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WARM-UP: 10/30/13 Find the standard form of the quadratic function. Identify the vertex and graph.
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WARM-UP: 10/26/12
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2.2 – POLYNOMIAL FUNCTIONS
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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
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DEFINITION OF A POLYNOMIAL FUNCTION:
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PROPERTIES OF POLYNOMIAL FUNCTIONS: 1) Polynomial functions are continuous Which of these is a polynomial function? a) b)
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PROPERTIES OF POLYNOMIAL FUNCTIONS: 2) Polynomial functions only have smooth curves, not sharp turns. Which of these is a polynomial function? a) b)
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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.
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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
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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).
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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).
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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).
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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).
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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
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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.
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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.
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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.
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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.
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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
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GRAPH THE FUNCTION:
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Since this is not factorable, we have to use the rational root theorem and synthetic division to solve for the zeros of the function.
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GRAPH THE FUNCTION: Rational Root Theorem: p: the factors of the constant q: the factors of the leading coefficient
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GRAPH THE FUNCTION: Synthetic Division:
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GRAPH THE FUNCTION: Synthetic Division: We now have to use the quadratic equation to solve for the remaining zeros.
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GRAPH THE FUNCTION: Quadratic Equation: Zeros of the function: y -intercept:
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GRAPH THE FUNCTION:
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