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9.2 Exponential Functions ■ Graphing Exponential Functions ■ Equations with x and y Interchanged ■ Applications of Exponential Functions

Slide 9- 3 Copyright © 2012 Pearson Education, Inc. Graphing Exponential Functions In Chapter 7, we studied exponential expressions with rational-number exponents such as 7 2/3. What about expressions with irrational exponents? To attach meaning to consider a rational approximation, r, of

Slide 9- 4 Copyright © 2012 Pearson Education, Inc. 1.4 < r < < r < < r < Any positive irrational exponent can be interpreted in a similar way. Negative irrational exponents are then defined using reciprocals.

Slide 9- 5 Copyright © 2012 Pearson Education, Inc. Exponential Function The function f (x) = a x, where a is a positive constant, is called the exponential function, base a.

Slide 9- 6 Copyright © 2012 Pearson Education, Inc. Example Graph the exponential function given by Solution xy, or f(x) 0 1 –1 2 – /3 9 1/9 27

Slide 9- 7 Copyright © 2012 Pearson Education, Inc. Example Graph the exponential function given by Solution xy, or f(x) 0 1 –1 2 –2 –3 1 1/3 3 1/ The curve is a reflection of y = 3 x.

Slide 9- 8 Copyright © 2012 Pearson Education, Inc. From the previous two examples, we can make the following observations. A. For a > 1, the graph of f (x ) = a x increases from left to right. The greater the value of a the steeper the curve. B. For 0 < a < 1, the graph of f (x ) = a x decreases from left to right. For smaller values of a, the graph becomes steeper. C. All graphs of f (x ) = a x go through the y-intercept (0, 1).

Slide 9- 9 Copyright © 2012 Pearson Education, Inc. D. All graphs of f (x) = a x have the x-axis as the asymptote. E. If f (x ) = a x, with a > 0, a ≠ 1, the domain of f is all real numbers, and the range of f is all positive real numbers. F. For a > 0, a not 1, the function given by f (x) = a x is one-to-one. Its graph passes the horizontal-line test.

Slide Copyright © 2012 Pearson Education, Inc. The graph of f(x) = a x – h + k looks like the graph of y = a x translated |h| units left or right and |k| units up or down. If h > 0, y = a x is translated h units right. If h < 0, y = a x is translated |h| units left. If k > 0, y = a x is translated k units up. If k < 0, y = a x is translated |k| units down.

Slide Copyright © 2012 Pearson Education, Inc. Equations with x and y Interchanged It will be helpful in later work to be able to graph an equation in which the x and y in y = a x are interchanged.

Slide Copyright © 2012 Pearson Education, Inc. Example Graph the exponential function given by Solution xy, or f(x) 1 3 1/3 9 1/ –1 2 –2 3

Slide Copyright © 2012 Pearson Education, Inc. Applications of Exponential Functions Example Interest compounded annually. The amount of money A that a principal P will be worth after t years at interest rate i, compounded annually, is given by the formula

Slide Copyright © 2012 Pearson Education, Inc. Example Solution Suppose that $60,000 is invested at 5% interest, compounded annually. a) Find a function for the amount in the account after t years. b) Find the amount of money in the account at t = 6. = $60000( ) t = $60000(1.05) t b) A(6) = $60000(1.05) 6

Slide Copyright © 2012 Pearson Education, Inc. continued The graph can be displayed using a graphing calculator.