2003-2004
Advanced Functions and Modeling
A Guide for Graphing Exponential Functions and Exploring e
Graphing (2 days/revisited when you do applications)
x
In general, exponential functions are of the form f ( x ) = a ib . Students may already be familiar with the
x
graph of f ( x ) = 2 , so you can use it as a place to begin your conversation. Have students sketch a
graph of this function by hand and describe its characteristics.
They should notice that the graph of
f ( x) = 2
x
is asymptotic to the x -axis and passes through the
points (0,1) and (1, 2) . This is an increasing function with a domain of all real numbers and a range of
y > 0 . The function is also concave up.
Now investigate changes in the graph of f ( x ) = a ib
x
as the base, b , changes and a = 1 . Initially, use
values of b > 1 then values of 0 < b < 1 . The only characteristic that changes is whether the function is
increasing or decreasing. If b > 1 , the graph increases. If 0 < b < 1 the graph decreases. For example,
x
1
x
.
have students graph, again by hand, f ( x ) = 3 and g ( x ) =
3
⎛1⎞
The function f ( x ) = 3 is increasing and the function g ( x ) = ⎜ ⎟
⎝3⎠
x
x
is decreasing. In fact, it seems that g ( x) is the function that results
from f ( x ) being reflected about the y -axis. Why would this be
true? Some algebraic manipulation can help us figure this out.
x
x
1
1
−x
−x
= 3
= 3 . If we write g ( x ) = 3 , we can see that it is a
3
x
reflection about the y -axis of f ( x ) = 3 .
( )
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Practice Problems. Sketch by hand a graph of each of the following functions. Label asymptotes and
important points such as intercepts and points that correspond to (1, b) . Check your graphs with the
calculator. [Answers shown to the right of the equation.]
x
⎛2⎞
a. y = ⎜ ⎟
⎝3⎠
b. y = 4
x
x
⎛3⎞
c. y = ⎜ ⎟
⎝2⎠
⎛1⎞
d. y = ⎜ ⎟
⎝5⎠
−x
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Next, fix a value for the base, such as 2, and change the value of the leading coefficient, a , noting the
changes. The value of a determines how slowly or quickly the function vertically increases or decreases.
x
For example, if a = 4 , then g ( x ) = 4 ⋅ 2 increases 4 times as quickly in a vertical direction as the function
x
f ( x) = 2 .
On the other hand, g ( x ) =
1
1 x
x
as quickly in a vertical direction as the function f ( x ) = 2 .
⋅ 2 increases
5
5
Consider functions of the form f ( x ) = b
x+d
. For example, examine g ( x ) = 2
x+3
x
together with f ( x ) = 2 .
Recognizing that the horizontal asymptotes are the same and comparing the relationship between the
x
point (1, 2) from f ( x) and the point ( −2, 2) from g ( x) should help you see the function f ( x ) = 2 has
x+3
been shifted 3 units to the left to arrive at the function g ( x ) = 2
. In general, if d > 0 , the function shifts
left d units. If d < 0 , the function shifts right d units. An interesting algebraic observation is that
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g ( x) = 2
x+3
x 3
x
= 2 ⋅ 2 = 8 ⋅ 2 . This means g ( x ) could be thought of as a shift to the left 3 units of the
function f ( x) or g ( x) could be thought of as increasing vertically 8 times as fast as the function f ( x) .
Practice Problems. Sketch by hand a graph of each of the following functions. Label asymptotes and
important points such as intercepts and points that correspond to (1, b) . Check your graphs with your
calculator. [Answers shown to the right of the equation.]
x−1
a. y = 2
b. y = 2 ⋅ 3
⎛1⎞
c. y = ⎜ ⎟
⎝4⎠
x
x+1 2
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⎛1⎞
d. y = 3 ⋅ ⎜ ⎟
⎝2⎠
e. y =
x
1 x
⋅4
2
f. y = 3
x+2
g. y = 0.2 ⋅ 4
−x
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x
You may also want to study functions of the type f ( x ) = a ib + c . If so, also investigate the changes in
x
1
1x
the graph as the value of c changes. Comparing g ( x ) =
− 2 to f ( x ) =
is another good example.
