Double Your Money Your math teacher believes that doing assignments consistently will improve your understanding and success in mathematics. At the beginning of the year, your parents tried to encourage this behavior by offering you the following incentive. ~ They would give you 5 cents for signing the contract. ~ After the first completed assignment, they would give you 10 cents ~ After each day of completing assignments, they would give you twice the amount of money that they gave you the day before. ~ If you don t do an assignment, you have to start over. 1. At first glance, does this incentive seem like a good deal? 2. Let x be the number of assignments completed. Complete the table. Remember: 2 0 = 1 # of Assign. Incentive (in cents) Pattern Using exponents 0 (initial) 5 5 1 5 2=10 5 2 2 10 2 = 5 2 2 5 2 2 3 4 x 3. Write a function equation for the amount of incentive given after x consecutive assignments are completed. f(x) = 4. Put the function in Y1 in your calculator and use the table to determine the amount your parents would pay for the 10 th completed assignment. 5. How much would your parents pay for the 30 th completed assignment? Is this surprising? 1
6. Set the window Xmin =0, Xmax = 10 and Zoom Fit. Sketch the graph below and label the vertical intercept. (Trace to X=0 to see the value) 7. How is the vertical intercept related to the function equation? 8. Why is there a 2 in the function equation? 9. Determine the practical domain and the practical range in this situation. Practical domain: Practical range: 10. How is this graph different from the graph of a quadratic function? Leave your equation in Y1 and enter Y2 = x 2 in your calculator. Investigate the graphs using a window setting of Xmin= 5, Xmax=5, Ymin = 0, Ymax = 50. Sketch and label the graphs: 2
New Terminology The type of function that we have just developed is called an exponential function. This is the type of function we will be investigating in this unit. The standard form of an exponential function is f(x) = a b x a is the vertical intercept of the graph b is the growth (or decay) factor 11. Suppose in the homework payment scenario your parents had initially given you 10 cents for signing the contract, and still agreed to double your money each day. a) Write a function equation for this situation. g(x) = b) Keep the original function in Y1. Y1= 5 * 2 x. Put your new equation in Y2. Set the window Xmin =0, Xmax = 5, Ymin = 0, and Ymax = 100. Sketch the graphs below and label the vertical (y) intercepts. (Trace to X=0 to see the values) f (x) = 5 2 x y-int = g(x) = y-int = 12. Suppose in the homework payment scenario your parents initially give you 2 cents for signing the contract, but agree to triple your money each day. a) Write a function equation for this situation. h(x) = b) Keep the original function in Y1. Y1= 5 * 2 x. Put your new equation in Y2. Set the window Xmin =0, Xmax = 5, Ymin = 0, and Ymax =100. Sketch the graphs below and label the vertical intercepts. Identify the growth factor in each. Which graph grows faster? f (x) = 5 2 x growth factor = h(x) = growth factor = grows faster 3
13. For each of the equations below, determine the vertical (y) intercept and the growth factor. a) f (x) = 15 4 x y- int = growth factor = x b) g(x) = 1.6 ( 5 ) y- int = growth factor = c) h(x) = 100(3) x y- int = growth factor = d) f (x) = 2 x y- int = growth factor = 14. For each set of equations below, circle the one with the faster growing graph. a) f (x) = 4 3 x or g(x) = 3 4 x b) f (x) = 2 10 x or g(x) = 100 2 x 15. Write an equation for each scenario. a) The initial amount of money in an account is $10 and its value triples every year. Write a function equation for the amount of money, m(x), after x years. m(x) = b) The initial population of a city is 15,000 and it doubles every year. Write a function equation for the population, p(x), after x years. p(x) = c) The number of people infected with a virus is currently 250, and that number quadruples every week. Write a function equation for the number of infected people, v(x), after x weeks. v(x) = d) Average computer processing speed is around 1.8 ghz and doubles every year. Write a function equation for the processing speed, s(x), after x years. s(x) = 4
Metabolizing Medicine When you take medicine, your body metabolizes and eliminates the medication until the amount is so small, it cannot be measured. Assume you are given a 100 mg dose of medication and that your body eliminates ¼ of the medicine every hour. 1. Estimate the number of hours it will take for your body to eliminate the medication from your bloodstream. (Just guess.) 2. How many mg should remain after 1 hour? 3. What number can be multiplied by 100 to get your answer to #2? This number is the decay factor. 4. When you are given the fraction that is removed, how can you find the fraction that remains? 5. Let x be the number of hours that pass. Complete the table: hours Medication Amt. Pattern Using exponents 0 (initial) 100 100 1 100 = 75 100 2 75 = 100 100 2 3 x 6. Write a function equation for the amount of medication that remains in the bloodstream after x hours. m(x) = 7. Put the function in Y1 in your calculator and determine the number of hours it takes to have less than 1 mg in the bloodstream. 5
