Download 2.2 Polynomial Function of Higher Degree Graphs of Polynomial

2.2 Polynomial Function of Higher Degree
Graphs of Polynomial Functions: are continuous.
• can be drawn without lifting your pencil
• No breaks, no holes, no gaps
• No sharp turns
Determine if the graph represents a polynomial function. Leading Coefficient Test
Degree is Odd
Positive Leading Coefficient
Negative Leading Coefficient
Degree is Even
Positive Leading Coefficient
Negative Leading Coefficient
Determine whether the graph represents an odd­degree or an even­degree polynomial and determine if the leading coefficient of the function is positive or negative. Use the leading coefficient test to determine the right­hand and left­hand behavior of the graph of the polynomial function. Zeros of a Polynomial Function
It can be shown that for a polynomial function f of degree n, the following statements are true: 1.) The function f has at most n real zeros. 2.) The graph of f has at most n­1 relative extrema (relative maxima or minima)
Real Zeros of Polynomial Functions
If f is a polynomial function and a is a real number, the following statements are equivalent. is a zero of the function f.
is a solution of the polynomial equation
is a factor of the polynomial is an x­intercept of the graph of f.
Use the equation to answer the following:
Describe the right hand/left hand behavior of the graph.
Factor to find the real zeros of the function. Determine the number of relative extrema.
Use a graphing utility to find the relative maxima and relative minima.
Factor to find the real zeros.
Repeated Zeros
In factored form: Graph: Sketch the graph by hand.
a.) Apply the leading coefficient test.
b.) Find the zeros of the function by factoring.
c.) Find additional points between the zeros.
d.) Sketch the graph. Sketch the graph by hand.
a.) Apply the leading coefficient test.
b.) Find the zeros of the function by factoring.
c.) Find additional points between the zeros.
d.) Sketch the graph. Intermediate Value Theorem
Use the Intermediate Value Theorem and a graphing utility to find intervals of length 1 in which the polynomial function is guarenteed to have a zero. Find a polynomial function that has the given zeros.