polynomial and rational functions lesson 2.3. animated cartoons note how mathematics are referenced...
TRANSCRIPT
Animated Cartoons
We need a wayto take a numberof pointsand makea smoothcurve
This lesson studies
polynomials
This lesson studies
polynomials
Polynomials
General polynomial formula
• 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
11 1 0( ) ...n n
n nP x a x a x a x a
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
Example:
( )( )
( )
P xR x
Q x Both polynomials
2500 2( )
xr x
x
Long Run Behavior
Given
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
11 1 0
11 1 0
...( )
...
n nn nm m
m m
a x a x a x aR x
b x b x b x b
nnm
m
a x
b x
Example
Given
Graph on calculator• Set window for -100 < x < 100, -5 < y < 5
2
2
3 8( )
5 2 1
x xr x
x x
Try This One
Consider
Which terms dominate as x gets large
What happens to as x gets large?
Note:• Degree of denominator > degree numerator• Previous example they were equal
2
5( )
2 6
xr x
x
2
5
2
x
x
When Numerator Has Larger Degree
Try
As x gets large, r(x) also gets large
But it is asymptotic to the line
22 6( )
5
xr x
x
2
5y x
Summarize
Given a rational function with leading terms
When m = n• Horizontal asymptote at
When m > n• Horizontal asymptote at 0
When n – m = 1• Diagonal asymptote
nnm
m
a x
b x
a
b
ay x
b
Vertical Asymptotes
A vertical asymptote happens when the function R(x) is not defined• This happens when the
denominator is zero
Thus we look for the roots of the denominator
Where does this happen for r(x)?
( )( )
( )
P xR x
Q x
2
2
9( )
5 6
xr x
x x
Vertical Asymptotes
Finding the roots ofthe denominator
View the graphto verify
2 5 6 0
( 6)( 1) 0
6 or 1
x x
x x
x x
2
2
9( )
5 6
xr x
x x