4.4SolvingExponentialFunctions
Before we can solve exponential functions, we need to make sure we can create an equation for any given
form of an exponential function including a graph, description, or table.
Creating an Equation from a Graph or Table
For our purposes, we will only be looking for a single transformation at a
time. When given the graph of an exponential function, the first thing to look for
would be the asymptote which will tell us if the graph has been translated up or
down. For example, consider the graph to the left. It has an asymptote at a
height of 4. So our function must be of the form () = @ 1 + 4. The only
remaining question is what the @ value is.
To determine this, we look for the growth factor. Notice that we get the
below table of values. The growth rate is going up by half, then one, then two.
This is a growth factor of two yielding the equation of () = 21 + 4.
()
−1
4.5
0
5
1
6
2
8
If we were just given a table, we would create the equation the same way. Look for the asymptote, check
for any other translation or stretching, and find the growth factor. For example, consider the following table.
()
2
3
1
1
0
1
3
1
1
9
2
1
27
It appears that the asymptote of this table is at a height of zero
because moving to the right on the table keeps getting systematically
C
closer to zero. The growth factor is clearly , so the only question is
whether there is a horizontal translation.
1
If we look at the function () = ACB and plug in = 0, we get that () = 1. Notice that we do not get
the value () = C which we see in the table. Therefore there must be a horizontal translation that moved that
1˜
value of one to the left one. This means our final equation is () = ACB
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.
Creating an Equation from a Description
More often than not, if we need to solve an exponential equation we will first be given a description of a
real world situation for which an exponential function is an appropriate model. In fact, the most common model
used is what is known as the general growth formula. The general growth formula is:
q = P(1 + 7¦
For this formula, q represents the new amount of whatever we’re talking about after the growth occurs.
The P represents the original amount of whatever we’re talking about. The 7 represents the growth rate per time
unit expressed as a decimal, and ) represents the amount of time that the growth has occurred.
For example, we know the population of the planet is growing by about 1.2% per year and is currently
about 7 billion. That would give us the following equation (knowing that the population would be in billions).
q = 71 + 0.012¦
or
q = 71.012¦
Let’s say that you had $50,000 invested in a stock that had been losing 3% per month. That would give
us the following equation:
q = 50,0001 − 0.03¦
or
q = 50,0000.97¦
Solving Exponential Equations
Now that we know how to create exponential equations, we can solve them. In some cases we will simply
be evaluating the function, while in others we will actually have to solve. Let’s begin with some evaluation
examples. Using the world population information from above, what would we expect the population to be in ten
years? What was the population twenty years ago? In both of these cases we are substituting values in for ).
Ten years from now
q = 7(1.012¦ = 71.0120 S 7.9 @;HH;6<
Twenty years ago
q = 7(1.012¦ = 71.012…0 S 5.5 @;HH;6<
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If we are truly solving, at this point the best method is to estimate a solution on a graph. For example, we
might ask the question of when the population of the world will hit 10 billion people. We could guess and check
several values (10 years from now, 20 years from now, 25 years from now, etc.), but that could take a long time.
Instead let’s use some technology to quickly produce a graph for us as follows:
Notice that when the population is about ten billion people, it is about
thirty years from now. When would we hit nine billion people? In about twentyone years. When would we hit eleven billion people? In about thirty-seven
years. The graph provides us a quick way to solve the exponential equation
without having to use inverse operations to isolate the variable.
While we’re looking at this graph, let’s take a quick side trip to relate the
domain to the graph. The domain in this case is time itself, so the portion of the
domain shown by this graph is mainly the future, fifty years of the future to be
precise. Notice that we’re not inputting the actual year that it is, but we consider
this year to be the initial value and then work backwards or forwards from the current year as necessary.
Let’s change our view of this function to look at a domain more about the past and estimate when the world
population was about 5.5 billion. We know from our evaluation earlier that it should be around twenty years ago,
but see that the graph confirms this.
Here is the same graph with a different viewing window. When was the
population about 5.5 billion? It appears it was indeed around twenty years ago.
In this case, the negative values in the domain correspond to years in the past.
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Lesson 4.4
Create an exponential function for the given graph, table, or description.
1.
2.
3.
4.
5.
6.
7.
8.
9.
−2
() 0.25
10.
2
3.25
−1
0.5
1
3.5
0
1
0
4
1
2
1
5
11.
−1
3
1
5.5
0
9
0
5
1
27
1
4
2
81
2
2
0
3.5
12.
1
2
1
3
2
2
3
0
0
1
1
0.5
2
0.25
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13. Your parents offer to pay you exponentially to study for your Algebra test. They say that if you study for one
hour you’ll get $6, two hours gets you a total of $7, three hours $9, four hours $13, etc. What equation are they
using to come up with those values?
14. In the game Fruit Slicer you get more points if you can slice multiple fruits in a single swipe. If you only get one
fruit in a swipe you get 10 points, two fruits in a swipe gives you 50 points, three fruits gives you 250 points, four
fruits 1250 points, etc. What equation does Fruit Ninja use to assign point values?
15. In general the stock return has averaged about 10% growth per year since its beginning. This is true even
through the Great Depression! You have been investing in the stock market for a while, and your stock is currently
worth $100,000. What equation models this situation?
16. Viruses spread exponentially. In fact, if you start with just 1 person infected with the zombie virus, after one
day a total of 10 people would be zombies. After two days a total of 100 people would be zombies. After three
days a total of 1000 people would be zombies. What equation models this situation?
17. India’s current is approximately 1.2 billion, almost as big as China! If India’s population is growing at a rate of
2% per year, what equation would model their population growth?
18. There are approximately 25,000 polar bears worldwide. Some scientists estimate that the polar bear
population has been declining at a yearly reduction rate of 1%. What equation would model this situation?
Solve the following problems using the polar bear example above.
19. Approximately what will the population of polar bears be in 20 years?
20. Approximately what will the population of polar bears be in 50 years?
21. Approximately what was the polar bear population 20 years ago?
22. Approximately what was the polar bear population 50 years ago?
23. Using technology, graph the polar bear equation. Approximately when was the polar bear population about
100,000 bears?
24. Using technology, graph the polar bear equation. Approximately when will the polar bear population be less
than 10,000 bears?
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Solve the following problems using the given investment information.
A man currently has stock worth a total of $5,000,000. It has been growing at an average rate of 10% per year.
25. What equation would model this situation?
26. If the man started investing 60 years ago and did not put any more money in, how much did he start with?
27. Assume the man was 20 years old when he put in that initial amount. Due to the miracles of technology this
80 year old man is still healthy and active enough to work. If he can work until he is 90 years old, how much will
his investment be worth then?
28. What domain would make sense for this problem? Why?
29. Using technology, graph the man’s investment equation. Approximately when was his investment worth about
$1,000,000?
30. Using technology, graph the man’s investment equation. Approximately how old would the man be when his
investment would be worth greater than $8,000,000?
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