4.5 Comparing Exponential Functions

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1 4.5 Comparing Exponential Functions So far we have talked in detail about both linear and exponential functions. In this section we ll compare exponential functions to other exponential functions and also compare exponential functions to linear functions. At times this comparison can lead us to solving a system of equations involving these functions. Comparing Exponentials When comparing one exponential function to another, we might think about the possible transformations or growth factors of those functions. For example, let s compare two exponential functions on a graph as follows: Which function is growing at a faster rate? In other words, which has the higher growth factor? While for a long time is above, we can see that eventually catches up to an surpasses. So in this case, must have a higher growth factor. In fact, we can calculate what that growth factor is for each function. The function has a growth factor of two while has a growth factor of three. Which function has the higher initial value? Remembering that the initial value is when 0, we see that has the higher initial value of 3. We should also be able to compare functions in different representations in the same way. For example, consider the following three exponential functions Of these three functions, which has the fastest growth rate? We can see that has a growth rate of two. The table for is probably the next easiest because see that it is in fact shrinking meaning it can t have the highest growth rate. The graph shows us that 0 6,1 4, and 2 2 meaning that it has a growth rate of three. Therefore has the highest growth rate. Notice that has the highest initial value of four. 203

2 Comparing in Context Let s say that after graduation you begin to seek employment, and two companies offer you jobs at the same salary. The only difference between the jobs is the retirement packages offered. Company A says that if you sign a lifetime contract to work for them for 50, they will put $10,000 in a bank account for your retirement that will grow at a rate of 10% yearly. Company B says they will open up a bank account for your retirement using the following formula 20, To access the money, both companies say that you must work for them for at least 15. Here are some interesting questions to explore: Which company puts the greater amount in your bank account to begin with? Company B is putting in $20,000 while Company A is only putting in $10,000. Which company gives the higher return rate on the initial investment? Company A offers a 10% return while Company B only offers a 5% return. If you plan to only work for 10, which company should you choose? For this problem we ll evaluate each function at the time 10 from now. Let 10, be Company A s equation. We ll rewrite Company B s equation in function notation as 20, Now see that 10 $25,937 and 10 $32,578. So Company B would be the better choice if you were only going to work for the next ten. If you plan to work for the next 50, which company should you choose? Following the same pattern we see that 50 $1,173,908 and 50 $229,348. Company A is now by far the better choice. How long would it take for your retirement account to be worth $1,000,000 in each company? To solve this comparison, we ll need to graph each function. Using the below graphs, we see that Company A s retirement account will be worth a million somewhere around 48 from now. Company B s account will take around 80 to reach a million dollars. Company A Company B 204

3 After how many would the accounts have the same amount of money? For this problem we ll need to see both graphs on the same coordinate plane as follows. The first shows us that it occurs somewhere before 20, so we ll refine the viewing window to zoom in and look closer. From the second graph we see that it close to 15 from now when they will be the same. Notice that they will be both be worth about 0.04 million dollars which is $40,000. Initial View Zoomed In View Retirement accounts worth the same amount. Comparing by Solving Systems Actually, what we just did was solving a system of equations. Since we re dealing with exponential functions, it will still be easiest to graph the systems and look for the point of intersection. Knowing how to graph, we can now compare not only exponentials to themselves but also an exponential to a linear equation. For example, we might be asked to find the solution to the following system of equations Notice there are two points of intersection: 4,8 and 0, 7. Will there always be two points of intersection? Could there be one or no points of intersection? How? 205

4 Lesson 4.5 Your financial advisor presents you with four plans for retirement as follows. All dollar amounts are given in millions of dollars. For example,. million is really $,. Answer the following questions about those retirement plans. : Put in an initial investment of $0.025 million and get a return rate of 5%. : : Years Money : int: 0, year: 1, : 2, List the retirement plans from the highest growth rate to the lowest growth rate. 2. List the retirement plans from the lowest initial investment to the highest initial investment. 3. How long will it take each retirement plan to be worth $1,000,000? (Hint: You will have to graph each plan.) 4. Fill out the following table evaluating each plan at specific points in time. Retire after 20 Retire after 30 Retire after 40 Retire after Which plan do you think is the best? Why do think that? What aspect of the function makes it the best retirement plan? 206

5 6. Fill out the following table showing when each plan is worth the same as every other plan in the future. If they are not equal to each other in the future (only in the past), then put for no solution. (Hint: You will have to graph each plan on the same coordinate plane.) You are deciding between different amounts of student loans and your college presents you with four possible plans each with different rates at which the loan is paid off. All dollar amounts for the remaining debt are given in thousands of dollars. For example,. thousand is really $,. Answer the following questions about those student loan plans. : Take out $250 thousand and have a payoff rate of 10%. : : int: 0, year: 1, : 2, : Years Money List the student loan plans from the fastest payoff rate to the slowest payoff rate. 8. List the student loan plans from the lowest initial debt to the highest initial debt. 9. How long will it take each student loan plan to be paid down to $1,000? (Hint: Graph each plan or guess and check.) 207

6 10. Fill out the following table evaluating each plan at specific points in time. after 10 after 15 after 20 after Which plan do you think is the best? Why do think that? 12. Fill out the following table showing when each plan is worth the same as every other plan in the future. If they are not equal to each other in the future (only in the past), then put for no solution. (Hint: You will have to graph each plan on the same coordinate plane.) Solve the following systems of equations

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