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11.1

11.1. POWER FUNCTIONS. Proportionality and Power Functions. Example 1 The area, A, of a circle is proportional to the square of its radius, r: A = π r 2 .

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11.1

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  1. 11.1 POWER FUNCTIONS

  2. Proportionality and Power Functions • Example 1 The area, A, of a circle is proportional to the square of its radius, r: A = πr2. • Example 2 The weight, w, of an object is inversely proportional to the square of the object’s distance, d, from the earth’s center: w =k/d2 = kd−2.

  3. A quantity y is directly proportional to a power of x if y = k xn, k and n are constants. A quantity y is inversely proportional to xn if y =k/xn , k and n are constants. A power function is a function of the form f(x) = k xp, where k and p are constants.

  4. Proportionality and Power Functions • Example 3 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 xp. (a) f(x) = (b) g(x) = 2 (x+ 5)3 (c) u(x) = (d) v(x) = 6 ・ 3x • Solution:The functions f (k=13, p=1/3) and u (k=5, p=-3/2) are power functions; the functions g and v are not power functions.

  5. The Effect of the Power p • Graphs of the Special Cases y = x0and y = x1 • The power functions corresponding to p = 0 and p = 1 are both linear. The function y= x0 = 1, except at x = 0. Its graph is a horizontal line with a hole at (0,1). Why is there a hole at (0,1)? The graph of y = x1 = x is a line through the origin with slope +1. Both graphs contain the point (1,1). y = x1 (1,1) y = x0 ●

  6. The Effect of the Power p Positive Integer Powers y = x5 y = x4 y = x2 y = x3 (1,1) ● (-1,-1) ● (-1,1) (1,1) ● ● Graphs of positive odd powers of x are “chair”-shaped Graphs of positive even powers of x are U-shaped

  7. The Effect of the Power p y = x-1 Negative Integer Powers y = x-2 (1,1) (1,1) (-1,1) (-1,-1) Both graphs have a horizontal asymptote of y = 0 Both graphs have a vertical asymptote of x = 0

  8. The Effect of the Power p Graphs of Positive Fractional Powers y = x1/2 y = x1/3 (1,1) (1,1) y = x1/4 y = x1/5 (-1,-1) Graphs of odd roots of x are defined for all values of x Graphs of even roots of x are not defined for x < 0

  9. Solution

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