1. 3.3: Rational Functions and Their Graphs
Rational functions are quotients of polynomial functions. This means that rational functions can be expressed as
where p(x) and q(x) are polynomial functions and q(x)  0. The domain of a rational function is the set of all real numbers except the x-values that make the denominator zero. For example, the domain of the rational function
is the set of all real numbers except 0, 2, and -5.
This is p(x).
This is q(x).

2. EXAMPLE: Finding the Domain of a Rational Function
3.3: Rational Functions and Their Graphs
EXAMPLE: Finding the Domain of a Rational Function
Find the domain of each rational function. a.
Solution Rational functions contain division. Because division by 0 is undefined, we must exclude from the domain of each function values of x that cause the polynomial function in the denominator to be 0.
a. The denominator of is 0 if x = 3. Thus, x cannot equal 3. The domain of f consists of all real numbers except 3, written {x | x  3}.
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3. EXAMPLE: Finding the Domain of a Rational Function
3.3: Rational Functions and Their Graphs
EXAMPLE: Finding the Domain of a Rational Function
Find the domain of each rational function. a.
Solution Rational functions contain division. Because division by 0 is undefined, we must exclude from the domain of each function values of x that cause the polynomial function in the denominator to be 0.
b. The denominator of is 0 if x = -3 or x = 3. Thus, the domain of g consists of all real numbers except -3 and 3, written {x | x  - {x | x  -3, x  3}.
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4. EXAMPLE: Finding the Domain of a Rational Function
3.3: Rational Functions and Their Graphs
EXAMPLE: Finding the Domain of a Rational Function
Find the domain of each rational function. a.
Solution Rational functions contain division. Because division by 0 is undefined, we must exclude from the domain of each function values of x that cause the polynomial function in the denominator to be 0.
c. No real numbers cause the denominator of to equal zero. The domain of h consists of all real numbers.

5. Rational Functions Arrow Notation
3.3: Rational Functions and Their Graphs
Rational Functions
Unlike the graph of a polynomial function, the graph of the reciprocal function has a break in it and is composed of two distinct branches. We use a special arrow notation to describe this situation symbolically:
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.

6. Vertical Asymptotes of Rational Functions
3.3: Rational Functions and Their Graphs
Vertical Asymptotes of Rational Functions
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.
f (x)   as x  a f (x)   as x  a - f a y x
x = a f a y x
x = a
Thus, f (x)   or f(x)  -  as x approaches a from either the left or the right.
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7. Vertical Asymptotes of Rational Functions
3.3: Rational Functions and Their Graphs
Vertical Asymptotes of Rational Functions
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.
f (x)  -  as x  a + f (x)  -  as x  a - f a y x
x = a
x = a f a y x
Thus, f (x)   or f(x)  -  as x approaches a from either the left or the right.

8. Vertical Asymptotes of Rational Functions
3.3: Rational Functions and Their Graphs
Vertical Asymptotes of Rational Functions
If the graph of a rational function has vertical asymptotes, they can be located by using the following theorem.
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.

9. Finding Vertical Asymptotes Example 1If
First simplify the
function. Factor
both numerator
and denominator
and cancel any
common factors.

10. The asymptote(s) occur where the simplified denominator equals 0.
The vertical line x=3 is the only vertical asymptote for this function.
As the input value x to this function gets closer and closer to 3 the function itself looks more and more like the vertical line x=3.

11. Graph of Example 1 The vertical dotted line at
x = 3 is the vertical asymptote

12. Finding Vertical Asymptotes Example 2If
Factor both the
numerator and
denominator and cancel
any common factors.
In this case there are no
common factors to
cancel.

13. The denominator equals zero whenever either
This function has two vertical asymptotes, one at x = -2 and the other at x = 3

14. Graph of Example 2 The two vertical dotted lines at
x = -2 and x = 3 are the vertical asymptotes

15. Horizontal Asymptotes of Rational Functions
3.3: Rational Functions and Their Graphs
Horizontal Asymptotes of Rational Functions
A rational function may have several vertical asymptotes, but it can have at most one horizontal asymptote.
Definition of a Horizontal Asymptote
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  

16. Horizontal Asymptotes of Rational Functions
3.3: Rational Functions and Their Graphs
Horizontal Asymptotes of Rational Functions
If the graph of a rational function has a horizontal asymptote, it can be located by using the following theorem.
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.
If n < m, the x-axis (y = 0) is the horizontal asymptote of the graph of f.
If n = m, the line y = is the horizontal asymptote of the graph of f.
If n = m + 1, the quotient is in the form ax + b , and the line y = ax + b is called an oblique (or slant) asymptote.
If n > m + 1, the graph of f has neither a horizontal asymptote or an oblique asymptote.

