Polynomial Polynomial Functions and End Functions and End

BehaviorBehavior

Polynomial Polynomial Functions and End Functions and End

BehaviorBehaviorOn to Section 2.3!!!On to Section 2.3!!!

Definitions: The Vocabulary Definitions: The Vocabulary of Polynomialsof Polynomials

Cubic Functions – polynomials of degree 3

Quartic Functions – polynomials of degree 4

Recall that a polynomial function of degree n can be writtenin the form:

1 21 2 1 0 ,

n nn nf x a x a x a x a x a

0na

Definitions: The Vocabulary Definitions: The Vocabulary of Polynomialsof Polynomials

• Each monomial is this sum is a

1 21 2 1 0

n nn nf x a x a x a x a x a

term of the polynomial. 1

1 0, , ,n nn na x a x a

• A polynomial function written in this way, with terms in descending degree, is written in standard form.

• The constants are the coefficients of the

polynomial.

1 0, , ,n na a a

• The term is the leading term, and is the constantn

na x 0aterm.

Practice ProblemsPractice ProblemsGraph the polynomial function, locate its extrema andzeros, and explain how it is related to the monomialsfrom which it is built. 3f x x x

Increasing on entire domain Increasing on entire domain No extrema!!! No extrema!!!

Graph with your grapher!!!

One zero One zero At x = 0!!! At x = 0!!!

General shape of the graph is much like that of the leadingGeneral shape of the graph is much like that of the leadingterm (cubing function), but near the origin behaves muchterm (cubing function), but near the origin behaves muchlike the its other term (identity function)like the its other term (identity function)

Function is odd, just like its two “building block” monomialsFunction is odd, just like its two “building block” monomials

Practice ProblemsPractice ProblemsGraph the polynomial function, locate its extrema andzeros, and explain how it is related to the monomialsfrom which it is built. 3g x x x

Local maximum of approx. 0.385 at about x = –0.577Local maximum of approx. 0.385 at about x = –0.577

Graph with your grapher!!!

Local minimum of approx. –0.385 at about x = 0.577Local minimum of approx. –0.385 at about x = 0.577

Zeros at x = –1, 0, 1 (factored version of the function?)Zeros at x = –1, 0, 1 (factored version of the function?)

Graph is generally similar to leading term, but near theGraph is generally similar to leading term, but near theorigin behaves like the other term, –xorigin behaves like the other term, –x

Function is odd, just like its two “building block” monomialsFunction is odd, just like its two “building block” monomials

Summarize these thoughts…Summarize these thoughts…

Theorem: Local Extrema and Zeros ofPolynomial Functions

A polynomial function of degree n has at mostn – 1 local extrema and at most n zeros.

Does this make sense with theDoes this make sense with thegraphs graphs we were just looking at???we were just looking at???

Leading Term Test for Polynomial End BehaviorLeading Term Test for Polynomial End Behavior

For any polynomial function, the end behavior is determined byits degree and leading coefficient:

1 0n

nf x a x a x a

0na n odd

limx

f x

limx

f x

0na n odd

limx

f x

limx

f x

Leading Term Test for Polynomial End BehaviorLeading Term Test for Polynomial End Behavior

For any polynomial function, the end behavior is determined byits degree and leading coefficient:

1 0n

nf x a x a x a

0na n even

lim limx x

f x f x

0na n even

lim limx x

f x f x

Practice ProblemsPractice ProblemsGraph the polynomial function in a window showing itsextrema and zeros and its end behavior. Describe theend behavior using limits.

3 22 11 12f x x x x Graph with your grapher!!!

0na n odd

One possible window: [–5, 5] by [–25, 25]One possible window: [–5, 5] by [–25, 25]

Function has 2 extrema and 3 zerosFunction has 2 extrema and 3 zeros

limx

f x

limx

f x

Practice ProblemsPractice ProblemsGraph the polynomial function in a window showing itsextrema and zeros and its end behavior. Describe theend behavior using limits.

4 3 22 2 22 18 35g x x x x x Graph with your grapher!!!

0na n even

One possible window: [–5, 5] by [–50, 50]One possible window: [–5, 5] by [–50, 50]

Function has 3 extrema and 4 zerosFunction has 3 extrema and 4 zeros

limxg x

lim

xg x

Zeros of Polynomial Functions

Recall that finding the real-number Recall that finding the real-number zeroszerosof a function of a function f f is equivalent to finding the is equivalent to finding thex-interceptsx-intercepts of the graph of of the graph of yy = = f f ((xx) or the) or thesolutions to the equationsolutions to the equation f f ((xx) = 0.) = 0.

A Quick Example…

Find the zeros of 3 2 6f x x x x We solve the equation f (x) = 0 by factoring:

3 2 6 0x x x 2 6 0x x x

3 2 0x x x x = 0, x – 3 = 0, or x + 2 = 0 Zero FactorZero Factor

Property!!!Property!!!Zeros of Zeros of ff : 0, 3, –2 : 0, 3, –2

Can we supportCan we supportour answerour answer

graphically???graphically???

Definition: Multiplicity of a Zero of a Polynomial Function

If f is a polynomial function and is mx ca factor of f but is not, then c is a 1m

x c

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 oddmultiplicity, then the graph of f crosses the x-axisat (c, 0) and the value of f changes sign at x = c.

If a polynomial function f has a real zero c of evenmultiplicity, then the graph of f does not cross thex-axis at (c, 0) and the value of f does not changesign at x = c.

Examples with our new info…

Consider the function 3 22 1f x x x

What are the zeros of the function? 2 and –12 and –1

What are the multiplicity of these zeros?

2 is a zero of multiplicity 3, –1 has multiplicity 22 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,At x = 2, the graph crosses the x-axis,and at x = –1, the graph “kisses” the x-axisand at x = –1, the graph “kisses” the x-axis

Verify all of this with your calculator!!!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 graphcrosses the x-axis at the corresponding x-intercept. Thensketch the graph of the function by hand.

3 22 1f x x x

Degree is 5, zeros are x = –2 (multiplicity 3) andDegree is 5, zeros are x = –2 (multiplicity 3) andx = 1 (multiplicity 2). The graph crosses the x-axisx = 1 (multiplicity 2). The graph crosses the x-axisat x = –2, but not at x = 1.at x = –2, but not at x = 1.

Intermediate Value TheoremEssentially, a sign change implies a real zero…

If a and b are real numbers with a < b and if f is continuouson the interval [a, b], then f takes on every value betweenf(a) and f(b). In other words, if y is between f(a) and f(b),then y = f(c) for some number c in [a, b].

00

In particular, if f(a) and f(b) have opposite signs (i.e., one isnegative and the other is positive), then f(c) = 0 for somenumber c in [a, b].

Intermediate Value TheoremLet’s “see” this theorem graphically:

0 0y

f a

f b

bc

a

If If f f ((aa) < 0 < ) < 0 < f f ((bb), then there is), then there isa zero a zero xx = = cc between between aa and and bb

Guided PracticeFind all of the real zeros of the given function.

4 3 20.1 6.5 7.9 2.4f x x x x x Because the function has degree 4, there are at most fourzeros Graph and find them!!!

Zeros at x = –3.100, 0.5, 1.133, and 1.367Zeros at x = –3.100, 0.5, 1.133, and 1.367

Homework: p. 203 17-27 odd, 33-47 oddHomework: p. 203 17-27 odd, 33-47 odd