rational functions - rational functions are quotients of polynomial functions: where p(x) and q(x)...
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Rational Functions
- Rational functions are quotients of polynomial functions:
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 for 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).
( )( )
( )P x
r xQ x
2 7 9( )( 2)( 5)x xf x
x x x
This is Q(x).
3.6: Rational Functions and Their Graphs
Rational FunctionsGraphs of rational functions have breaks in them and can have distinct branches. We use a special arrow notation to help describe this situation symbolically:
Arrow NotationArrow NotationSymbol 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.
Arrow NotationArrow NotationSymbol 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.
3.6: Rational Functions and Their Graphs
3.6 Rational Functions and Graphs
Consider the graph of :
1. What happens as ?
2. What happens as
3. What happens as
4. What happens as
1
x
x ?x
0 ?x 0 ?x
Definition of a Vertical AsymptoteDefinition 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
Definition of a Vertical AsymptoteDefinition 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
Vertical Asymptotes of Rational Functions
Thus, 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
3.6: Rational Functions and Their Graphs
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Definition of a Vertical AsymptoteDefinition 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.
Definition of a Vertical AsymptoteDefinition 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.
Vertical Asymptotes of Rational Functions
Thus, 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
3.6: Rational Functions and Their Graphs
Vertical Asymptotes of Rational Functions
If the graph of a rational function has vertical asymptotes, they can be located in the following way:
Locating Vertical AsymptotesLocating 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), then x a is a vertical asymptote of the graph
of f(x).
Locating Vertical AsymptotesLocating 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), then x a is a vertical asymptote of the graph
of f(x).
( )( )
( )P x
f xQ x
3.6: Rational Functions and Their Graphs
Definition of a Horizontal AsymptoteDefinition 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.
Definition of a Horizontal AsymptoteDefinition 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.
Horizontal Asymptotes of Rational Functions
A rational function may have several vertical asymptotes, but it can have at most one horizontal asymptote.
ff
y
x
y = by = b
x
y
ff
y = by = b
ff
y
x
y = by = b
f (x) b as x f (x) b as x f (x) b as x
3.6: 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 in the following way:
Locating Horizontal AsymptotesLocating Horizontal AsymptotesLet 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 is the horizontal asymptote of the graph of f.
2. If n m, the line y is the horizontal asymptote of the graph of f.
3. If n m, the graph of f has no horizontal asymptote.
Locating Horizontal AsymptotesLocating Horizontal AsymptotesLet 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 is the horizontal asymptote of the graph of f.
2. If n m, the line y is the horizontal asymptote of the graph of f.
3. If n m, the graph of f has no horizontal asymptote.
11 1 0
11 1 0
( ) , 0, 0n n
n nn mm m
m m
a x a x a x af x a b
b x b x b x b
n
m
ab
3.6: Rational Functions and Their Graphs
3.6: Rational Functions and Their Graphs
Strategy for Graphing a Rational FunctionStrategy for Graphing a Rational FunctionSuppose that
where P(x) and Q(x) are polynomial functions with no common factors.
1. Factor: Factor numerator and denominator. 2. Intercepts: Find the y-intercept by evaluating f (0). Find the
x-intercepts by finding the zeros of the numerator. 3. Asymptotes: Find the horizontal asymptote by using the rule for determining the
horizontal asymptote of a rational function. Find the vertical asymptote(s) by finding the zeros of the denominator. 4. Behavior: check the behavior on each side of vertical asymptote(s).5. Sketch graph. (Plot at least one point between and beyond each x-intercept and vertical asymptote.)
Strategy for Graphing a Rational FunctionStrategy for Graphing a Rational FunctionSuppose that
where P(x) and Q(x) are polynomial functions with no common factors.
1. Factor: Factor numerator and denominator. 2. Intercepts: Find the y-intercept by evaluating f (0). Find the
x-intercepts by finding the zeros of the numerator. 3. Asymptotes: Find the horizontal asymptote by using the rule for determining the
horizontal asymptote of a rational function. Find the vertical asymptote(s) by finding the zeros of the denominator. 4. Behavior: check the behavior on each side of vertical asymptote(s).5. Sketch graph. (Plot at least one point between and beyond each x-intercept and vertical asymptote.)
( )( )
( )P x
f xQ x
EXAMPLE: Finding the Slant Asymptoteof a Rational Function
Find the slant asymptotes of f (x) 2 4 5 .
3x x
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:
2 1 4 51 3 3 1 1 8
3
2
81 13
3 4 5
xx
x x x
Remainder
3.6: Rational Functions and Their Graphs
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EXAMPLE: Finding the Slant Asymptoteof a Rational Function
Find the slant asymptotes of f (x) 2 4 5 .
3x x
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 4 53
x xx
-2 -1 4 5 6 7 8321
7
6
5
4
3
1
2
-1
-3
-2
Vertical asymptote: x = 3
Vertical asymptote: x = 3
Slant asymptote: y = x - 1
Slant asymptote: y = x - 1
3.6: Rational Functions and Their Graphs
Locating Horizontal Asymptotes:
Divide the numerator and denominator by the highest power of x that appears in denominator and then let
Horizontal Asymptotes of Rational Functions
If the graph of a rational function has a horizontal asymptote, it can be located in the following way:
3.6: Rational Functions and Their Graphs
x .