Polynomial Functions 1 Definitions 2 Degrees 3 Graphing

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

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

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

Terms of a Polynomial Cubic Term Linear Term Quadratic 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

Up and Up

Down and Down

Down and Up

Up and Down

Determining End Behavior Types Leading Term n is even n is odd a is positive a is negative Up and Up Down and Up Down and Down Up and Down

Leading Coefficient: + END BEHAVIOR f(x) = x2 Degree: Even Leading Coefficient: + End Behavior: Up and Up

Leading Coefficient: – END BEHAVIOR f(x) = -x2 Degree: Even Leading Coefficient: – End Behavior: Down and Down

Leading Coefficient: + END BEHAVIOR f(x) = x3 Degree: Odd Leading Coefficient: + End Behavior: Down and Up

Leading Coefficient: – END BEHAVIOR f(x) = -x3 Degree: Odd Leading Coefficient: – End Behavior: 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

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

Quadratic Function f(x) = x2 + 3x + 2 Degree = 2 2-1=1 Turning Points (1) Quadratic Function f(x) = x2 + 3x + 2 Degree = 2 2-1=1

f(x) = x3 + 4x2 + 2 Cubic Functions Degree = 3 3-1=2 Turning Points (0 or 2) f(x) = x3 f(x) = x3 + 4x2 + 2 Cubic Functions Degree = 3 3-1=2

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

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

Finding the Degree From a Table x y -3 -1 -2 -7 5 1 11 2 9 3 1st 2nd 3rd -6 10 -6 4 4 -6 8 -2 -6 6 -8 -6 -2 -14 -16 3rd Degree Polynomial