chapter 3 – polynomial and rational functions 3.7 - rational functions
TRANSCRIPT
3.7 - Rational Functions
Section 3.7 Rational
Functions
Chapter 3 – Polynomial and Rational Functions
3.7 - Rational Functions
ExampleRational functions are quotients of polynomials. For
example, functions that can be expressed as
where P(x) and Q(x) are polynomials and Q(x) 0.
Note: We assume that P(x) and Q(x) have no factors in common.
( )( )
( )
P xr x
Q x
3.7 - Rational Functions
Basic Rational Function
We want to identify the characteristics of rational functions
1( )r x
x
Domain Range
x-intercepts y-intercepts
Asymptotes (VA, HA, SA) Directional Limits
MaxMin
IncreaseDecrease
3.7 - Rational Functions
DomainIn order to find the domain of a rational function, we
must set the denominator equal to zero. These values are where our function does not exist.
Hint: If possible, always factor the denominator first before finding the domain.
3.7 - Rational Functions
Arrow NotationWe will be using the following arrow notation for
asymptotes:
3.7 - Rational Functions
Vertical AsymptotesThe line x = a is a vertical asymptote of the function
y = f (x) if y approaches ∞ as x approaches a from the right or left.
3.7 - Rational Functions
Vertical Asymptotes (VA)
To find the VA
1. Set the denominator = 0 and solve for x.
2. Check using arrow notation.
3.7 - Rational Functions
Horizontal Asymptotes
The line y = b is a horizontal asymptote of the function y = f (x) if y approaches b as x approaches ∞.
3.7 - Rational Functions
Horizontal Asymptotes (HA)To find the HA, we let r be the rational function
1. If n < m, then r has the horizontal asymptote y=0.
2. If n = m, then r has the horizontal asymptote .
3. If n > m, then r has no horizontal asymptotes. We need to check for a slant asymptote (SA).
11 1 0
11 1 0
...( )
...
n nn n
n nn n
a x a x a x ar x
b x b x b x b
n
n
ay
b
3.7 - Rational Functions
Slant Asymptotes To find the SA, we perform long division and get
where R(x)/Q(x) is the remainder and the SA is y = ax + b.
( ) ( )( )
( ) ( )
P x R xr x ax b
Q x Q x
3.7 - Rational Functions
Example
Given the above equation, find the characteristics of rational functions and sketch a graph of the function.
Domain Range
x-intercepts y-intercepts
Asymptotes (VA, HA, SA)
Directional Limits
MaxMin
IncreaseDecrease
2 1( )
1
xf x
x
Domain Range
x-intercepts y-intercepts
Asymptotes (VA, HA, SA) Directional Limits
MaxMin
IncreaseDecrease
3.7 - Rational Functions
Example
Given the above equation, find the characteristics of rational functions and sketch a graph of the function.
Domain Range
x-intercepts y-intercepts
Asymptotes (VA, HA, SA)
Directional Limits
MaxMin
IncreaseDecrease
2( )
1
xf x
x
Domain Range
x-intercepts y-intercepts
Asymptotes (VA, HA, SA) Directional Limits
MaxMin
IncreaseDecrease
3.7 - Rational Functions
Example
Given the above equation, find the characteristics of rational functions and sketch a graph of the function.
Domain Range
x-intercepts y-intercepts
Asymptotes (VA, HA, SA)
Directional Limits
MaxMin
IncreaseDecrease
3
2
27( )
5 6
xf x
x x
Domain Range
x-intercepts y-intercepts
Asymptotes (VA, HA, SA) Directional Limits
MaxMin
IncreaseDecrease
3.7 - Rational Functions
Example
Given the above equation, find the characteristics of rational functions and sketch a graph of the function.
Domain Range
x-intercepts y-intercepts
Asymptotes (VA, HA, SA) Directional Limits
MaxMin
IncreaseDecrease
3 2
2
3 4 12( )
2 8
x x xf x
x