 16x x

xx x

f

Step 1 To find the vertical asymptote(s), solve x2 + 8x + 16 = 0.

2 8 16 0x x Set the denominator equal to 0.

2( 4) 0x Factor.

4x Zero-factor property.

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Solution

Graph2

2

3 3 6( ) .

8 16x x

xx x

f

Since the numerator is not 0 when x = – 4, the vertical asymptote has the equation x = – 4.

4x Zero-factor property.

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Solution

Graph2

2

3 3 6( ) .

8 16x x

xx x

f

Step 2 We divide all terms by x2 to get the equation of the horizontal asymptote.

Leading coefficient of numerator

Leading coefficient of denominator

31

y

or 3.y

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Solution

Graph2

2

3 3 6( ) .

8 16x x

xx x

f

Step 3 The y-intercept is (0) = – 3/8.

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Solution

Graph2

2

3 3 6( ) .

8 16x x

xx x

f

Step 4 To find the x-intercept(s), if any, we solve (x) = 0.

2

2

3 3 68 16

0x x

x x

23 3 6 0x x Set the numerator equal

to 0.

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Solution

Graph2

2

3 3 6( ) .

8 16x x

xx x

f

Step 4 23 3 6 0x x Set the numerator equal to 0.

2 2 0x x Divide by 3.

( 2)( 1) 0x x Factor.

2 or 1x x Zero-factor property

The x-intercepts are – 1 and 2.

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Solution

Graph2

2

3 3 6( ) .

8 16x x

xx x

f

Step 5 We set (x) = 3 and solve to locate the point where the graph intersects the horizontal asymptote.

Multiply by x2 + 8x + 16.

2

2

3 3 68 16

3x x

x x

2 23 3 6 3 24 48x x x x

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Solution

Graph2

2

3 3 6( ) .

8 16x x

xx x

f

Step 5

Multiply by x2 + 8x + 16.2 23 3 6 3 24 48x x x x

3 6 24 48x x Subtract 3x2.

27 54x Subtract 24x; add 6.

2x Divide by – 27.

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Solution

Graph2

2

3 3 6( ) .

8 16x x

xx x

f

Step 52x Divide by – 27.

The graph intersects its horizontal asymptote at (– 2, 3).

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Solution

Graph2

2

3 3 6( ) .

8 16x x

xx x

f

Step 6 and 7 Some of the other points that lie on the graph are

These are used to complete the graph.

1 210,9 , 8,13 , and 5, .

8 3

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Behavior of Graphs of Rational Functions Near Vertical Asymptotes

Suppose that (x) is defined by a rational expression in lowest terms. If n is the largest positive integer such that (x – a)n is a factor of the denominator of (x), the graph will behave in the manner illustrated.

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Example 8 GRAPHING A RATIONAL

FUNCTION WITH AN OBLIQUE ASYMPTOTE

Solution

Graph 2 1

( ) .2

xx

x

f

3.5 - 1 Rational function A function of the form where p(x) and q(x) are polynomials, with q(x) ≠ 0, is called a rational function.

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