Polynomial Functions

3

2

1Definitions

Degrees

Graphing

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Definitions

Polynomial Monomial Sum of monomials

Terms Monomials that make up the polynomial Like Terms are terms that can be combined

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Degree of Polynomials

Simplify the polynomial Write the terms in descending order The largest power is the degree of the

polynomial

)1)(52( 2 aa

5252 23 aaa

Polynomial Degree 3rd

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A LEADING COEFFICIENT is the coefficient of the term with the highest degree.

(must be in order)

What is the degree and leading coefficient of 3x5 – 3x + 2 ?

Degree of Polynomials

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Degree of Polynomials

Polynomial functions with a degree of 1 are called LINEAR POLYNOMIAL FUNCTIONS

Polynomial functions with a degree of 2 are called QUADRATIC POLYNOMIAL FUNCTIONS

Polynomial functions with a degree of 3 are called CUBIC POLYNOMIAL FUNCTIONS

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Cubic Term

Terms of a Polynomial

2534)( 23 xxxxP

Quadratic Term

Linear Term

Constant Term

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End Behavior Types

Up and Up Down and Down Down and Up Up and Down These are “read” left to right Determined by the leading coefficient &

its degree

Up and Up

xxxy 53 34

Down and Down

Down and Up

Up and Down

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Determining End Behavior Types

n is even n is odda is positivea is negative

nax Leading Term

Up and Up

Down and Down

Down and Up

Up and Down

END BEHAVIOR

Degree: Even

Leading Coefficient: +

f(x) = x2

End Behavior:

Up and Up

END BEHAVIOR

Degree: Even

Leading Coefficient: –

End Behavior:

f(x) = -x2

Down and Down

END BEHAVIOR

Degree: Odd

Leading Coefficient: +

End Behavior:

f(x) = x3

Down and Up

END BEHAVIOR

Degree: Odd

Leading Coefficient: –

End Behavior:

f(x) = -x3

Up and Down

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Turning Points

Number of times the graph “changes direction”

Degree of polynomial-1 This is the most number of turning points

possible Can have fewer

Turning Points (0)

f(x) = x + 2

LinearFunction

Degree = 1

1-1=0

Turning Points (1)

f(x) = x2 + 3x + 2QuadraticFunction

Degree = 2

2-1=1

Turning Points (0 or 2)

f(x) = x3 + 4x2 + 2

CubicFunctions

Degree = 3 3-1=2

f(x) = x3

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Graphing From a Function

Create a table of values More is better Use 0 and at least 2 points to either side

Plot the points Sketch the graph No sharp “points” on the curves

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Finding the Degree From a Table

List the points in order Smallest to largest (based on x-values) Find the difference between y-values Repeat until all differences are the same Count the number of iterations (times you

did this) Degree will be the same as the number of

iterations

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Finding the Degree From a Table

x y

-3 -1

-2 -7

-1 -3

0 5

1 11

2 9

3 -7

-6

4

8

6

-2

-16

10

4

-2

-8

-14

-6

-6

-6

-6

1ST

2ND3RD

3rd Degree Polynomial