Rational Functions and Their Graphs

Example

Find the Domain of this Function.

Solution:

The domain of this function is the set of all real numbers not equal to 3.

3

7)(

x

xxf

Arrow Notation

Symbol Meaning

x a x approaches a from the right.

x a x approaches a from the left.

x x approaches infinity; that is, x increases without bound.

x x approaches negative infinity; that is, x decreases without bound.

The line x a is a vertical asymptote of the graph of a function f if f (x) increases or decreases without bound as x approaches a.

f (x) as x a f (x) as x a

The line x a is a vertical asymptote of the graph of a function f if f (x) increases or decreases without bound as x approaches a.

f (x) as x a f (x) as x a

Thus, f (x) or f(x) as x approaches a from either the left or the right.

f

a

y

x

x = a

f

a

y

x

x = a

Definition of a Vertical Asymptote

The line x a is a vertical asymptote of the graph of a function f if f (x)

increases or decreases without bound as x approaches a. The line x a is a vertical asymptote of the graph of a function f if f (x)

increases or decreases without bound as x approaches a.

Thus, f (x) or f(x) as x approaches a from either the left or the right.

x = a

fa

y

x

f

a

y

x

x = a

f (x) as x a f (x) as x a

Definition of a Vertical Asymptote

Locating Vertical Asymptotes

If is a rational function in which

p(x) and q(x) have no common factors and a is a zero of q(x), the denominator, then x = a is a vertical asymptote of the graph of f.

( )( )

( )

p xf x

q x

The line y = b is a horizontal asymptote of the graph of a function f if f (x) approaches b as x increases or decreases without bound.The line y = b is a horizontal asymptote of the graph of a function f if f (x) approaches b as x increases or decreases without bound.

f

y

x

y = b

x

y

f

y = b

f

y

x

y = b

f (x) b as x f (x) b as x f (x) b as x

Definition of a Horizontal Asymptote

Locating Horizontal Asymptotes

Let f be the rational function given by

The degree of the numerator is n. The degree of the denominator is m.

1. If n<m, the x-axis, or y=0, is the horizontal asymptote of the graph of f.

2. If n=m, the line y = an/bm is the horizontal asymptote of the graph of f.

3. If n>m,t he graph of f has no horizontal asymptote.

f (x) an xn an 1x

n 1 ... a1x a0

bm xm bm 1xm 1 ... b1x b0

, an 0,bm 0

Strategy for Graphing a Rational Function

Suppose that where p(x) and q(x) are polynomial functions with no common factors.

1. Find any vertical asymptote(s) by solving the equation q (x) 0.

2. Find the horizontal asymptote (if there is one) using the rule for determining the horizontal asymptote of a rational function.

3. Use the information obtained from the calculators graph and sketch the graph labeling the asymptopes.

( )( )

( )

p xf x

q x

Sketch the graph of

105

32)(

x

xxf

• The vertical asymptote is x = -2

• The horizontal asymptote is y = 2/5

105

32)(

x

xxf

105

32)(

x

xxf

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10