4.5ComparingExponentialFunctions
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 F, we can see that
eventually F catches up to an surpasses . So in this case, F 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 F 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.
21
F
()
−2
7
−1
5
0
4
1
3.5
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 F0 6, F(1) = −4, and F(2) = 2 meaning that it has a growth
rate of three. Therefore F() has the highest growth rate.
Notice that () has the highest initial value of four.
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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 years, 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 4 20,0001.05¦ . To access the money, both companies say that you must work for them
for at least 15 years.
Here are some interesting questions to explore:
•
•
•
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 years, which company should you choose? For this problem we’ll evaluate
each function at the time 10 years from now. Let P) 10,000(1.1)¦ be Company A’s equation. We’ll
rewrite Company B’s equation in function notation as Q()) = 4 = 20,000(1.05¦ . Now see that P10 S
$25,937 and Q10 S $32,578. So Company B would be the better choice if you were only going to work
for the next ten years.
If you plan to work for the next 50 years, which company should you choose? Following the same pattern
we see that P50 S $1,173,908 and Q50 S $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 years from now. Company B’s account
will take around 80 years to reach a million dollars.
Company A
Company B
$ in Millions
•
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.
$ in Millions
•
Years from Now
Years from Now
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•
After how many years 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 years, so we’ll refine the viewing window to zoom in and look closer. From the second graph we
see that it close to 15 years 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
$ in Millions
$ in Millions
Retirement accounts
worth the same amount.
Years from Now
Years from Now
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.
1 1
Š ‹ 8
2
F 15
7
4
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?
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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, V. Vi million is really $i, VVV. Answer the following questions about those
retirement plans.
Plan A:
-int: 0, 0.05
Plan D:
Put in an initial investment of $0.025 million and get a
return rate of 5%.
Plan B:
1 year:
1, 0.06)
F) 0.011.15
¦
2 years:
(2, 0.072
Plan C:
Years
0
1
2
3
4
Money 0.02 0.022 0.0242 0.02662 0.029282
1. 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
years
Retire after 30
years
Retire after 40
years
Retire after 50
years
Plan A
Plan B
Plan C
Plan D
5. Which plan do you think is the best? Why do think that? What aspect of the function makes it the best retirement
plan?
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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.)
Plan A
Plan B
Plan B
Plan C
Plan C
Plan D
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, V. i thousand is really $V, iVV. Answer the following questions about
those student loan plans.
Plan A:
Plan D:
-int: 0, 300)
Take out $250 thousand and have a payoff rate of
10%.
1 year: (1, 240)
Plan B:
2 years: (2, 192)
Plan C:
Years
Money
0
150
F) 1000.85¦
1
135
2
121.5
3
4
109.35 98.415
7. 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 years.)
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10. Fill out the following table evaluating each plan at specific points in time.
Remaining debt
after 10 years
Remaining debt
after 15 years
Remaining debt
after 20 years
Remaining debt
after 25 years
Plan A
Plan B
Plan C
Plan D
11. 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.)
Plan A
Plan B
Plan B
Plan C
Plan C
Plan D
Solve the following systems of equations.
13.
21… 9
F 2 3
14.
1…
AB
F C
9
2
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15.
17.
214
21
F 2 2
1˜D
ACB
F 4
D
16.
7
18.
31˜D 7
F 4 14
41 2
F 0 2