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

E VALUATING P OLYNOMIAL F UNCTIONS A polynomial function is a function of the form f (x) = a n x n + a n – 1 x n – 1 +· · ·+ a 1 x + a 0 Where a n  0 and the exponents are all whole numbers. A polynomial function is in standard form if its terms are written in descending order of exponents from left to right. For this polynomial function, a n is the leading coefficient, a 0 is the constant term, and n is the degree. a n  0 anan anan leading coefficient a 0a 0 a0a0 constant term n n degree descending order of exponents from left to right. n n – 1

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

Graphs of Polynomial Functions Graphs of polynomial functions are continuous. That is, they have no breaks, holes, or gaps. Polynomial functions are also smooth with rounded turns. Graphs with points or cusps are not graphs of polynomial functions. x y x y continuousnot continuouscontinuous smoothnot smooth polynomialnot polynomial x y f (x) = x 3 – 5x 2 + 4x + 4

Decide whether the function is a polynomial function. If it is, write the function in standard form and state its degree, type and leading coefficient. Identifying Polynomial Functions The function is not a polynomial function because the term 3 x does not have a variable base and an exponent that is a whole number. S OLUTION f (x) = x x

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

f (x) = x 2 – 3 x 4 – Identifying Polynomial Functions f (x) = x x f (x) = 6x x – 1 + x Polynomial function? f (x) = – 0.5x +  x 2 – 2

Polynomial Functions can be classified by degree

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

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

End Behavior Task

Let’s Summarize

G RAPHING P OLYNOMIAL F UNCTIONS END BEHAVIOR FOR POLYNOMIAL FUNCTIONS C ONCEPT S UMMARY > 0even f (x)+  f (x) +  > 0odd f (x)–  f (x) +  < 0even f (x)–  f (x) –  < 0odd f (x)+  f (x) –  a n n x –  x + 

Ex. Determine the left and right behavior of the graph of each polynomial function. f(x) = -x 5 +3x 4 – x f(x) = x 4 + 2x 2 – 3x f(x) = 2x 3 – 3x 2 + 5

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

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

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

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