Polynomial Functions n r a e l l

Polynomial Functions n r a e l l

Polynomial Functions n r a e l l l ou y t a h W To classify polynomials. To graph polynomial functions and describe end behavior. r a l u b a Voc yMonomial, degree of a monomial, polynomial, degree of a polynomial, polynomial function, standard form of a polynomial function, turning point, end behavior. Take a note You can classify a polynomial it its degree or by its number of terms. Polynomial with a degree 0 to 5 have specific names. Remember: monomial is a number, a variable, or a

product of a real number with one or more variables with whole-number exponents. The degree of a monomial in Polynomial one variable is is a monomial or a sum of monomial the exponent of the variable. The degree of a polynomial in one variable is the greatest degree among its monomials terms. For exampl e: 4 2 a )7 x 5 x x 9 2 b )8 x 3 xy 2 y 1 2 3 5 c) x 2 x x 2

1 d )7 x 4 x x 6 3 2 Polynomial in one variable: X Degree of the polynomial:4 Not a polynomial in one variable Organize the1expression 5 3 2 x 2x 2 x Polynomial in one variable X Degree of the polynomial 5 Not a polynomial, the term 1 x can not be writen in the form x n Your turn: Classifying Polynomials What is the classification of each polynomial by degree By number of terms?

A. 3 x 9 x 2 5 B. 4 x 6 x 2 x 4 10 x 2 12 Answers: x 4 4 x 2 4 x 12 2 9x 3x 5 Degree 2 and has 3 terms Quadratic trinomial Degree 4 and has 4 terms. Quartic polynomial of 4 terms ote: polynomial with the variable x defines a polynomial function f x. The standard form of a polynomial function arranges the erms by degree in descending order. n P x a n x a n 1 x n 1 ... a1 x a0 Where n 0 and a real numbers Take a note: A polynomial function has distinguishing behaviors.You can look at its algebraic form and know something about

its graph. You can look at its graph and know something about its algebraic form. Constant function Degree 0 Cubic function Degree 3 Linear function Degree 1 Quadratic function Degree 2 Quartic funcion Degree 4 Quintic function Degree 5 The degree of a polynomial function affects the shape of its graph and determines the maximum numbers of turning points. TURNING POINTS are the points where the graphs changes directions. It also affects the end behavior. END BEHAVIOR is the directions of the graph to the far left and the far right. End behavior of a n even n odd Polynomial a positive up and up down and

n Function up ax of Degree a down and up and negative down down n with Leading n n 1 InTerm general the graph of a polynomial function of degree n 1 Has at most turning points. The graph of a polynomial function of odd degree has an even number of turning points. The graph of a polynomial function of Your turn: Describing end behavior of polynomial functions onsider the leading term of each polynomial functio What is the end behavior of the graph? Check your nswer with a graphing calculator. 4 3 2 3 B. f x 2 x 8 x 8 x 2 A. f ( x) 4 x 3x

The leading term is 4x . n is odd and a is positive The end behavior is down and up. 3 4 The leading term is 2x . n is even and a is negative The end behavior is down and down Your turn: Graphing cubic functions What is the graph of each cubic function? Describe the graph. If you have a graphing calculator use it. 1 3 a ) f x x 2 Step 1: a) b)f x 3 x x 3 c)f x x 3 2 x 2 x 2 d)f x x 3 1 Step 3: the end behavior is down and up. No turning points x f(x)

-2 -4 -1 -0.5 0 0 1 0.5 2 4 Step 2: b c The end behavior is up and down. There are two turning points d The end behavior is down and up; No turning points

The end behavior is up and down two turning points Your turn: Using differences to determine degree X f(x) What is the degree of the polynomial a) -3 -1 function that generates the data -2 -7 show? -1 -3 b) 0 5 1 11 2 9 3 -7

y-value First -1 Difference Second -6 Third Difference -7 Difference 10 4 -6 -3 4 8 -6 5 -2 6 -6 11 -8 -2 -6 9 -14 -16 -7 x f(x) -3

23 -2 -16 -1 -15 0 -10 1 -13 2 -12 3 29b)Degree The third difference is a constant so 4 the degree of the polynomial function is 3 Classwork odd TB pg 285 exercises 8-49 Homework even

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