6 2 Exponential Functions Objectives Classify an exponential

6. 2 Exponential Functions Objectives: • Classify an exponential function as representing exponential growth or exponential decay • Calculate the growth of investments under various conditions

Exponential Functions y = x 2 y = 2 x The function f(x) = bx is an exponential function with base b, where b is a positive real number other than 1 and x is any real number. An asymptote is a line that a graph approaches (but does not reach) as its x- or y-values become very large or very small.

Exponential Functions Graph y 1 = 2 x and y 2 = When b > 1, the function f(x) = bx represents exponential growth. When 0 < b < 1, the function f(x) = bx represents exponential decay.

Example 1 Graph f(x) = 2 x along with each function below. Tell whether each function represents exponential growth or exponential decay. Then give the y-intercept. y = 4(2 x) exponential growth, since the base, 2, is > 1 y-intercept is 4 because the graph of f(x) = 2 x, which has a y-intercept of 1, is stretched by a factor of 4 exponential decay, since the base, ½, is < 1 y-intercept is 6 because the graph of f(x) = 2 x, which has a y-intercept of 1, is stretched by a factor of 6

Practice Graph f(x) = 2 x along with each function below. Tell whether each function represents exponential growth or exponential decay. Then give the y-intercept.

Critical Thinking What transformation of f occurs when a < 0 in The graph is reflected across the x-axis.

Compound Interest The total amount of an investment, A, earning compound interest is where P is the principle, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the time in years.

Example 2 Find the final amount of a $500 investment after 8 years at 7% interest compounded annually, quarterly, and monthly. compounded annually: = $859. 09 compounded quarterly: = $871. 11 compounded monthly: = $873. 91

Practice Find the final amount of a $2200 investment at 9% interest compounded monthly for 3 years.

Effective Yield The effective yield is the annually compounded interest rate that yields the final amount of an investment. Suppose you buy a motorcycle for $10, 000 and sell it one year later for $13, 000. The effective yield would be 30% because you made 30% more ($3, 000) than the original price you paid. You can determine the effective yield by fitting an exponential regression equation to two points.

Example 3 A collector buys an antique stove for $500 at the beginning of 1990 and sells it for $875 at the beginning of 1998. Find the effective yield. Step 1: Find two points that represent the information after 0 years the stove was worth $500 (0, 500) after 8 years the stove was worth $875 (8, 875) Step 2: Enter the two points on a list and find the exponential regression equation that fits the points. The multiplier is about 1. 0725 – 1 = 0. 0725 = 7. 25%