8. Set the window Xmin =0, Xmax = 15 and Zoom Fit. Sketch the graph below and label the vertical intercept. 9. Is there an x intercept? According to the equation, the amount of medication will never actually be zero, but the value will get closer to zero as time goes on. New Terminology A horizontal asymptote is a horizontal line that a graph approaches as the input values get very far from 0 (in a positive or negative direction). For exponential functions of the form f(x) = a b x, the horizontal asymptote is the x axis. 10. What is the equation for the horizontal asymptote for this graph? 11. Determine the practical domain and the practical range in this situation. Practical domain: Practical range: 12. Suppose that in the medication scenario you were given an initial dose of 50 mg. a) Write a function equation for this situation. g(x) = 6
b) Keep the original function in Y1. Y1 = 100(0.75) x. Put your new equation in Y2. Set the window Xmin =0, Xmax = 20, Ymin = 0, and Ymax = 100. Sketch the graphs below and label the vertical intercepts. Also label the horizontal asymptote and write its equation. f (x) = 100(0.75) x y-int = g(x) = y-int = horizontal asymptote c) Use your calculator table or graph to find the amount of medication remaining after 20 hours. f(20) = g(20) = 13. Suppose that in the medication scenario, your initial dose was 100 mg, but your body metabolized 1/10 of the medicine every hour. a) If you lose 1/10 of the medicine every hour, what fraction of the medicine remains in your body after one hour? (Write your answer as a fraction and a decimal. This is the decay factor) b) Write a function equation for this situation. h(x) = c) Keep the original function in Y1. Y1 = 100(0.75) x. Put your new equation in Y2. Set the window Xmin =0, Xmax = 20, Ymin = 0, and Ymax = 100. Sketch the graphs below and label the vertical intercepts. Label the horizontal asymptote. Also write the decay factors. Which decays faster? f (x) = 100(0.75) x decay factor= g(x) = decay factor = decays faster 7
14. For each set of equations below, circle the one with the faster decaying graph. a) f (x) = 4(0.8) x or g(x) = 8(0.2) x b) f (x) = 0.1(0.1) x or g(x) = 0.9(0.9) x 15. We said before that the standard form of an exponential function is f(x) = a b x where a is the vertical intercept of the graph and b is the growth (or decay) factor. Examine the examples that we have had so far and determine the factor and whether or not the graph is growth or decay. Example Equation Factor Growth or Decay? Pg 1 # 3 Pg 3 # 11 Pg 3 # 12 Pg 5 # 6 Pg 6 # 12 Pg 7 # 13 How can you determine whether the factor is a growth factor or a decay factor? 16. Put the following equations in your calculator: Y1 = 5(2) x Y2 = 5(0.5) x Set the window Xmin = 5, Xmax = 5, Ymin = 0, and Ymax = 50. Sketch the graphs below. Label the y intercepts and the horizontal asymptotes. Which equation shows growth? Which equation shows decay? Growth equation: Growth Factor: Decay equation: Decay factor = 17. Determine whether each equation will be exponential growth or exponential decay. a) f (x) = 0.5(3) x b) g(x) = 5(0.3) x c) h(x) = 0.8(0.95) x d) f (x) = 1000(1.06) x 8
Factor vs. Rate When modeling situations that grow or decay exponentially, sometimes we talk about a factor for example, when the values double we know to multiply by a factor of 2. But when we describe the values as increasing or decreasing by a percentage rate for example, the values increase by 5%, by what factor should we multiply? We need to figure out how factor and rate are related. 1. When something doubles, by what percent is it increasing? If we keep 100% and increase by 100%, we re really multiplying by 200%, which equals 2. 2. Practice converting from percent to decimal and from decimal to percent: Percent Decimal 20% 1.5% 0.04 0.5 300% 1.25 3. If we are increasing a value by 5%, we are keeping 100% and adding 5% for a total of 105%. But the factor should be in decimal form, not percent form. By what factor should we multiply? 4. So, to find the growth factor from the growth rate we add 100% and change to a decimal. Perhaps it is easier to use the formula: 1 + growth rate(in decimal form) = growth factor or 1 + r = f Where did the 1 come from? 9
5. Complete the table: Growth rate Growth rate in decimal form Growth factor 20% 3.5% 100% 1.025 6. When a quantity decreases by a certain % rate, we call that the decay rate. If a quantity loses ¼ of its value, what is the decay rate? 1.44 2.34 7. Remember that when the quantity lost ¼ of the value, we used the fraction that was remaining (¾) as the factor. This was found by subtracting the fraction from 1. We will do the same thing with the decay rate. To find the decay factor: 1 decay rate (in decimal form) = decay factor or 1 r = f How is this different from the formula relating growth rate and growth factor? 8. Complete the table: Decay rate Decay rate in decimal form Decay factor 20% 3.5% 15.8% 9. Look back at the tables for growth rate/factor and decay rate/factor. There is a way to tell whether a factor represents growth or decay. How can you tell? 0.8 0.36 0.04 10
