Name: Date: Eponential Growth & Decay Warm-Up: Evaluate the following eponential functions over the given domain. Graph the function over the given domain on the coordinate plane below. Determine the average rate of change over the domain. #1a. f ( ) = 2 2 ;{ 3 4} f ( ) = 2 2 f ( ) f ( ) -3-3 -2-2 -1-1 0 0 1 1 2 2 3 3 4 4 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 y 1 2 b. f ( ) = 4 ;{ 3 4} = 1 2 4 f ( ) 4 3 2 1 1 2 3 4
Vocabulary: y = a b y = a b base growth base decay The graph below shows eponential growth and eponential decay. For eponential growth, as increases, y increases eponentially. For eponential decay, as increases, y decreases eponentially. 2
The concept of eponential growth and decay when applied to word problems, normally involve a quantity (some number) growing or decaying at a constant percentage over time. Typically, eponential growth ( A = p( 1 + r) t ) focuses on populations, financial investments, or biological A = p 1 r ) is often found in the diminishing price of a product (as in the yearly loss of value of a car, called depreciation), the steady decrease in population, or the radioactive decay of isotopes. entities growing in size, while eponential decay ( ( ) t Write an eponential function to model each situation. Find each amount after the specified time. 1. Suppose the population of 10 animals quadruples every year. Write a function that would determine how many animals there would be after 6 years. 2. If a population of 100 grows by 6% a year, how large will the population be in 20 years? Write a function and determine the outcome. 3. The population of a town, which is now 6,000, increases 10% each year for five years. a. To the nearest integer, what will be the total population after 5 years? b. To the nearest integer, what will be the percent increase in population over the five year? 3
4. If the population of a town in 1920 was 1,000 and statistically seemed to double every 12 years, what should its epected population be in the year 2016? 5. Suppose you deposit $1,000 in a fund that pays 7.2% interest compounded annually. Find the account balance after 5 years. How much interest was earned? 6. A bank offers a savings account annual interest rate of 4.5% compounded monthly. The formula for simple compound interest is A = p( 1 + r) t, where A represents the final amount, p is the principal or initial investment, and r is the annual rate. If $5,000 is placed in that account and no money is withdrawn, how much money (to the nearest dollar) would be in the account after 2 years? 4
7. A piece of machinery that cost $8,000 depreciates each year by an amount equal to 10 1 A = p 1 r. To the nearest dollar, how much will it be worth at the end of the fifth year? of its value at the beginning of the year. The formula for depreciation is ( ) t 8. Suppose you deposit $1,000 in a fund that pays 7.2% interest compounded quarterly. Find the account balance after 5 years. Summary When interest is compounded quarterly (four times per year), you divide the interest rate by 4, the number of interest periods per year. To find the number of payment periods, you multiply the number of years by the number of interest periods per year. Eample: An Annual Interest Rate of 8% Compounded Periods per Year Interest Rate per Period Annually 1 8% every year Semi-annually 2 8% = 4% every 6 months 2 Quarterly 4 8% = 2% every 3 months 4 Monthly 12 8% = 0.6% every month 12 5
9) A man invests $10,000 at an annual interest rate of 5% compounded annually. What is the value of this investment after 20 years? 10) The population of a town is 300,000. It grows at an annual rate of 2.5%. What is the population of the town after 15 years? 11) The value of a car depreciates 20% every year. If the value of a new car is $30,000, what is the value of the car after 5 years? 12) $50,000 is invested at an annual interest rate of 8% compounded quarterly. What is the value of the investment after 10 years? 6
13) A couple had $200,000 in their retirement account. They set up a plan to withdraw 5% of the (remaining) balance every 3 months. What will be the balance in the account after 12 years? 14) Iodine-131 is used to destroy thyroid tissue in the treatment of an overactive thyroid. The halflife of iodine -131 is 8 days. If a hospital receives a shipment of 200 g of iodine-131, how much would remain after 32 days? 15) Mercury-197 is used for kidney scans and has a half-life of 3 days. If 32 grams of mercury-197 is ordered, but takes 15 days to arrive, how much would arrive in the shipment? 7