1. Polynomial and Rational Functions
Lesson 2.3

2. Animated Cartoons
Note how mathematics are referenced in the creation of cartoons

3. This lesson studies polynomials
Animated Cartoons
We need a way to take a number of points and make a smooth curve
This lesson studies polynomials

4. 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

5. 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

6. 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

7. 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

8. Properties of Polynomial Functions
Even degree
Leading coefficient positive
Leading coefficient negative
Odd degree

9. Rational Function: Definition
Consider a function which is the quotient of two polynomials
Example:
Both polynomials

10. 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. Example Given Graph on calculator
Set window for -100 < x < 100, -5 < y < 5

12. Example Note the value for a large x
How does this relate to the leading terms?

13. 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

14. When Numerator Has Larger Degree
Try
As x gets large, r(x) also gets large
But it is asymptotic to the line

15. 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

16. 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)?

17. Vertical Asymptotes Finding the roots of the denominator
View the graph to verify

18. Assignment
Lesson 2.3
Page 91
Exercises 3 – 59 EOO