f(x) = x 2 + 4x – 11 g(x) = x 3 + 4 h(x) = 7 – 3x 5 + 6x

1. Warm-Up Feb. 10thState the end behavior (no calculator) and the domain and range for each polynomial. • f(x) = x2 + 4x – 11 • g(x) = x3 + 4 • h(x) = 7 – 3x5 + 6x

2. Graphing & Writing Polynomial Functions

3. Vocab & Background • The maximum number of Extrema (max and mins) of a graph can be found from the degree. • If the degree of the polynomial is n, then the number of extrema is at most n – 1. (there may be less) • The degree will also tell you the maximum number of possible x-intercepts. (there may be less)

4. Zeros (aslo known as roots or x-intercepts) are where the graph crosses the x-axis. • In some cases there may be a double zero, meaning the graph bounces at that value. • There may also be imaginary roots which don’t show up on the graph but still count toward the maximum number of zeros. • Y-intercept is where the graph crosses the y-axis (where x = 0)

5. P(x) = 5x4 + 3x3 – 4x5 + x + 6x2 – 2 Degree: Leading Coefficient: Max Number of Extrema: Max Number of Zeros/X-intercepts: Y-intercept:

6. Increasing/Decreasing • Reading the graph from left to right (very important!)… • The graph is increasing where the y-values are increasing • The graph is decreasing where the y-values are decreasing • It changes at its extrema. • We use interval notation to write where the graph is increasing or decreasing.

7. Parts of a Polynomial Graph • Extrema~ Max/Min • absolute vs. local (relative) • Y-intercept: • X-intercepts/Zeros: • Increasing: • Decreasing:

8. Parts of a Polynomial Graph • Extrema~ Max/Min • absolute vs. local (relative) • Y-intercept: • X-intercepts/Zeros: • Increasing: • Decreasing:

9. Examples • For the function below find the following: y-intercept, end behavior, domain, zeros, range, extrema and intervals of increasing/decreasing. *Hint: there may be multiplicity of zeros. Also if you must round, round to the nearest hundredth.

12. You Try:

13. Writing Polynomials • Zeros at -2, 3 and 5 • Zeros at 4, -6, and 0. • Zeros at 1 and ¾.

14. Double Roots • A zero with a multiplicity of 1 will cross the x-axis. • A zero with a multiplicity of 2 will “bounce/touch” the x-axis.

15. From a Graph

16. Special Roots • An irrational root of a polynomial contains a radical. • For example: • They always travel in pairs with their conjugate. • An imaginary root contains an imaginary number, i. • For example: • They also travel in pairs with their conjugate.

17. Examples with Special Roots • Write a polynomial for the given roots