Exponential Functions Using Growth and Decay Factors

Clara invests $5000 in an account that pays 6.25% interest per year. After how many years will her investment be worth $10,000? Example 2: Economics Application Step 1 Write a function to model the growth in value of her investment. f(t) = a(1 + r) t Exponential growth function. Substitute 5000 for a and for r. f(t) = 5000( ) t f(t) = 5000(1.0625) t Simplify.

Example 2 Continued Step 2 When graphing exponential functions in an appropriate domain, you may need to adjust the range a few times to show the key points of the function.

Example 2 Continued Step 3 Use the graph to predict when the value of the investment will reach $10,000. Use the feature to find the t-value where f(t) ≈ 10,000. The function value is approximately 10,000 when t ≈ The investment will be worth $10,000 about years after it was purchased.

In 1981, the Australian humpback whale population was 350 and increased at a rate of 14% each year since then. Write a function to model population growth. Use a graph to predict when the population will reach 20,000. P(t) = a(1 + r) t Substitute 350 for a and 0.14 for r. P(t) = 350( ) t P(t) = 350(1.14) t Simplify. Exponential growth function. Check It Out! Example 2

Graph the function. Use to find when the population will reach 20,000. It will take about 31 years for the population to reach 20,000. Check It Out! Example 2 Continued

A city population, which was initially 15,500, has been dropping 3% a year. Write an exponential function and graph the function. Use the graph to predict when the population will drop below Example 3: Depreciation Application f(t) = a(1 – r) t Substitute 15,500 for a and 0.03 for r. f(t) = 15,500(1 – 0.03) t f(t) = 15,500(0.97) t Simplify. Exponential decay function.

Example 3 Continued Graph the function. Use to find when the population will fall below It will take about 22 years for the population to fall below ,

A motor scooter purchased for $1000 depreciates at an annual rate of 15%. Write an exponential function and graph the function. Use the graph to predict when the value will fall below $100. f(t) = a(1 – r) t Substitute 1,000 for a and 0.15 for r. f(t) = 1000(1 – 0.15) t f(t) = 1000(0.85) t Simplify. Exponential decay function. Check It Out! Example 3