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# 3_3 - from a decreasing function to an increasing function...

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Section 3.3: Polynomial Functions and Models Instructor: Ms. Hoa Nguyen 1 Graphs Polynomial of DEGREE n where n is a non-negative integer and a n 6 = 0: f ( x ) = a n x n + a n - 1 x n - 1 + . . . + a 1 x + a 0 (1) a n , a n - 1 , . . . , a 1 , a 0 : real numbers. The DOMAIN consists of ALL REAL NUMBERS . Example 1 Which of the following is a polynomial function? For those that are, state the degree; for those are not, tell why not. f ( x ) = 2 + 4 x - 10 x 3 f ( x ) = x 1 2 = x f ( x ) = 0 f ( x ) = 18 f ( x ) = x +3 x 9 = ( x + 3) x - 9 f ( x ) = x - 3 + x The graph of every polynomial function is: CONTINUOUS (no gaps or holes; can be drawn without lifting pencil from the paper) SMOOTH (no sharp corners or cusps) End behavior (for large | x | , the graph of f ( x ) = a n x n + a n - 1 x n - 1 + . . . + a 1 x + a 0 behaves like the graph of f ( x ) = a n x n ) Figure 1: The graph of y = f ( x ) = ( x + 1)( x )( x - 1)( x - 2) Turning point : local maximum (the point where the graph changes from an increasing function to a decreasing function) or local minimum point (the point where the graph changes

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Unformatted text preview: from a decreasing function to an increasing function). 1 2 The Zeros of a polynomial r a (real) zero of f , or root of f , or an x-intercept of the graph of f means that • f ( r ) = 0, r is a REAL number. • ( x-r ) is a factor of f . Example 2 Example 3 Example 4 2 3 Multiplicity r is a zero of multiplicity m of f if • ( x-r ) m is a factor of f . • ( x-r ) m +1 is NOT a factor of f . Then ﬁnd the degree of the polynomial. r is a zero of EVEN multiplicity : • Graph TOUCHes the x-axis at r . • Sign of f does NOT change from one side to the other side of r . r is a zero of ODD multiplicity : • Graph CROSSes the x-axis at r . • Sign of f changes from one side to the other side of r . Figure 2: The graph of y = f ( x ) = ( x + 1) 2 ( x )( x-1)( x-2) 2 3 Example 6 Example 7 Example 8 4...
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3_3 - from a decreasing function to an increasing function...

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