1. Polynomial Functions
Lesson 9.2

2. Power Function Definition
Recall from the chapter on shifting and stretching, what effect the k will have?
Vertical stretch or compression
for k < 1

3. Special Power Functions
Parabola y = x2
Cubic function y = x3
Hyperbola y = x-1

4. Special Power Functions
y = x-2

5. Special Power Functions
Most power functions are similar to one of these six
xp with even powers of p are similar to x2
xp with negative odd powers of p are similar to x -1
xp with negative even powers of p are similar to x -2
Which of the functions have symmetry?
What kind of symmetry?

6. Polynomials Definition: The sum of one or more power function
Each power is a non negative integer

7. Polynomials General 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”

8. Polynomial Properties
Consider what happens when x gets very large negative or positive
Called “end behavior”
Also “long-run” behavior
Basically the leading term anxn takes over
Compare f(x) = x3 with g(x) = x3 + x2
Look at tables
Use standard zoom, then zoom out

9. Polynomial Properties
Compare tables for low, high values

10. Polynomial Properties
Compare graphs ( -10 < x < 10)
The leading term x3 takes over
For 0 < x < 500 the graphs are essentially the same

11. Zeros of Polynomials We seek values of x for which p(x) = 0
We have the quadratic formula
There is a cubic formula, a quartic formula

12. Zeros of Polynomials We will use other methods Consider
What is the end behavior?
What is q(0) = ?
How does this tell us that we can expect at least two roots?

13. Methods for Finding Zeros
Graph and ask for x-axis intercepts
Use solve(y1(x)=0,x)
Use zeros(y1(x),x)
When complex roots exist, use cSolve() or cZeros()

14. Practice Given y = (x + 4)(2x – 3)(5 – x)
What is the degree?
How many terms does it have?
What is the long run behavior?
f(x) = x3 +x + 1 is invertible (has an inverse)
How can you tell?
Find f(0.5) and f -1(0.5)

15. Assignment
Lesson 9.2
Page 400
Exercises 1 – 29 odd