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Polynomial Functions

Polynomial Functions. 1. Definitions. 2. Degrees. 3. Graphing. Definitions. Polynomial Monomial Sum of monomials Terms 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

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Polynomial Functions

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  1. Polynomial Functions 1 Definitions 2 Degrees 3 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

  4. Degree of Polynomials A LEADING COEFFICIENTis the coefficient of the term with the highest degree. (must be in order) What is the degree and leading coefficient of 3x5 – 3x + 2 ?

  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. Terms of a Polynomial Cubic Term Linear Term Quadratic 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

  8. Up and Up

  9. Down and Down

  10. Down and Up

  11. Up and Down

  12. Determining End Behavior Types Leading Term Down and Up Up and Up Up and Down Down and Down

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

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

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

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

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

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

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

  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 1st 2nd 3rd -6 10 -6 4 4 -6 8 -2 -6 6 -8 -6 -2 -14 -16 3rd Degree Polynomial

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