Exponential Functions Definition of the Exponential Function The

Exponential Functions

Definition of the Exponential Function 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. Here are some examples of exponential functions. f (x) = 2 x Base is 2. g(x) = 10 x Base is 10. h(x) = 3 x+1 Base is 3.

Example The exponential function f (x) = 13. 49(0. 967)x – 1 describes the number of O-rings expected to fail, when the temperature is x°F. On the morning the Challenger was launched, the temperature was 31°F, colder than any previous experience. Find the number of Orings expected to fail at this temperature. Solution. Because the temperature was 31°F, substitute 31 for x and evaluate the function at 31. f (x) = 13. 49(0. 967)x – 1 f (31) = 13. 49(0. 967)31 – 1=3. 77 This is the given function. Substitute 31 for x.

Characteristics of Exponential Functions 1. 2. 3. 4. 5. 6. The domain of f (x) = bx consists of all real numbers. The range of f (x) = bx consists of all positive real numbers. The graphs of all exponential functions pass through the point (0, 1) because f (0) = b 0 = 1. If b > 1, f (x) = bx has a graph that goes up to the right and is an increasing function. If 0 < b < 1, f (x) = bx has a graph that goes down to the right and is a decreasing function. f (x) = bx is a one-to-one function and has an inverse that is a function. The graph of f (x) = bx approaches but does not cross the x-axis. The xaxis is a horizontal asymptote. f (x) = bx 0<b<1 f (x) = bx b>1

Transformations Involving Exponential Functions Transformation Equation Description Horizontal translation g(x) = bx+c • Shifts 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 Vertical translation g(x) = -bx + c the graph of f (x) = bx left c units if c > 0. • Shifts the graph of f (x) = bx right c units if c < 0. the graph of f (x) = bx about the x-axis. • Reflects the graph of f (x) = bx about the y-axis. • Shifts the graph of f (x) = bx up c units if c > 0. • Shifts the graph of f (x) = bx down c units if c < 0.

Example Use the graph of f (x) = 3 x to obtain the graph of g(x) = 3 x+1. Solution. Examine the table below. Note that the function g(x) = 3 x+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) = 3 x one unit to the left. We construct a table showing some of the coordinates for the parent function f to build the graphs. x f(x) g(x) = 3 x+1 -2 f (x) = 3 x -1 0 1 1 3 2 9 (-1, 1) -5 -4 -3 -2 -1 (0, 1) 1 2 3 4 5 6

Problems Sketch a graph using transformation of the following: 1. 2. 3. Recall the order of shifting: horizontal, reflection (horz. , vert. ), vertical.

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

Formulas for Compound Interest •

Example: Choosing Between Investments You want to invest $8000 for 6 years, and you have a choice between two accounts. The first pays 7% per year, compounded monthly. The second pays 6. 85% per year, compounded continuously. Which is the better investment? Solution. The better investment is the one with the greater balance in the account after 6 years. Let’s begin with the account with monthly compounding. We use the compound interest model with P = 8000, r = 7% = 0. 07, n = 12 (monthly compounding, means 12 compoundings per year), and t = 6. The balance in this account after 6 years is $12, 160. 84. more

Example: Choosing Between Investments You want to invest $8000 for 6 years, and you have a choice between two accounts. The first pays 7% per year, compounded monthly. The second pays 6. 85% per year, compounded continuously. Which is the better investment? Solution. For the second investment option, we use the model for continuous compounding with P = 8000, r = 6. 85% = 0. 0685, and t = 6. The balance in this account after 6 years is $12, 066. 60, slightly less than the previous amount. Thus, the better investment is the 7% monthly compounding option.

Example 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= Pert A = 2000 e(. 07)(8) A = 2000 e(. 56) A = $3501. 35

One-to-One Property of Exponential Functions To solve these equations, 1 st Rewrite each side with the same base 2 nd Set the exponents equal to each other 3 rd Solve the equation for the variable

Half-Life b) Determine the quantity present after 1000 years.