Polynomial Functions Definitions Degrees Graphing

Definitions  Polynomial  Monomial  Sum of monomials  Terms  Monomials that make up the polynomial  Like Terms are terms that can be combined 2

Degree of Polynomials  Simplify the polynomial  Write the terms in descending order  The largest power is the degree of the polynomial 3

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 3x 5 – 3x + 2 ? Degree of Polynomials

5 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 Quadratic Term Linear Term Constant Term

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 7

Up and Up

Down and Down

Down and Up

Up and Down

Determining End Behavior Types n is evenn is odd a is positive a is negative 12

END BEHAVIOR Degree: Even Leading Coefficient: + f(x) = x 2 End Behavior: Up and Up

END BEHAVIOR Degree: Even Leading Coefficient: – End Behavior: f(x) = -x 2 Down and Down

END BEHAVIOR Degree: Odd Leading Coefficient: + End Behavior: f(x) = x 3 Down and Up

END BEHAVIOR Degree: Odd Leading Coefficient: – End Behavior: f(x) = -x 3 Up and Down

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 17

Turning Points (0) f(x) = x + 2 Linear Function Degree = 1 1-1=0

Turning Points (1) f(x) = x 2 + 3x + 2 Quadratic Function Degree = 2 2-1=1

Turning Points (0 or 2) f(x) = x 3 + 4x Cubic Functions Degree = 33-1=2 f(x) = x 3

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 21

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 22

Finding the Degree From a Table xy