17. Finding Horizontal Asymptotes Example 3If
then there is a horizontal asymptote at the line y=0 because the degree of the numerator (2) is less than the degree of the denominator (3). This means that as x gets larger and larger in both the positive and negative directions (x → ∞ and x → -∞)
the function itself looks more and more like the horizontal line y = 0

18. Graph of Example 3
The horizontal line y = 0 is the horizontal asymptote.

19. Finding Horizontal Asymptotes Example 4If
then because the degree of the numerator (2) is equal to the degree of the denominator (2) there is a horizontal asymptote at the line y=6/5. Note, 6 is the leading coefficient of the numerator and 5 is the leading coefficient of the denominator. As x→∞ and as x→-∞ g(x) looks more and more like the line y=6/5

20. Graph of Example 4
The horizontal dotted line at y = 6/5 is the horizontal asymptote.

21. f (-x) = f (x): y-axis symmetry f (-x) = -f (x): origin symmetry
3.3: Rational Functions and Their Graphs
Strategy for Graphing a Rational Function
Suppose that
where p(x) and q(x) are polynomial functions with no common factors.
1. Determine whether the graph of f has symmetry.
f (-x) = f (x): y-axis symmetry f (-x) = -f (x): origin symmetry
2. Find the y-intercept (if there is one) by evaluating f (0). 3. Find the x-intercepts (if there are any) by solving the equation p(x) = Find any vertical asymptote(s) by solving the equation q (x) = Find the horizontal asymptote (if there is one) using the rule for determining the horizontal asymptote of a rational function. 6. Plot at least one point between and beyond each x-intercept and vertical asymptote. 7. Use the information obtained previously to graph the function between and beyond the vertical asymptotes.

22. EXAMPLE: Graphing a Rational Function
3.3: Rational Functions and Their Graphs
EXAMPLE: Graphing a Rational Function
Solution
Step 1 Determine symmetry: f (-x) = = = f (x):
Symmetric with respect to the y-axis.
Step 2 Find the y-intercept: f (0) = = 0: y-intercept is 0.
Step 3 Find the x-intercept: 3x2 = 0, so x = 0: x-intercept is 0.
Step 4 Find the vertical asymptotes: Set q(x) = 0.
x2 - 4 = Set the denominator equal to zero.
x2 = 4
x = 2
Vertical asymptotes: x = -2 and x = 2.
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23. EXAMPLE: Graphing a Rational Function
3.4: Rational Functions and Their Graphs
EXAMPLE: Graphing a Rational Function
Solution
Step 5 Find the horizontal asymptote: y = 3/1 = 3.
Step 6 Plot points between and beyond the x-intercept and the vertical asymptotes. With an x-intercept at 0 and vertical asymptotes at x = 2 and x = -2, we evaluate the function at -3, -1, 1, 3, and 4. -5 -4 -3 -2 -1 1 2 3 4 5 7 6
Vertical asymptote: x = 2
Vertical asymptote: x = -2
Horizontal asymptote: y = 3
x-intercept and y-intercept x -3 -1 1 3 4
f(x) =
The figure shows these points, the y-intercept, the x-intercept, and the asymptotes.
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24. EXAMPLE: Graphing a Rational Function
3.4: Rational Functions and Their Graphs
EXAMPLE: Graphing a Rational Function
Solution
Step 7 Graph the function. The graph of f (x) is shown in the figure. The y-axis symmetry is now obvious. -5 -4 -3 -2 -1 1 2 3 4 5 7 6
Vertical asymptote: x = 2
Vertical asymptote: x = -2
Horizontal asymptote: y = 3
x-intercept and y-intercept -5 -4 -3 -2 -1 1 2 3 4 5 7 6
x = -2
y = 3
x = 2

25. EXAMPLE: Finding the Slant Asymptote of a Rational Function
3.3: Rational Functions and Their Graphs
EXAMPLE: Finding the Slant Asymptote of a Rational Function
Find the slant asymptotes of f (x) =
Solution Because the degree of the numerator, 2, is exactly one more than the degree of the denominator, 1, the graph of f has a slant asymptote. To find the equation of the slant asymptote, divide x - 3 into x2 - 4x - 5: 3
Remainder
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26. EXAMPLE: Finding the Slant Asymptote of a Rational Function
3.3: Properties of Rational Function and Their Graphs
EXAMPLE: Finding the Slant Asymptote of a Rational Function
Find the slant asymptotes of f (x) =
Solution The equation of the slant asymptote is y = x - 1. Using our strategy for graphing rational functions, the graph of f (x) = is shown. -2 -1 4 5 6 7 8 3 2 1 -3
Vertical asymptote:
x = 3
Slant asymptote:
y = x - 1