How do I analyze a polynomial function?

Daily Questions:1) What is polynomial function?2)How do I determine end behavior?

EVALUATING POLYNOMIAL FUNCTIONS

A polynomial function is a function of the form

f (x) = an x n + an – 1 x

n – 1 +· · ·+ a 1 x + a 0

Where an 0 and the exponents are all whole numbers.

A polynomial function is in standard form if its terms are written in descending order of exponents from left to right.

For this polynomial function, an is the leading coefficient,

a 0 is the constant term, and n is the degree.

an 0

an

anleading coefficient

a 0

a0constant term n

n

degree

descending order of exponents from left to right.

n n – 1

927323 2345 xxxxx

14612 23 xxx

32827 234 xxxx

Examples of Polynomial Functions

What do you notice about all these equations?

All exponents must be whole numbers and coefficients are all real numbers…

Graphs of polynomial functions are continuous. That is, they have no breaks, holes, or gaps.

Polynomial functions are also smooth with rounded turns. Graphs with points or cusps are not graphs of polynomial functions.

x

y

x

y

continuous not continuous continuoussmooth not smooth

polynomial not polynomial not polynomial

x

y f (x) = x3 – 5x2 + 4x + 4

Decide whether the function is a polynomial function. If it is,write the function in standard form and state its degree, typeand leading coefficient.

Identifying Polynomial Functions

The function is not a polynomial function because the

term 3

x does not have a variable base and an exponentthat is a whole number.

SOLUTION

f (x) = x 3 + 3

x

Identifying Polynomial Functions

Decide whether the function is a polynomial function. If it is,write the function in standard form and state its degree, typeand leading coefficient.

SOLUTION

f (x) = 6x 2 + 2 x

–1 + x

The function is not a polynomial function because the term2x

–1 has an exponent that is not a whole number.

f (x) = x 2 – 3 x

4 – 712

Identifying Polynomial Functions

f (x) = x 3 + 3x

f (x) = 6x2 + 2 x– 1 + x

Polynomial function?

f (x) = – 0.5x + x2 – 2

Polynomial Functions can be classified by degree

3)( xf

83)( xxf

183)( 2 xxxf

CONSTANT, MONOMIAL

LINEAR, BINOMIAL

QUADRATIC, TRINOMIAL

Polynomial Functions can be classified by degree and by the number of terms

3 2( ) 2 4 3 12f x x x x CUBIC, POLYNOMIAL

Given f(x) find f(-3).

3 2( ) 2 4 3 12f x x x x

-69

End Behavior Task

Let’s Summarize

GRAPHING POLYNOMIAL FUNCTIONS

END BEHAVIOR FOR POLYNOMIAL FUNCTIONSCONCEPT

SUMMARY

> 0 even f (x) + f (x) +

> 0 odd f (x) – f (x) +

< 0 even f (x) – f (x) –

< 0 odd f (x) + f (x) –

an n x – x +

Ex. as x _____ ( ) _____

as x _____ ( ) _____

f x

f x

Determine the left and right behavior of the graph of each polynomial function.

f(x) = -x5 +3x4 – x

f(x) = x4 + 2x2 – 3x

f(x) = 2x3 – 3x2 + 5 as x _____ ( ) _____

as x _____ ( ) _____

f x

f x

as x _____ ( ) _____

as x _____ ( ) _____

f x

f x

Tell me what you know about the equation…

Odd exponent

Positive leading coefficient

Tell me what you know about the equation…

Even exponent

Positive leading coefficient

Tell me what you know about the equation…

Odd exponent

Positive leading coefficient

Tell me what you know about the equation…

Even exponent

Negative leading coefficient