polynomial functions 33 22 11 definitions degrees graphing

Polynomial Functions

3

2

1Definitions

Degrees

Graphing

2

Definitions

Polynomial Monomial Sum of monomials

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

3

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

4

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

5

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

6

Cubic Term

Terms of a Polynomial

2534)( 23 xxxxP

Quadratic Term

Linear Term

Constant Term

7

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

12

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

17

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

21

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

22

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

23

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