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Unformatted text preview: Chapter 11 Polynomials and Rational Functions 11.1 Power functions Key Points • Proportionality and power functions • The general form of a power function • The classification of power functions into six basic types Power function ( ) p f x kx = ( ) p f x kx = Proportionality and Power Functions • Example Which of the following functions are power functions? For each power function, state the value of the constants k and p in the formula y = k x p . (a) f(x) = (b) g(x) = 2 ( x + 5) 3 (c) u(x) = (d) v(x) = 6 ・ 3 x Functions Modeling Change: A Preparation for Calculus, 4th 3 13 x 3 / 25 x The Effect of the Power p • Graphs of the Special Cases y = x and y = x 1 • The power functions corresponding to p = 0 and p = 1 are both linear. The function y = x = 1, except at x = 0. Its graph is a horizontal line with a hole at (0,1). The graph of y = x 1 = x is a line through the origin with slope +1. Both graphs contain the point (1,1). Functions Modeling Change: A Preparation for Calculus, 4th 1 1 y = x 1 y = x (1,1) ● The Effect of the Power p Positive Integer Powers Functions Modeling Change: A Preparation for Calculus, 4th 4 2 2 4 x 2 4 6 8 10 12 y (1,1) y = x 4 y = x 2 (1,1) ● ● 2 1 1 2 x 4 2 2 4 y y = x 3 y = x 5 (1,1) (1, 1) ● ● Graphs of positive even powers of x are Ushaped Graphs of positive odd powers of x are “chair”shaped 3 2 1 1 2 3 x 1 2 3 4 5 6 y 3 2 1 1 2 3 x 4 2 2 4 y The Effect of the Power p Negative Integer Powers Functions Modeling Change: A Preparation for Calculus, 4th (1,1) (1,1) y = x 2 (1,1) (1,1) Both graphs have a vertical asymptote of x = Both graphs have a vertical asymptote of x = Both graphs have a horizontal asymptote of y = 0 Both graphs have a horizontal asymptote of y = 0 y = x1 x 0.1 0.05 0.01 0.001 0.0001 0 1/x 10 20 100 1000 10000 undefined 1/x 2 100 400 10,000 1,000,000 100,000,000 undefined x 0 10 20 30 40 50 1/x undefined 0.1 0.05 0.033333 0.025 0.02 1/x 2 undefined 0.01 0.0025 0.001111 0.000625 0.0004 4 2 2 4 x 2 1 1 2 y 4 2 2 4 x 2 1 1 2 y The Effect of the Power p Graphs of Positive Fractional Powers Functions Modeling Change: A Preparation for Calculus, 4th (1,1) y = x 1/4 y = x 1/2 y = x 1/3 y = x 1/5 (1,1) (1, 1) Graphs of even roots of x are not defined for x < 0 Graphs of odd roots of x are defined for all values of x 24,like 26, like 28,18,38 Functions Modeling Change: A Preparation for Calculus, 4th A quantity y is (directly) proportional to a power of x if y = k x n , k and n are constants. A quantity y is (directly) proportional to a power of x if y = k x n , k and n are constants....
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 Spring '08
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 Polynomials, Rational Functions, lim P

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