Pre-Calculus Section 2.2 Polynomial Functions of Higher Degree

Characteristics of Polynomial Functions Graphs are continuous - there are no breaks in the graph, you could draw them without lifting your pencil

They have only smooth rounded turns, not sharp ones such as the absolute value function.

Transformations: They have the same transformations as a quadratic function: horizontal, vertical shrink, stretch reflection on x-axis, reflection on y-axis

Use a graphing calculator to describe the right-hand and left-hand behavior of the graph of each polynomial function. 1. f(x) = - x3 + 4x

Use a graphing calculator to describe the right-hand and left-hand behavior of the graph of each polynomial function. 2. f(x) = x4 - 5x2 + 4 3. f(x) = x5 - x

Zeros of a Polynomial Function The function f has at most n real zeros The graph of f has at most (n - 1) relative extrema (min or max) A zero is where f(x) = 0 Factor Quadratic formula Graphing calculator - zero

Real Zeros of Polynomial Function If f is a polynomial function and a is a real number, the following statements are equivalent. x = a is a zero of the function f x = a is a solution of the polynomial equations f(x) = 0 (x - a) is a factor of the polynomial f(x) (a, 0) is an x-intercept of the graph of f

Find all real zeros and relative extrema of: 4. f(x) = x3 - x2 - 2x 5 Find all real zeros and relative extrema of: 4. f(x) = x3 - x2 - 2x 5. f(x) = - 2x4 + 2x2

Repeated Zeros For a polynomial function, a factor of (x - a)k, k >1, yields a repeated zero x = a of multiplicity k. if k is odd, the graph crosses the x-axis at x = a If k is even, the graph touches the x-axis, but does not cross, at x = a

Find all real zeros and determine the multiplicity of each zero: 6 Find all real zeros and determine the multiplicity of each zero: 6. f(x) = x2 - 25

7. f(x) = x2 - 6x + 9 8. f(x) = x4 + x3 - 2x2

9. f(x) = x3 - 4x2 + 4x 10. f(x) = ½ x 4(x2 - 25)

Real Zeros of Polynomial Function If a is a zero of the function then (x - a) is a factor of the polynomial f(x) When given a zero and asked to write the polynomial function 1. write each zero as a factor 2. multiply and simplify

Write a polynomial function with the following zeros. 11. - ½ , 3, 3

12. 0, -2, 3, 5

13. 4, 2 + 11 , 2 − 11

Multiplicity refers to single root, double root, triple root, … Write each root as a factor paying attention to the multiplicity Simplify

Write a polynomial function with the given zeros, multiplicities, and degree. 14. zero: -2, multiplicity 2 zero: -1, multiplicity 1 degree: 3

15. zero: -4, multiplicity 2 zero: 3, multiplicity 2 degree: 4

Use your graphing calculator to graph the function Use your graphing calculator to graph the function. Identify any symmetry. Determine the number of x-intercepts of the graph. 16. f(x) = x2 (x + 6)

Use your graphing calculator to graph the function Use your graphing calculator to graph the function. Identify any symmetry. Determine the number of x-intercepts of the graph. 17. f(x) = - ½ (a - 4)2 (a + 4)2

Use your graphing calculator to graph the function Use your graphing calculator to graph the function. Identify any symmetry. Determine the number of x-intercepts of the graph. 18. f(x) = x3 - 4x

Pre Calc Sec 2-2 Use your graphing calculator to graph the function. Identify any symmetry. Determine the number of x-intercepts of the graph. 19. f(x) = 1/5 (x + 1)2(x - 3)(2x - 9)