Polynomial Functions and End Behavior On to Section

Polynomial Functions and End Behavior On to Section 2. 3!!!

Definitions: The Vocabulary of Polynomials Cubic Functions – polynomials of degree 3 Quartic Functions – polynomials of degree 4 Recall that a polynomial function of degree n can be written in the form:

Definitions: The Vocabulary of Polynomials • Each monomial is this sum term of the polynomial. is a • A polynomial function written in this way, with terms in descending degree, is written in standard form. • The constants polynomial. • The term. are the coefficients of the is the leading term, and is the constant

Practice Problems Graph the polynomial function, locate its extrema and zeros, and explain how it is related to the monomials from which it is built. Graph with your grapher!!! Increasing on entire domain No extrema!!! One zero At x = 0!!! General shape of the graph is much like that of the leading term (cubing function), but near the origin behaves much like the its other term (identity function) Function is odd, just like its two “building block” monomials

Practice Problems Graph the polynomial function, locate its extrema and zeros, and explain how it is related to the monomials from which it is built. Graph with your grapher!!! Local maximum of approx. 0. 385 at about x = – 0. 577 Local minimum of approx. – 0. 385 at about x = 0. 577 Zeros at x = – 1, 0, 1 (factored version of the function? ) Graph is generally similar to leading term, but near the origin behaves like the other term, –x Function is odd, just like its two “building block” monomials

Summarize these thoughts… Theorem: Local Extrema and Zeros of Polynomial Functions A polynomial function of degree n has at most n – 1 local extrema and at most n zeros. Does this make sense with the graphs we were just looking at? ? ?

Leading Term Test for Polynomial End Behavior For any polynomial function, the end behavior is determined by its degree and leading coefficient: n odd

Leading Term Test for Polynomial End Behavior For any polynomial function, the end behavior is determined by its degree and leading coefficient: n even

Practice Problems Graph the polynomial function in a window showing its extrema and zeros and its end behavior. Describe the end behavior using limits. n odd Graph with your grapher!!! One possible window: [– 5, 5] by [– 25, 25] Function has 2 extrema and 3 zeros

Practice Problems Graph the polynomial function in a window showing its extrema and zeros and its end behavior. Describe the end behavior using limits. n even Graph with your grapher!!! One possible window: [– 5, 5] by [– 50, 50] Function has 3 extrema and 4 zeros

Zeros of Polynomial Functions Recall that finding the real-number zeros of a function f is equivalent to finding the x-intercepts of the graph of y = f (x) or the solutions to the equation f (x) = 0.

A Quick Example… Find the zeros of We solve the equation f (x) = 0 by factoring: Can we support our answer graphically? ? ? x = 0, x – 3 = 0, or x + 2 = 0 Zeros of f : 0, 3, – 2 Zero Factor Property!!!

Definition: Multiplicity of a Zero of a Polynomial Function If f is a polynomial function and a factor of f but is is not, then c is a zero of multiplicity m of f. A zero of multiplicity m > 2 is a repeated zero.

Zeros of Odd and Even Multiplicity If a polynomial function f has a real zero c of odd multiplicity, then the graph of f crosses the x-axis at (c, 0) and the value of f changes sign at x = c. If a polynomial function f has a real zero c of even multiplicity, then the graph of f does not cross the x-axis at (c, 0) and the value of f does not change sign at x = c.

Examples with our new info… Consider the function What are the zeros of the function? 2 and – 1 What are the multiplicity of these zeros? 2 is a zero of multiplicity 3, – 1 has multiplicity 2 What does the graph of the function look like at these zeros? At x = 2, the graph crosses the x-axis, and at x = – 1, the graph “kisses” the x-axis Verify all of this with your calculator!!!

Examples with our new info… State the degree and list the zeros of the given function. State the multiplicity of each zero and whether the graph crosses the x-axis at the corresponding x-intercept. Then sketch the graph of the function by hand. Degree is 5, zeros are x = – 2 (multiplicity 3) and x = 1 (multiplicity 2). The graph crosses the x-axis at x = – 2, but not at x = 1.

Intermediate Value Theorem Essentially, a sign change implies a real zero… If a and b are real numbers with a < b and if f is continuous on the interval [a, b], then f takes on every value between f(a) and f(b). In other words, if y 0 is between f(a) and f(b), then y 0 = f(c) for some number c in [a, b]. In particular, if f(a) and f(b) have opposite signs (i. e. , one is negative and the other is positive), then f(c) = 0 for some number c in [a, b].

Intermediate Value Theorem Let’s “see” this theorem graphically: If f (a) < 0 < f (b), then there is a zero x = c between a and b

Guided Practice Find all of the real zeros of the given function. Because the function has degree 4, there at most four zeros Graph and find them!!! Zeros at x = – 3. 100, 0. 5, 1. 133, and 1. 367 Homework: p. 203 17 -27 odd, 33 -47 odd