Summary The standard form of an exponential function is f(x) = a b x a is the vertical intercept of the graph b is the growth (or decay) factor If b > 1, b is a growth factor and can be found using: b = 1 + r (where r is the growth rate in decimal form) If 0 < b < 1, b is a decay factor and can be found using: b = 1 r (where r is the decay rate in decimal form) 10. A town population is currently 15,000 and is increasing by 2% per year. a) Change the rate to a decimal. b) What is the growth factor? c) Write a function to model the population x years from now. P(x) = d) What would the population of the town be after 10 years? 11. A town population is currently 15,000 and is decreasing by 2% per year. a) Change the rate to a decimal. b) What is the decay factor? c) Write a function to model the population x years from now. P(x) = d) What would the population of the town be after 10 years? 11
12. When an object increases in value by a certain rate, it is said to appreciate. When an object decreases in value, it is said to depreciate. a) Your car has a current value of $25,000 and its value is depreciating by 20% per year. Write a function equation for the value of your car after x years, and find the predicted value after 5 years. V(x) = V(5) = b) Your rare coin has a current value of $200 and its value is appreciating by 5% per year. Write a function equation for the value of your coin after x years, and find the predicted value after 5 years. V(x) = V(5) = 12
Compound Interest When you deposit money into a savings account, it earns interest at a certain yearly percentage rate. But usually that interest is added several times each year, so our standard exponential function form will not be useful here. In order to calculate the accumulating value of the account we need a special formula: Compound Interest Formula ( ) nt A = P 1+ r n A is the accumulating amount in the account P is the principal (or original amount deposited) r is the annual interest rate (in decimal form) n is the number of times per year that interest is compounded, and t is the time in years the money is invested If interest is compounded ( ) Quarterly, n = Semi- annually, n = Monthly, n = Daily, n = Annually, n = 1. You invest $100 at 4% annual interest compounded monthly for 5 years. a. Find the value for each variable in the formula: P = r = n = t = b. Substitute all of the values in the formula and evaluate with your calculator. A = c. Which variable would change if interest were compounded daily? d. Rewrite the equation to find the amount accumulated if you invest $100 at 4% annual interest compounded daily for 5 years and evaluate it. A = e. How does changing n (the number of compoundings per year) affect the accumulated value? 13
2. Find the amount accumulated in each scenario: a) $5000 is invested at 3.5% annual interest compounded quarterly for 10 years. b) $400 is invested at 8% annual interest compounded annually for 5 years. c) $100,000 is invested at 4.25% annual interest compounded semi- annually for 30 years. 3. Suppose you deposit $10,000 into an account that has a 4.5% annual interest rate (usually referred to as APR, for annual percentage rate), and whose interest is compounded quarterly. a. Substitute the appropriate values for P, n, and r into the formula to get the accumulate amount, A, as a function of t. A(t) = b. Put the function in Y1 in your calculator. Complete the table. t 0 10 20 A(t) c. Sketch a graph of the function over the first 20 years. Plot and label the points from the table above. d. Does the graph look exponential? e. Use your table to determine the number of years it will take for the investment to triple. 14
4. Keep the equation from #3 in Y1 in your calculator. Y1 = In Y2, put a function that will model the growth of $10,000 invested at 4.5% annual interest compound monthly. (Note that we only change the value of n.) Y2 = In Y3, put a function that will model the growth of $10,000 invested at 4.5% annual interest compound daily. Y3 = a. Complete the table below: t Quarterly, n = 4 Monthly, n = 12 Daily, n = 365 0 10 20 b. Compare the 3 columns. Do the numbers compare as you expected? Explain. c. Is there a larger difference between the quarterly and monthly columns, or monthly and daily columns? 15
It turns out that as the number of compoundings increase, the increase in interest accumulation levels off. There is a formula for this: Continuously Compounded Interest Formula ( A = Pe rt ) A is the accumulating amount in the account P is the principal (or original amount deposited) e is a constant approximately equal to 2.7183 (use 2 nd LN on your calculator) r is the annual interest rate (in decimal form) t is the time in years the money is invested d. Find the amount that accumulates when 10,000 is invested at 4.5% interest compounded continuously for 20 years. How does this compare? A = 5. Find the amount accumulated in each scenario: a) $5000 is invested at 6.25% annual interest compounded continuously for 10 years. d) $50,000 is invested at 2.5% annual interest compounded continuously for 30 years. 6. Historically, investments in the stock market have yielded an average rate of 11.7% per year. Suppose you invest $1,000 in an account at an 11.7% annual interest rate compounded continuously. a. Write a function for the accumulated amount in terms of t years. A(t) = b. Use your calculator table to determine the time it takes for the investment to triple in value. c. Determine the accumulated amount after 35 years. 16