1. Graphing polynomials
To graph a polynomial we need to know three things
1) Type of polynomial
2) Roots
3) y-intercept

2. Types of Polynomials Positive odd Negative odd Positive even
Starts down ends up
Negative odd
Starts up ends down
Positive even
Up on both ends
Negative even
Down on both ends

3. We can find the type from the sign and power of the leading term.
The sign tells us if it is positive or negative and the power (degree) tells us if it is even or odd.
The degree is even
Examples:
The coefficient is positive
Type: positive even

4. State the type for each. Negative even Negative odd Positive odd
Positive even
Negative even

5. Roots
The degree of the polynomial, tells us how many roots we will have.
Example: How many roots will the following functions have?
4 roots
5 roots
4 roots
6 roots

6. Roots Multiplicity of roots Single root Example: Factor (x+2) root: -2
graph goes straight through root
double root
Example: Factor (x-1)2
root: 1 (M2)
graph goes through the root like a quadratic
Indicates a double root
triple root
Example: Factor (x+4)3
root: -4 (M3)
graph goes through the root like a cubic
Indicates a triple root

7. Roots Roots: -5, 3, -2 Roots: 0, 4 (M2), Roots: 0 (M2), 3, -4 (M3)
Real roots are x-intercepts.
To find the roots, we let y = 0 and solve for x.
Example: Find the roots for the following.
Roots: -5, 3, -2
Roots: 0, 4 (M2),
Roots: 0 (M2), 3, -4 (M3)

8. Y-intercept y-intercept: (0, -60) y-intercept: (0, 20)
To find the y-intercept let x = 0
Always write the y-intercept as a point.
Example: Find the y-intercept for each
y-intercept: (0, -60)
y-intercept: (0, 20)
y-intercept: (0,0)

9. First, plot the roots and label
Now we are ready to graph.
State the type, roots, y-intercept and graph.
First, plot the roots and label
positive odd
Type: ______________________
3, -4, 1
roots: ______________________
(0, 24)
y-intercept: __________
Last, sketch the graph (remember the type helps us with the shape)
Next, plot and label the y-intercept

10. State the type, roots, y-intercept and graph.
-3 is a double root, so it the graph looks like a quadratic here
Negative even
Type: ______________________
0, -3 (M2),
roots: ______________________
(0, 0)
y-intercept: __________

11. State the type, roots, y-intercept and graph.
-3 is a triple root, so it the graph looks like a cubic here
positive odd
Type: ______________________
2, -4 (M3), -1
roots: ______________________
y-intercept: __________

12. State the type, roots, y-intercept and graph.

13. State the type, roots, y-intercept and graph.

14. State the type, roots, y-intercept and graph.

15. Graphing Polynomials Day 2

16. State the type, roots, y-intercept and graph.
Expanded Form
Partially Factored
Sometimes we need to factor in order to find the roots.
Factored form: ______________________________________________
Positive odd
Type: ______________________
-2 (M2), 1 (M3)
roots: ______________________
(0, -2)
y-intercept: __________
Note: the type and y-intercept are both easy to find from the expanded form.

17. State the type, roots, y-intercept and graph.
Remember to factor first
Type: ______________________
roots: ______________________
y-intercept: __________

18. State the type, roots, y-intercept and graph.

19. Remember real roots are x-intercepts, but imaginary roots are not.
Imaginary roots always come in pairs.
Sketch a graph for each description.
A negative even function with 3 real roots.
A positive even function with no real roots
A negative odd function with 5 roots.