An Intro to Polynomials

1. An Intro to Polynomials Objectives: Identify, evaluate, add, and subtract polynomials Classify polynomials, and describe the shapes of their graphs

2. Classification of a Polynomial n = 0 constant 3 linear n = 1 5x + 4 quadratic n = 2 2x2 + 3x - 2 cubic n = 3 5x3 + 3x2 – x + 9 quartic 3x4 – 2x3 + 8x2 – 6x + 5 n = 4 n = 5 -2x5 + 3x4 – x3 + 3x2 – 2x + 6 quintic

3. Example 1 Classify each polynomial by degree and by number of terms. a) 5x + 2x3 – 2x2 cubic trinomial b) x5 – 4x3 – x5 + 3x2 + 4x3 quadratic monomial

4. Example 2 Add (-3x4y3 + 6x3y3 – 6x2 + 5xy5 + 1) + (5x5 – 3x3y3 – 5xy5) -3x4y3 + 6x3y3 – 6x2 + 5xy5 + 1 5x5 - 3x3y3 - 5xy5 5x5 – 3x4y3 + 3x3y3 – 6x2 + 1

5. Graphs of Polynomial Functions Graph each function below. 2 1 y = x2 + x - 2 3 2 y = 3x3 – 12x + 4 3 2 y = -2x3 + 4x2 + x - 2 4 3 y = x4 + 5x3 + 5x2 – x - 6 4 3 y = x4 + 2x3 – 5x2 – 6x Make a conjecture about the degree of a function and the # of “U-turns” in the graph.

6. Graphs of Polynomial Functions Graph each function below. 3 0 y = x3 3 0 y = x3 – 3x2 + 3x - 1 4 1 y = x4 Now make another conjecture about the degree of a function and the # of “U-turns” in the graph. The number of “U-turns” in a graph is less than or equal to one less than the degree of a polynomial.

7. Example 6 Graph each function. Describe its general shape. a) P(x) = 2x3 - 1 a curve that always rises to the right b) Q(x) = -3x4 + 2 a U-shape that has one turn

8. Polynomial Functions and Their Graphs Objectives: Identify and describe the important features of the graph of a polynomial function Use a polynomial function to model real-world data

9. f(a) is a local maximum if there is an interval around a such that f(a) > f(x) for all values of x in the interval, where x = a. f(a) is a local minimum if there is an interval around a such that f(a) < f(x) for all values of x in the interval, where x = a. Graphs of Polynomial Functions

10. Increasing and Decreasing Functions Let x1 and x2 be numbers in the domain of a function, f. The function f is increasing over an open interval if for every x1 < x2 in the interval, f(x1) < f(x2). The function f is decreasing over an open interval if for every x1 < x2 in the interval, f(x1) > f(x2).

11. Example 1 Graph P(x) = -2x3 – x2 + 5x + 6. a) Approximate any local maxima or minima to the nearest tenth. minimum: (-1.1,2.0) maximum: (0.8,8.3) b) Find the intervals over which the function is increasing and decreasing. increasing: x > -1.1 and x < 0.8 decreasing: x < -1.1 and x > 0.8

12. Exploring End Behavior of f(x) = axn Graph each function separately. For each function, answer parts a-c. 1) y = x2 2) y = x4 3) y = 2x2 4) y = 2x4 5) y = x3 6) y = x5 7) y = 2x3 8) y = 2x5 9) y = -x2 10) y = -x4 11) y = -2x2 12) y = -2x4 13) y = -x3 14) y = -x5 15) y = -2x3 16) y = -2x5 a. Is the degree of the function even or odd? b. Is the leading coefficient positive or negative? c. Does the graph rise or fall on the left? on the right?

13. Exploring End Behavior of f(x) = axn Rise or Fall rise rise fall fall fall rise rise fall

14. Example 2 Describe the end behavior of each function. a) V(x) = x3 – 2x2 – 5x + 3 falls on the left and rises on the right b) R(x) = 1 + x – x2 – x3 + 2x4 rises on the left and the right