1. Exponential Functions Topic 3: Applications of Exponential Functions

2. Solve problems that involve the application of exponential equations to loans, mortgages, and investments. Graph data, and determine the exponential function that best approximates the data.

3. Explore… The rate of depreciation of a new vehicle will vary depending on the make and model of the vehicle. New cars depreciate about 20% the moment they are driven off of the lot. The accepted average car depreciation rate is roughly 15% per year after the first year. Jodi bought a new car one year ago, and today it is worth $20 000. Assume a constant depreciation rate of 15% /a. Interest rates are usually expressed as a percent per annum (/a).

4. Explore… 1. Calculate the amount that Jodi’s car will depreciate over the next year. It will depreciate by $3 000 in the next year. 2. Using the amount from question 1, calculate the value of Jodi’s car one year from today. It will be worth $17 000 one year from today.

5. Explore… 3.Explain how to calculate the value from question 2 using only one step. depreciation so 1- 0.15 = 0.85 of the value remaining 4. Show how to calculate the amount that Jodi’s car will be worth a. two years from today b. three years from today

6. Explore… 5.Determine an equation that you can use to calculate the value of the car x years from today. a is the initial value of the car b is the common ratio, depreciation so 1- 0.15

7. Information The general form of exponential equations is y represents the final amount, a represents the initial amount, b represents the common ratio, and x represents the input or independent variable

8. Information

9. There are many real-life applications of exponential equations. Some of the most common are listed: Compound interest is the interest earned on both the original amount of money that was invested and any interest that has accumulated over time. It is sometimes referred to as earning interest on top of interest. When money earns interest or is charged interest, an exponential function can be used to model the situation. The formula for compound interest is where A(n) is the future value, P is the principal, i is the interest rate per compounding period, and n represents the number of compounding periods.

10. Information The interest rate per compounding period, i, is the interest rate, expressed as a decimal, that is charged or earned during one compounding period. The compounding period, n, is the time over which interest is calculated and paid on an investment or loan. Some common compounding periods are daily, weekly, monthly, quarterly, semi-annually, and annually. Interest is calculated at the end of every compounding period and added to the balance. The new total becomes the starting value for the next period.

11. Information The half-life exponential function is used to calculate exponential decay using the half-life (the amount of time it takes something to decrease to half of the original amount). It is represented by the function where A(t) represents the final amount, A o represents the initial amount, t represents the time, and h represents the half life.

12. Information The doubling period exponential function is used to calculate exponential growth using the doubling period (the amount of time it takes something to double fro its original amount). It is represented by the function where N(t) represents the final amount, N o represents the initial amount, t represents the time, and d represents the doubling period.

13. Example 1 The number of movie tickets that are sold in Edmonton on Friday night is increasing at a rate of 5% each week. In week 0, 2500 tickets were sold. The number of tickets sold, N(t), after t weeks can be represented by an exponential equation. a) How many tickets, to the nearest whole ticket will be sold in week 12? Determining and solving an equation in context An increase by 5% each week means that the constant ratio (b) is 1.05. The initial population (y-intercept or a-value) is 2500. 12 weeks later, so x = 12. Solve for y.

14. Example 1 b) In how many weeks, to the nearest tenth, would Friday night ticket sales reach 5000 tickets? Determining and solving an equation in context Divide both sides by 2500. There is no common base, so I must solve graphically! Graph left side in Y1 and right side in Y2. Window X: [0, 20, 2] Y: [0, 1, 0.2] Using 2 nd Trace 5: intersect, x = 14.2 weeks

15. Example 2 When diving underwater, the light decreases by 10% for every metre that the diver dives. On a sunny day off the coast of Vancouver Island, a diving team recorded a 100% visibility at the surface. a) Write the exponential equation that represent this situation where V(d) is the visibility at any given depth, d, in metres. Determining and solving an equation in context Initial visibility (at 0m below the surface). Visibility decrease by 10% per metre, so rate of decrease is 1 – 0.10.

