WARM-UP: 10/30/13 Find the standard form of the quadratic function. Identify the vertex and graph.

WARM-UP: 10/26/12

2.2 – POLYNOMIAL FUNCTIONS

I N THIS SECTION, YOU WILL LEARN TO  use transformation to sketch graphs  use the Leading Coefficient Test to determine the end behavior of polynomial graphs  use zeros of polynomial functions as sketching aids

DEFINITION OF A POLYNOMIAL FUNCTION:

PROPERTIES OF POLYNOMIAL FUNCTIONS: 1) Polynomial functions are continuous Which of these is a polynomial function? a) b)

PROPERTIES OF POLYNOMIAL FUNCTIONS: 2) Polynomial functions only have smooth curves, not sharp turns. Which of these is a polynomial function? a) b)

LEADING COEFFICIENT TEST: Definition of a leading coefficient: the coefficient of the highest degree The Leading Coefficient Test determines the end-behavior of any polynomial function. It determines whether the graph falls or rises depending on the highest degree of the polynomial and its leading coefficient.

LEADING COEFFICIENT TEST: The Leading Coefficient Test is dependent on both of the following two values: a) Highest degree of the polynomial: n can be odd or even b) Leading coefficient: a can be positive or negative

LEADING COEFFICIENT TEST FOR ODD DEGREE POLYNOMIAL FUNCTIONS: 1) If a > 0 and n is odd, then the graph increases without bound on the right (rises) and decreases without bound on the left (falls).

LEADING COEFFICIENT TEST FOR ODD DEGREE POLYNOMIAL FUNCTIONS: 2) If a < 0 and n is odd, then the graph decreases without bound on the right (falls) and increases without bound on the left (rises).

LEADING COEFFICIENT TEST FOR EVEN DEGREE POLYNOMIAL FUNCTIONS: 1) If a > 0 and n is even, then the graph increases without bound on the right and on the left (rises).

LEADING COEFFICIENT TEST FOR EVEN DEGREE POLYNOMIAL FUNCTIONS: 2) If a < 0 and n is even, then the graph decreases without bound on the right and on the left (falls).

LEADING COEFFICIENT TEST SUMMARY: Degree n Sign of a Left- End Right- End odd a > 0FallsRises odd a < 0RisesFalls even a > 0Rises even a < 0Falls

LEADING COEFFICIENT TEST: Note: The Leading Coefficient Test only determines the end behavior of the function, but does not tell you how the graph behaves between the end behavior.

ZEROS OF THE FUNCTION: The zeros of the function can help determine certain properties of the polynomial graph. a) The function can have at most n - 1 turns: points at which the graph can change from increasing to decreasing.

ZEROS OF THE FUNCTION: 2) The function can have at most n real zeros. Since the highest degree is 6, this function can have most 6 real zeros.

ZEROS OF THE FUNCTION: 3) In general, multiple roots will behave in two different ways. a) If k is even, the graph will only touch the x -axis and not cross it. b) If k is odd, the graph will cross the x -axis.

STEPS TO GRAPH A FUNCTION: 1) Solve for the zeros 2) Solve for the y -intercept 3) Use the information from the multiple roots to determine where it touches and crosses 4) Use the Leading Coefficient Test to determine the end behavior 5) Plot a few points between the zeros

GRAPH THE FUNCTION:

 Since this is not factorable, we have to use the rational root theorem and synthetic division to solve for the zeros of the function.

GRAPH THE FUNCTION:  Rational Root Theorem: p: the factors of the constant q: the factors of the leading coefficient

GRAPH THE FUNCTION:  Synthetic Division:

GRAPH THE FUNCTION:  Synthetic Division:  We now have to use the quadratic equation to solve for the remaining zeros.

GRAPH THE FUNCTION:  Quadratic Equation:  Zeros of the function:  y -intercept:

GRAPH THE FUNCTION: