Polynomial and Rational
Functions
Lesson 2.3
Animated Cartoons
Note how
mathematics
are referenced
in the creation
of cartoons
Animated Cartoons
We need a way
to take a number
of points
and make
a smooth
curve
This lesson
studies
polynomials
Polynomials
General polynomial formula
P( x)  an x  an 1 x
n
n 1
 ...  a1 x  a0
• a0, a1, … ,an are constant coefficients
• n is the degree of the polynomial
• Standard form is for descending powers of x
• anxn is said to be the “leading term”
Note that each term is a power function
Family of Polynomials
Constant polynomial functions
• f(x) = a
Linear polynomial functions
• f(x) = m x + b
Quadratic polynomial functions
• f(x) = a x2 + b x + c
Family of Polynomials
Cubic polynomial functions
• f(x) = a x3 + b x2 + c x + d
• Degree 3 polynomial
Quartic polynomial functions
• f(x) = a x4 + b x3 + c x2+ d x + e
• Degree 4 polynomial
Properties of Polynomial Functions
If the degree is n then it will have at most
n – 1 turning points
•
•
•
End behavior
• Even degree
• Odd degree
or
or
Properties of Polynomial Functions
Even degree
• Leading coefficient positive
• Leading coefficient negative
Odd degree
• Leading coefficient positive
• Leading coefficient negative
Rational Function: Definition
Consider a function which is the quotient of
two polynomials
P( x)
R( x) 
Q( x)
Example:
2500  2 x
r ( x) 
x
Both polynomials
Long Run Behavior
n 1
an x  an 1 x  ...  a1 x  a0
Given R( x) 
m
m 1
bm x  bm1 x  ...  b1 x  b0
n
The long run (end) behavior is determined
by the quotient of the leading terms
• Leading term dominates for
large values of x for polynomial
• Leading terms dominate for
the quotient for extreme x
an x n
bm x m
Example
Given
3x  8 x
r ( x)  2
5x  2 x  1
2
Graph on calculator
• Set window for -100 < x < 100, -5 < y < 5
Example
Note the value for a large x
2
3x
2
5x
How does this relate to the leading terms?
Try This One
5x
Consider r ( x)  2
2x  6
Which terms dominate as x gets large
5x
What happens to 2
2x
as x gets large?
Note:
• Degree of denominator > degree numerator
• Previous example they were equal
When Numerator Has Larger Degree
2
2
x

6
Try r ( x) 
5x
As x gets large, r(x) also gets large
But it is asymptotic to the line
2
y x
5
Summarize
n
an x
Given a rational function with
m
bm x
leading terms
When m = n
a
• Horizontal asymptote at
b
When m > n
• Horizontal asymptote at 0
When n – m = 1
• Diagonal asymptote
a
y x
b
Vertical Asymptotes
A vertical asymptote happens when the
function R(x) is not defined P( x)
• This happens when the
denominator is zero
Q( x)
Thus we look for the roots of the
denominator
x2  9
r ( x) 
x2  5x  6
Where does this happen for r(x)?
 R( x)
Vertical Asymptotes
Finding the roots of
x2  9
r ( x)  2
the denominator
x  5x  6
x2  5x  6  0
( x  6)( x  1)  0
x  6 or x  1
View the graph
to verify
Assignment
Lesson 2.3
Page 91
Exercises 3 – 59 EOO