How do I analyze a polynomial function?

1. How do I analyze a polynomial function? Daily Questions: 1) What is polynomial function? 2)How do I determine end behavior?

2. EVALUATING POLYNOMIAL FUNCTIONS an n n n– 1 a0 an 0 leading coefficient an constant term degree a0 n descending order of exponents from left to right. A polynomial function is a function of the form f(x) = an xn+ an– 1xn– 1+· · ·+ a1x + a0 Where an 0 and the exponents are all whole numbers. For this polynomial function, an is the leading coefficient, a0 is the constant term, and nis the degree. A polynomial function is in standard form if its terms are written in descending order of exponents from left to right.

3. Examples of Polynomial Functions What do you notice about all these equations? All exponents must be whole numbers and coefficients are all real numbers…

4. y y y f(x) = x3 – 5x2+4x + 4 x x x Graphs of polynomial functions are continuous. That is, they have no breaks, holes, or gaps. Graphs of Polynomial Functions continuous not continuous continuous smooth not smooth polynomial not polynomial not polynomial Polynomial functions are also smooth with rounded turns. Graphs with points or cusps are not graphs of polynomial functions.

5. Identifying Polynomial Functions f(x) = x3+ 3x Decide whether the function is a polynomial function. If it is, write the function in standard form and state its degree, typeand leading coefficient. SOLUTION The function is not a polynomial function because the term 3xdoes not have a variable base and an exponentthat is a whole number.

6. Identifying Polynomial Functions f(x) = 6x2+ 2x–1+ x Decide whether the function is a polynomial function. If it is, write the function in standard form and state its degree, typeand leading coefficient. SOLUTION The function is not a polynomial function because the term2x–1has an exponent that is not a whole number.

7. Identifying Polynomial Functions 1 f(x) = x2– 3x4– 7 2 f(x) = –0.5x+ x2– 2 Polynomial function? f(x) = x3+ 3x f(x) = 6x2+ 2x–1+ x

8. Polynomial Functions can be classified by degree

9. Polynomial Functions can be classified by degree and by the number of terms CONSTANT, MONOMIAL LINEAR, BINOMIAL QUADRATIC, TRINOMIAL CUBIC, POLYNOMIAL

10. Given f(x) find f(-3). -69

11. End Behavior Task

12. Let’s Summarize

13. CONCEPT END BEHAVIOR FOR POLYNOMIAL FUNCTIONS SUMMARY annx – x + > 0 even f(x) + f(x) +  > 0 odd f(x) – f(x) +  < 0 even f(x) – f(x) –  < 0 odd f(x) + f(x) –  GRAPHING POLYNOMIAL FUNCTIONS

14. Determine the left and right behavior of the graph of each polynomial function. Ex. f(x) = x4 + 2x2 – 3x f(x) = -x5 +3x4 – x f(x) = 2x3 – 3x2 + 5

15. Tell me what you know about the equation… Odd exponent Positive leading coefficient

16. Tell me what you know about the equation… Even exponent Positive leading coefficient

17. Tell me what you know about the equation… Odd exponent Positive leading coefficient

18. Tell me what you know about the equation… Even exponent Negative leading coefficient