Section 5.2 Exponential Functions and Graphs Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.

Objectives  Graph exponential equations and exponential functions.  Solve applied problems involving exponential functions and their graphs.

Exponential Function The function f(x) = a x, where x is a real number, a > 0 and a  1, is called the exponential function, base a. The base needs to be positive in order to avoid the complex numbers that would occur by taking even roots of negative numbers. The following are examples of exponential functions:

Example Graph the exponential function y = f (x) = 2 x.

As x increases, y increases without bound. As x decreases, y decreases getting close to 0; as x  ∞, y  0. Example (continued) The x-axis, or the line y = 0, is a horizontal asymptote. As the x-inputs decrease, the curve gets closer and closer to this line, but does not cross it.

Example Graph the exponential function Note This tells us the graph is the reflection of the graph of y = 2 x across the y-axis. Selected points are listed in the table.

Example (continued) As x increases, the function values decrease, getting closer and closer to 0. The x- axis, y = 0, is the horizontal asymptote. As x decreases, the function values increase without bound.

Graphs of Exponential Functions Observe the following graphs of exponential functions and look for patterns in them.

Example Graph y = 2 x – 2. The graph is the graph of y = 2 x shifted to right 2 units.

Graph y = 5 – 0.5 x. Example The graph is a reflection of the graph of y = 2 x across the y-axis, followed by a reflection across the x-axis and then a shift up 5 units.

Application The amount of money A that a principal P will grow to after t years at interest rate r (in decimal form), compounded n times per year, is given by the formula

Example Suppose that $100,000 is invested at 6.5% interest, compounded semiannually. a. Find a function for the amount to which the investment grows after t years. b. Graph the function. c. Find the amount of money in the account at t = 0, 4, 8, and 10 yr. d. When will the amount of money in the account reach $400,000?

Example (continued) Since P = $100,000, r = 6.5%=0.65, and n = 2, we can substitute these values and write the following function

Example (continued) b)Use the graphing calculator with viewing window [0, 30, 0, 500,000].

Example (continued) We can compute function values using function notation on the home screen of a graphing calculator.

Example (continued) We can also calculate the values directly on a graphing calculator by substituting in the expression for A(t):

Example (continued) d) Set 100,000(1.0325) 2t = 400,000 and solve for t, which we can do on the graphing calculator. Graph the equations y 1 = 100,000(1.0325) 2t y 2 = 400,000 Then use the intersect method to estimate the first coordinate of the point of intersection.

Example (continued) Or graph y 1 = 100,000(1.0325) 2t – 400,000 and use the Zero method to estimate the zero of the function coordinate of the point of intersection. Regardless of the method, it takes about years, or about 21 yr, 8 mo, and 2 days.

The Number e e is a very special number in mathematics. Leonard Euler named this number e. The decimal representation of the number e does not terminate or repeat; it is an irrational number that is a constant; e  …

Example Find each value of e x, to four decimal places, using the e x key on a calculator. a) e 3 b) e  0.23 c) e 2 d) e 1 a) e 3 ≈ b) e  0.23 ≈ c) e 0 = 1d) e 1 ≈

Graphs of Exponential Functions, Base e Example Graph f (x) = e x and g(x) = e –x. Use the calculator and enter y 1 = e x and y 2 = e –x. Enter numbers for x.

Graphs of Exponential Functions, Base e - Example (continued) The graph of g is a reflection of the graph of f across they-axis.

Example Graph f (x) = e x + 3. The graph f (x) = e x + 3 is a translation of the graph of y = e x left 3 units.

Example Graph f (x) = e –0.5x. The graph f (x) = e –0.5x is a horizontal stretching of the graph of y = e x followed by a reflection across the y-axis.

Example Graph f (x) = 1  e  2x. The graph f (x) = 1  e  2x is a horizontal shrinking of the graph of y = e x followed by a reflection across the y-axis and then across the x-axis, followed by a translation up 1 unit.