2
2
In general, if c > 0 , the function moves up c units. If c < 0 , the function moves down c units. When
graphing by hand, it is important to show the asymptote to indicate the vertical shift.
x
In summary, for the function f ( x ) = b , some of the generalizations we hope they see would be that the
function is always asymptotic to the x -axis and the graph passes through the points
(0,1) and (1, b) . If
the value of b > 1 , the graph is increasing. If the value of b < 1 , the graph is decreasing. For the
x
function f ( x ) = a ib + c , the value of c determines the horizontal asymptote. If b > 1 the function is
increasing. If 0 < b < 1 the function is decreasing. If c > 0 the function moves up c units. If c < 0 the
function moves down c units. In a real-world application, the asymptote is particularly important. For
example, it may tell you the room temperature at which a warm drink is leveling off or the amount of
medicine in the bloodstream after you’ve been taking it for a while.
Once students have processed this information and have had a chance to do some homework, sending
them to the board to practice graphing is a useful activity. Have as many students as possible go to the
board. Call out sample functions to have them graph. This activity gives you and the students an
opportunity for assessment in a public, yet non-threatening way. Some of the characteristics they should
focus on are domain, range, symmetry, asymptotes, increasing/decreasing, concavity, shifts, stretches
and shrinks, reflections, intercepts, and the function value when x = 1 . Some examples might be:
a. f ( x ) = 4i 2
x
Graphing Exponential Functions and Exploring e
Advanced Functions and Modeling
b. f ( x ) = −5
6
x
x
c. f ( x ) = 4 ⋅ 2 + 1
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⎛1⎞
d. f ( x ) = ⎜ ⎟
⎝ 3⎠
x
x
g. f ( x ) = 2 − 3
e. f ( x ) = 4
x−1
⎛2⎞
h. f ( x ) = 3i⎜ ⎟
⎝3⎠
f. f ( x ) = 3
x
i. f ( x ) =
x+2
1
⋅2
x+1
−3
4
For their future reference, students should return to their desks periodically to take notes throughout this
activity.
x
References for graphing functions of the type f ( x ) = a ib :
Contemporary Precalculus Through Applications,
Corporation, © 2000, pages 66, 97-100, and 197-203.
Second
edition,
Everyday
Learning
Functions Modeling Change, Second Edition, John Wiley & Sons, Inc., ©2004, pages 118-119.
Comap’s Mathematics: Modeling Our World, Precalculus, W. H. Freeman and Company, ©2000,
pages 82-85.
Algebra and Trigonometry, Brooks/Cole Thomson Learning, ©2001, pages 379-383.
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The Shodor Education Foundation, Inc. is a wonderful web site. Go to this address
(http://www.shodor.org/interactivate/activities/tools.html#fun) and select Function Flyer. It
will
allow you to input a function and “slide” the parameters. It is a good way to immediately see how
a particular parameter is changing the function.
Good problems to use for examples, practice, or homework:
Contemporary Precalculus Through Applications, Second edition, Everyday Learning
Corporation, © 2000, page 101 #1dk, #3, #6dj, page 109 #1lq, page 200 class practice, page 203
#1, #2, #3a-f, #4-7.
Functions Modeling Change, Second Edition, John Wiley & Sons, Inc., ©2004, page 122
Exercise #1-10, #11-14, #20-30. This set of problems particularly has some good writing
questions. Often the written responses give teachers a better indication of the level of
understanding that the skills practice.
Comap’s Mathematics: Modeling Our World, Precalculus, W. H. Freeman and Company, ©2000,
page 89 #1, #8, #9.
Algebra and Trigonometry, Brooks/Cole Thomson Learning, ©2001, page 384 #1-40. The
problems titled Discovery-Discussion, page 385 #53,54 are good ones for group work and class
discussion. These may give you more of an idea about whether or not the students understand
how exponential functions work as opposed to whether or not they can mimic the process of
making a graph. The Discovery Project on page 386 is also a nice problem that you might want
to use as a culminating activity. Students could write a formal solution to this real-world problem,
documenting their thinking and understanding of exponential functions.
Exploring e (3 days)
If students are not already familiar with compound interest, begin the conversation at this point.
Suppose you put $750 into an account that earns 2% annual interest and make no additional deposits or
withdrawals. What is the balance of that account at the end of 5 years?
At the end of year 1 the balance is
At the end of year 2 the balance is
At the end of year 3 the balance is
At the end of year 4 the balance is
At the end of year 5 the balance is
$750(1.02) = $765.00 .
$750(1.02)(1.02) = $750(1.02) = $780.30 .
$750(1.02)(1.02)(1.02) = $750(1.02) = $795.91 .
$750(1.02)(1.02)(1.02)(1.02) = $750(1.02) = $811.82 .