16. Example 2 b) At what depth with the visibility be half the visibility at the surface? Round to the nearest tenth. The visibility at the depth we are finding is 50% (half of 100%), so this is our V(d) value. Now we solve for d. Since the two sides cannot be written with a common base, solve graphically. Enter left side into Y1 and the right side into Y2. Solve for the x-value of the intersection point. Depth: 6.6 metres

17. Example 3 Olga invested $8 000 in an investment for 5 years compounded annually at a rate of 7.25%/a. a) Algebraically determine the value of her investment at the end of the term. Determining the value of an investment that is compounded annually Interest rates are usually expressed as a percent per annum (/a).

18. Example 3 b) Calculate the interest earned on her investment. Since she started with $8000 and ends up with $11352.11, she has earned $3352.11 in interest (11352.11 – 8000).

19. Example 4 Jarrod invests $5000. The bond he chooses has an annual appreciation rate of 2.8%/a, compounded annually. a) How long, to the nearest tenth of a year, will it take for his investment to double? Determining the initial investment Appreciation is the increase in the value of an asset over time. The accumulated amount (A) is 10 000. The interest rate per period (i) is 0.028 The principal is $5000. Solve for n.

20. Example 4 Continued… Determining the initial investment Appreciation is the increase in the value of an asset over time. Since the two sides cannot be written with the same base, solve graphically. Enter left side into Y1 and the right side into Y2. Solve for the x-value of the intersection point. His money reaches a value of $10 000 after 25.1 years.

21. Example 4 b) Suppose the loan was instead compounded monthly. i.Determine how long it will now take for the investment to double in value, to the nearest month. Use the equation where x is the number of months. Determining the initial investment Appreciation is the increase in the value of an asset over time. The accumulated amount (A) is 10 000. The interest rate per period (i) is The principal is $5000. Solve for x.

22. Example 4 ii. How long will it take in years, to the nearest year? Determining the initial investment Appreciation is the increase in the value of an asset over time. Since the two sides cannot be written with the same base, solve graphically. Enter left side into Y1 and the right side into Y2. Solve for the x-value of the intersection point. His money reaches a value of $10 000 after 297 months. Continued…

23. Example 4 Determining the initial investment Continued… c) Explain how changing a compounding period affects how an investment will grow. As the compounding period increases the length of time for the investment to grow decreases.

24. Example 5 When an animal dies, the amount of radioactive carbon-14 (C-14) in its bones decreases. Archaeologists use this fact to determine the age of a fossil based on the amount of C-14 remaining. The half-life of C-14 is 5 730 years. Head-Smashed-In Buffalo Jump in southwestern Alberta is recognized as the best example of a buffalo jump in North America. The oldest bones unearthed at the site had 49.5% of the C-14 left. Use the formula, where t is the number of years, and y is the remaining percentage of C-14, to determine the age of the bones when they were found, to the nearest year. Using half life to solve a contextual problem

25. Example 5 Using half life to solve a contextual problem I can solve graphically! Graph left side in Y1 and right side in Y2. Window X: [0, 7000, 500] Y: [0, 100, 10] Using 2 nd Trace 5: intersect, x = 5813 years.

26. The general form of exponential equations is y represents the final amount, a represents the initial amount, b represents the common ratio, and x represents the input or independent variable Need to Know

28. Compound interest is the interest earned on both the original amount of money that was invested and any interest that has accumulated over time. It is sometimes referred to as earning interest on top of interest. When money earns or is charged interest, an exponential function can be used to model the situation. The formula for compound interest is

29. Need to Know For,

30. Need to Know interest earned = final closing balance – total amount invested (commonly referred to as profit) interest paid = total amount paid – amount borrowed The higher the number of compounding periods, the faster an investment will grow and the faster the interest will accumulate. Appreciation is the increase in the value of an asset over time. Depreciation is the decrease in the value of an asset over time. You’re ready! Try the homework from this section.