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### How do I analyze a polynomial function?

Daily Questions:

1) What is polynomial function?

2)How do I determine end behavior?

EVALUATING POLYNOMIAL FUNCTIONS

an

n

n

n– 1

a0

an 0

leading coefficient

an

constant term

degree

a0

n

descending order of exponents from left to right.

A polynomial function is a function of the form

f(x) = an xn+ an– 1xn– 1+· · ·+ a1x + a0

Where an 0 and the exponents are all whole numbers.

For this polynomial function, an is the leading coefficient,

a0 is the constant term, and nis the degree.

A polynomial function is in standard form if its terms are

written in descending order of exponents from left to right.

Examples of Polynomial Functions

What do you notice about all these equations?

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

y

y

f(x) = x3 – 5x2+4x + 4

x

x

x

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

Graphs of Polynomial Functionscontinuous

not continuous

continuous

smooth

not smooth

polynomial

not polynomial

not polynomial

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

Identifying Polynomial Functions

f(x) = x3+ 3x

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

The function is not a polynomial function because the term 3xdoes not have a variable base and an exponentthat is a whole number.

Identifying Polynomial Functions

f(x) = 6x2+ 2x–1+ x

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

The function is not a polynomial function because the term2x–1has an exponent that is not a whole number.

Identifying Polynomial Functions

1

f(x) = x2– 3x4– 7

2

f(x) = –0.5x+ x2– 2

Polynomial function?

f(x) = x3+ 3x

f(x) = 6x2+ 2x–1+ x

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

CONSTANT, MONOMIAL

LINEAR, BINOMIAL

QUADRATIC, TRINOMIAL

CUBIC, POLYNOMIAL

END BEHAVIOR FOR POLYNOMIAL FUNCTIONS

SUMMARY

annx – x +

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

> 0 odd f(x) – f(x) +

< 0 even f(x) – f(x) –

< 0 odd f(x) + f(x) –

GRAPHING POLYNOMIAL FUNCTIONS

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

Ex.f(x) = x4 + 2x2 – 3x

f(x) = -x5 +3x4 – x

f(x) = 2x3 – 3x2 + 5

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