$750(1.02)(1.02)(1.02)(1.02)(1.02) = $750(1.02) = $828.06 .
2
3
4
5
Each year you earn 2% interest on the current balance. In general, the balance of the account for n
n
years is given by balance = initial amount (1 + rate )
It is more likely that interest will be compounded more often than once a year. Suppose we have the
same $750 in an account that earns 2% annual interest, but the interest is compounded quarterly. That
is, each quarter the account earns
0.02
= 0.005 or 0.5% interest. Now what is the balance of that
4
account at the end of 5 years?
⎛
⎝
At the end of quarter 1 the balance is $750 ⎜ 1 +
Graphing Exponential Functions and Exploring e
Advanced Functions and Modeling
0.02 ⎞
⎟ = $753.75 .
4 ⎠
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⎛ 0.02 ⎞ ⎛ 0.02 ⎞
⎛ 0.02 ⎞
At the end of quarter 2 the balance is $750 ⎜ 1 +
⎟ ⎜1 +
⎟ = $750 ⎜ 1 +
⎟ = $757.52 .
4 ⎠⎝
4 ⎠
4 ⎠
⎝
⎝
2
At the end of quarter 3 the balance is
⎛ 0.02 ⎞ ⎛ 0.02 ⎞ ⎛ 0.02 ⎞
⎛ 0.02 ⎞
$750 ⎜ 1 +
⎟ ⎜1 +
⎟ ⎜1 +
⎟ = $750 ⎜ 1 +
⎟ = $761.31 .
4 ⎠⎝
4 ⎠⎝
4 ⎠
4 ⎠
⎝
⎝
3
At the end of quarter 4, which is 1 year, the balance is
⎛ 0.02 ⎞ ⎛ 0.02 ⎞ ⎛ 0.02 ⎞ ⎛ 0.02 ⎞
⎛ 0.02 ⎞
$750 ⎜ 1 +
⎟ ⎜1 +
⎟ ⎜1 +
⎟ ⎜1 +
⎟ = $750 ⎜ 1 +
⎟ = $765.11
4 ⎠⎝
4 ⎠⎝
4 ⎠⎝
4 ⎠
4 ⎠
⎝
⎝
4
At the end of quarter 5 the balance is
⎛ 0.02 ⎞ ⎛ 0.02 ⎞ ⎛ 0.02 ⎞ ⎛ 0.02 ⎞ ⎛ .02 ⎞
⎛ 0.02 ⎞
$750 ⎜ 1 +
⎟ ⎜1 +
⎟ ⎜1 +
⎟ ⎜1 +
⎟ ⎜1 +
⎟ = $750 ⎜ 1 +
⎟ = $768.94 .
4 ⎠⎝
4 ⎠⎝
4 ⎠⎝
4 ⎠⎝
4 ⎠
4 ⎠
⎝
⎝
5
If we continue this process to quarter 20, which is the end of year 5, the balance is
⎛ 0.02 ⎞
$750 ⎜ 1 +
⎟ = $828.67 .
4 ⎠
⎝
20
In general, the balance of the account for n years is given by an equation of the form
n icompounding periods
yearly rate
⎛
⎞
. Certainly more examples need
balance = initial amount ⎜ 1 +
⎟
⎝ compounding periods ⎠
to be done, but this is the general idea.
Now the connection to e . Begin by looking at a very basic example. Suppose we have an account with
$1 that earns 100% annual interest which is equal to a yearly rate of 1. How does the balance of the
account change after 1 year if compounding takes place annually, quarterly, monthly, daily, hourly, and
minutely?
compounding periods
annually
account balance
1
⎛ 1⎞
1⎜1 + ⎟ = 2
⎝ 1⎠
quarterly
⎛ 1⎞
1⎜1 + ⎟
⎝ 4⎠
monthly
2.44140625
12
⎛ 1⎞
1⎜1 + ⎟
⎝ 12 ⎠
daily
Graphing Exponential Functions and Exploring e
Advanced Functions and Modeling
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2.61303529022
365
1 ⎞
⎛
1⎜1 +
⎟
⎝ 365 ⎠
9
2.71456748202
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hourly
1 ⎞
⎛
1⎜1 +
⎟
⎝ 365i 24 ⎠
365i 24
2.71812669063
365i 24i60
minutely
1
⎛
⎞
1⎜1 +
⎟
⎝ 365i 24i60 ⎠
2.7182792154
As we add more and more compounding periods in a year we can see that the value of the expression
k
⎛ 1⎞
⎜1 + ⎟ approaches some limiting value. Mathematicians define this limiting value, which is 2.71828…,
⎝ k⎠
as the number e in honor of the Swiss mathematician Leonhard Euler. In mathematical terms we can
write
⎛ 1⎞
lim
1+ ⎟
k →∞ ⎜
⎝ k⎠
k
=e
Be aware! As you use some calculators to find estimates for e , as k gets larger, errors in precision may
occur.
