Modeling Exponential Functions

Warm-Up Write as a decimal. 1) 8% 2) 2.4% 3) 0.01% Evaluate. 4) 36 1) 8% 2) 2.4% 3) 0.01% Evaluate. 4) 36 5) (34)(43) 6) (24)(23)

Exponential Model The basic form of any exponential model is Where a = initial amount b = multiplier

Modeling Bacteria Growth Time (hr) 1 2 3 4 5 6 Population 25 50 100 200 400 800 1600 Write an algebraic expression that represents the population of bacteria after n hours. The expression is called an exponential expression because the exponent, n is a variable and the base, 2, is a fixed number. The base of an exponential expression is commonly referred to as the multiplier.

Example 1 Find the multiplier for each rate of exponential growth or decay. a) 9% growth 100% + 9% = 109% = 1.09 b) 0.08% growth 100% + 0.08% = 100.08% = 1.0008 c) 2% decay 100% - 2% = 98% = 0.98 d) 8.2% decay 100% - 8.2% = 91.8% = 0.918

Example 2 Suppose that you invested $1000 in a company’s stock at the end of 1999 and that the value of the stock increased at a rate of about 15% per year. Predict the value of the stock, to the nearest cent, at the end of the years 2004 and 2009. Since the value of the stock is increasing at a rate of 15%, the multiplier will be 115%, or 1.15 = $2011.36 = $4045.56

Example 3 Suppose that you buy a car for $15,000 and that its value decreases at a rate of about 8% per year. Predict the value of the car after 4 years and after 7 years. Since the value of the car is decreasing at a rate of 8%, the multiplier will be 92%, or 0.92 = $10,745.89 = $8,367.70

Practice A vitamin is eliminated from the bloodstream at a rate of about 20% per hour. The vitamin reaches a peak level in the bloodstream of 300 mg. Predict the amount, to the nearest tenth of a milligram, of the vitamin remaining 2 hours after the peak level and 7 hours after the peak level.

Compound Interest The total amount of an investment, A, earning compound interest is where P is the principal, 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 1 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.

Assignment Pg. 430, 431 #14-32 even, 44, 45, 46