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 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. Examples: The coefficient is positive The degree is even 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 rootExample: Factor (x+2) root: -2 graph goes straight through root double rootExample: Factor (x-1) 2 root: 1 (M2) graph goes through the root like a quadratic Indicates a double root triple rootExample: Factor (x+4) 3 root: -4 (M3) graph goes through the root like a cubic Indicates a triple root

7. Roots Multiplicity of roots Roots can have higher multiplicity as well. The graph goes through roots with even multiplicity (M4, M6, M8, etc.) like a quadratic only as the multiplicity increases the graph looks flatter at the root. Example: If the root is 3 (M4)If the root is 3 (M12)

8. Roots Multiplicity of roots The graph goes through roots with odd multiplicity (M5, M6, M9, etc.) like a cubic only as the multiplicity increases the graph looks flatter at the root. Example: If the root is 3 (M5)If the root is 3 (M11)

9. Roots 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) Roots: 1, Roots: 5 (M4), -1+i, -1- i

10. Y-intercept 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)

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

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

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

14. State the type, roots, y-intercept and graph. Type: ______________________ roots: ______________________ y-intercept: __________

15. State the type, roots, y-intercept and graph. Type: ______________________ roots: ______________________ y-intercept: __________

16. State the type, roots, y-intercept and graph. Type: ______________________ roots: ______________________ y-intercept: __________

17. Graphing Polynomials Day 2

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

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

20. State the type, roots, y-intercept and graph. Type: ______________________ roots: ______________________ y-intercept: __________

21. State the type, roots, y-intercept and graph. Type: ______________________ roots: ______________________ y-intercept: __________

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