⎛
⎝
k
1+
Now consider the value of lim
k →∞ ⎜
I⎞
⎟ for any value of I . Use your calculator to compare the following
k⎠
values for large values of k :
e
e
e
e
⎛ 0.02 ⎞
to ⎜1 +
⎟
k ⎠
⎝
k
0.02
⎛ 0.08 ⎞
to ⎜1 +
⎟
k ⎠
⎝
k
0.08
0.035
⎛ 0.035 ⎞
to ⎜1 +
⎟
k ⎠
⎝
0.0125
⎛
⎝
to ⎜1 +
k
0.0125 ⎞
⎟
k
⎠
k
What we see supports the result
⎛ I⎞
lim
1+ ⎟
k →∞ ⎜
⎝ k⎠
If
we
have
a
situation
with
continuous
k
I
=e .
compounding,
we
can
use
the
equation
balance = initial amount ⋅ e I ⋅N , where I is the annual interest rate and N is the number of years
compounding.
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Practice Problems (taken from Contemporary Precalculus Through Applications, Second edition,
Everyday Learning Corporation, © 2000, page 204). [Answers are
shown.]
1. A typical worker at a supermarket bakery can decorate f (t ) cakes per
(
hour after t days on the job, where f (t ) = 10 1 − e
−0.25t
).
a. Sketch a graph of f (t ) . Restrict the domain to meaningful values of t .
b. How many cakes can a newly employed worker decorate in an hour?
f (0) = 0 (use t = 0 since a new employee has worked 0 days)
c. After eight days, how many cakes can a worker decorate in an hour?
f (8) 8.6 so approximately 8.6 cakes
d. Based on this graph, after a worker has decorated cakes for a very long time, how many cakes can he
or she decorate in an hour?
As t → ∞, f (t ) levels off to 10
2. Rheumatoid arthritis patients are treated with large doses of aspirin. Research has shown that the
concentration of aspirin in the bloodstream increases for a short period of time after the drug is
administered and then decreases exponentially. For a typical patient, this relationship is given by
a =14.91e−0.18t , where t represents the number of hours since peak concentration and a represents the
concentration of aspirin measured in milligrams per cubic centimeter of
blood.
a. Graph the function over an appropriate domain. Label the coordinates
of the points that correspond to t = 0 and t = 1 .
b. Determine the peak concentration of aspirin.
14.91 mg per cc
c. Determine the amount of aspirin remaining four hours after peak concentration.
f (4) 7.26 , so approximately 7.26 mg per cc
d. Use graphing technology to determine the time at which the concentration of aspirin is 5mg per cc of
blood.
−0.18t
. The concentration reaches 5 mg per cc in slightly more than
By graphing, find t so that 5 = 14.91e
six hours.
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References for graphing exploring e :
Contemporary Precalculus Through
Corporation, © 2000, pages 190-193.
Applications,
Second
edition,
Everyday
Learning
Functions Modeling Change, Second Edition, John Wiley & Sons, Inc., ©2004, pages 125-129.
Comap’s Mathematics: Modeling Our World, Precalculus, W. H. Freeman and Company, ©2000,
page 87-88.
Algebra and Trigonometry, Brooks/Cole Thomson Learning, ©2001, pages 387-392.
Good problems to use for examples, practice, or homework:
Contemporary Precalculus Through Applications,
Corporation, © 2000,
Second
edition,
Everyday
Learning
Functions Modeling Change, Second Edition, John Wiley & Sons, Inc., ©2004, page 130
Exercises #1-10, #11-30, page 133 #5. On page 137 there is a Check Your Understanding
section. All statements are true or false. The most useful part of this exercise would be that
students are to give an explanation for their answer.
Comap’s Mathematics: Modeling Our World, Precalculus, W. H. Freeman and Company, ©2000,
page 85 Activity 2.2 leads students through the process of discovering e rather than having you
lead them through it, which could be a more interesting process.
Algebra and Trigonometry, Brooks/Cole Thomson Learning, ©2001, page 395 #1-30.
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