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section32

# section32 - between the right and left-end behaviors of the...

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MAT 111 - College Algebra Section 3.2 Polynomial Functions of Higher Degree Graphs of polynomial functions are: 1. Continuous. 2. Smooth. Power functions are functions of the form f ( x ) = ax n where n 0 is a non-negative integer and a 6 = 0. To sketch the graph of a general polynomial function we need to determine the following: 1. The end-behavior of the polynomial. 2. The zeros of the polynomial and the multiplicity of each zero. 3. Use of the Intermediate Value Theorem . End-Behavior of a Polynomial: Consider the function f ( x ) = x 5 - x 3 + 7. Determining the end-behavior of f means determining the behavior of the values of the dependent variable, f ( x ), when values of the independent variable, x , become very large or very small. In symbols f ( x ) ? when x → ±∞ The end-behavior of a polynomial is determined by the leading term.

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Examples: Determine the right-hand and left-hand behavior of the graph of the polynomial function: 1. f ( x ) = 1 + x 6 2. g ( x ) = x 4 - 4 x 2 3. h ( x ) = - 4 x 3 + 4 x 2 + 15 x Zeros of a Polynomial: These are the x -intercepts of the graph of the polynomial and they tell us what happens
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Unformatted text preview: between the right- and left-end behaviors of the polynomial. They are also solutions to the equation f ( x ) = 0. Moreover, they determine factors of the functions: if c , a real number, is a zero of f , then x-c is a factor of f ( x ). There are two types of zeros of a polynomial: 1. Zeros of even multiplicity. 2. Zeros of odd multiplicity. Examples: Find the zeros of each polynomial and determine the multiplicity of each zero: 1. g ( x ) = x 4-4 x 2 2. h ( x ) =-4 x 3 + 4 x 2 + 15 x Fact: A polynomial of degree n has at most n-1 turning points. Intermediate Value Theorem: Let a and b be real numbers such that a < b . If f is a polynomial function such that f ( a ) 6 = f ( b ), then in the interval [ a, b ] f takes on every value between f ( a ) and f ( b ). Examples: Sketch the graph of the function: 1. g ( x ) = x 4-4 x 2 2. h ( x ) =-4 x 3 + 4 x 2 + 15 x...
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section32 - between the right and left-end behaviors of the...

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