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Exponential Functions

Exponential Functions. The exponential function f with base b is defined by: f ( x ) = b x or y = b x Where b is a positive constant other than 1 and x is any real number. Definition of the Exponential Function. Base is 2. Base is 10. Base is 3.

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Exponential Functions

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  1. Exponential Functions

  2. The exponential function f with base b is defined by: f (x) = bx or y = bx Where b is a positive constant other than 1 and x is any real number. Definition of the Exponential Function

  3. Base is 2. Base is 10. Base is 3. Examples of exponential functions. f (x) = 2xg(x) = 10xh(x) = 3x+1 Definition of the Exponential Function

  4. The exponential function: f (x) = 13.49(0.967)x – 1 describes the number of O-rings expected to fail, f (x), when the temperature is x°F. On the morning the Challenger was launched, the temperature was 31°F, (colder than any previous launch). Find the number of O-rings expected to fail at this temperature. Text Example

  5. This is the given function. f (31) = 13.49(0.967)31 – 1 f (x) = 13.49(0.967)x – 1 Substitute 31 for x. Press .967 ^ 31 on a graphing calculator to get .353362693426. Multiply this by 13.49 and subtract 1 to obtain f (31) = 13.49(0.967)31 – 1=3.77 Solution Because the temperature was 31°F, substitute 31 for x and evaluate the function at 31. Text Example

  6. The domain of f (x) = bxconsists of all real numbers. • The range of f (x) = bxconsists of all positive real numbers. • The graphs of all exponential functions pass through the point (0, 1) because f (0) = b0 = 1. • If b > 1, f (x) = bxhas a graph that goes up to the right and is an increasing function. • If 0 < b < 1, f (x) = bxhas a graph that goes down to the right and is a decreasing function. • f (x) = bxis a one-to-one function and has an inverse that is a function. (a logarithm) Characteristics of Exponential Functions

  7. The graph of f (x) = bxapproaches but does not cross the x-axis. The x-axis is a horizontal asymptote. Characteristics of Exponential Functions f (x) = bx 0 < b < 1 f (x) = bx b > 1

  8. Transformation Equation Description Horizontal translation g(x) = bx+c • Shifts the graph of f (x) = bx to the left c units if c > 0. • Shifts the graph of f (x) = bx to the right c units if c < 0. Vertical stretching or shrinking g(x) = c bx • Multiplying y-coordintates of f (x) = bx by c, • Stretches the graph of f (x) = bx if c > 1. • Shrinks the graph of f (x) = bx if 0 < c < 1. Reflecting g(x) = -bx g(x) = b-x • Reflects the graph of f (x) = bx about the x-axis. • Reflects the graph of f (x) = bx about the y-axis. Vertical translation g(x) = bx + c • Shifts the graph of f (x) = bx upward c units if c > 0. • Shifts the graph of f (x) = bxdownward c units if c < 0. Transformations Involving Exponential Functions

  9. x f (x) = 3x g(x) = 3x+1 g(x) = 3x+1 f (x) = 3x -2 3-2 = 1/9 3-2+1 = 3-1 = 1/3 -1 3-1 = 1/3 3-1+1 = 30 = 1 0 30 = 1 30+1 = 31 = 3 1 31 = 3 31+1 = 32 = 9 (-1, 1) (0, 1) 2 32 = 9 32+1 = 33 = 27 1 2 3 4 5 6 -5 -4 -3 -2 -1 Use the graph of f (x) = 3x to obtain the graph of g(x) = 3 x+1. Solution Examine the table below. Note that the function g(x) = 3x+1 has the general form g(x) = bx+c, where c = 1. Because c > 0, we graph g(x) = 3 x+1 by shifting the graph of f (x) = 3x one unit to the left. We construct a table showing some of the coordinates for f and g to build their graphs. Text Example

  10. An irrational number, symbolized by the letter e, appears as the base in many applied exponential functions. This irrational number is approximately equal to 2.72. More accurately, The number e is called the natural base. The function f (x) = ex is called the natural exponential function. The Natural Base e -1

  11. f (x) = 3x f (x) = ex 4 (1, 3) f (x) = 2x 3 (1, e) 2 (1, 2) (0, 1) 1 The Natural Base e -1

  12. Formulas for Compound Interest • After t years, the balance, A, in an account with principal P and annual interest rate r (in decimal form) is given by the following formulas: • For n compoundings per year: A = P(1 + r/n)nt • For continuous compounding: A = Pert.

  13. Use A= Pert to solve the following problem: Find the accumulated value of an investment of $2000 for 8 years at an interest rate of 7% if the money is compounded continuously Solution: A= PertA = 2000e(.07)(8)A = 2000 e(.56)A = 2000 * 1.75A = $3500 Example

  14. Exponential Functions

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