8.1.3WhatpatternscanIsee?
Total
Interest
Earned
0
20
40
60
Money
Owed
1000
1020
1040
1060
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Linear and Exponential Growth
Patternsintables,graphs,andexpressionsgivecluesaboutthekindof
growththatisbeingrepresented.Today,youwillworkwithyourteamto
identifywhetheratable,graph,orsituationrepresentssimpleinterestor
compoundinterestbasedonthekindofgrowththatisshown.Youwillalso
comparethedifferentkindsofinterestinsituationstoseewhichoneisa
betterdeal.Bytheendofthislesson,youshouldbeabletoanswer
thesequestions:
Whatarethepatternsinthetables,graphs,and
expressionsforeachkindofinterest?
Whatarethedifferencesbetweenthetwokindsofinterest?
Howdoesthepatternrelatetohowtheamountsaregrowing?
8-24.
John needs to borrow $250.
Moneybags Municipal Bank will
charge him 8% simple interest each
week to borrow the money. Scrooge
Savings will charge him 6%
compound interest each week. With
your team, help John decide which
loan is the better choice.
a.
Make a table for each bank. The
tables should show the amount John
owes after each week for 12 weeks.
b. Graph both tables on the same set
of axes.
c.
Which bank will charge him more if he pays the loan back in 12
weeks? Should he choose the same bank if he plans to pay back the
loan in just 4 weeks?
d. Simple interest is an example of linear growth, and compound
interest is an example of exponential growth. Talk with your team
about how you could explain to someone else the difference
between the two kinds of growth.
8-26.
The table at right represents the balance in Devin’s
bank account for the last five months. Has Devin
been earning simple interest or compound interest?
How do you know?
Time Balance
(months
($)
)
Imari borrowed money from her uncle to
buy a new bicycle. She made the graph at
right to show how much interest she will
need to pay him if she pays back the loan at
different points in time.
a.
257.50
1
263.25
2
269.00
3
274.75
4
280.50
5
286.25
y
About how much interest will
Imari owe her uncle if she pays
him back in 3 months? When will
she owe him $40 in interest?
b. Is Imari’s uncle charging her
simple or compound interest?
Justify your answer. How would
the graph be different if he was
charging her the other type of
interest (simple or compound)?
0
Interest($)
8-25.
x
Time(months)
8-27.
Tom has a choice between borrowing
$250 from his grandmother, who will
charge him simple interest, or
borrowing $250 from the bank, which
will charge him compound interest at
the same percentage rate. He plans to
pay off the loan in 8 months. He
wrote the two expressions below to
figure out how much he would owe
for both loan options.
250(1.02)8
a.
250  8(5.00)
What percent interest will he be charged?
Where can you see the rate (percent) in
each expression?
b. Which expression represents the amount he would owe his
grandmother? Which one represents what he would owe the bank?
c.
Evaluate each expression. Should he borrow the money from his
grandmother or from the bank?
8-28.
Additional Challenge: Four years ago, Spencer borrowed $50 from his
cousin, who charged him simple interest each year. If Spencer pays his
cousin back $62 today, what annual interest rate did his cousin charge?
Could you figure this out if he had been charged compound interest
instead?
8-29.
LEARNING LOG
How can you describe the differences between simple and
compound interest? In your Learning Log, write down
what each kind of interest looks like in a table, on a graph,
and in an expression. Be sure to answer the questions given
at the beginning of the lesson (reprinted below). Title this
entry “Simple and Compound Interest” and include today’s date.
Whatarethepatternsinthetables,graphs,andexpressionsforeach
kindofinterest?
Whatarethedifferencesbetweenthetwokindsofinterest?
Howdoesthepatternrelatetohowtheamountsaregrowing?
7.1.1
What do exponential graphs look like?
••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
Investigating
In this lesson you will investigate the characteristics of the family of functions
. As a team, you will generate data for various functions in this family,
form questions about your data, and answer each of these questions using multiple
representations. Your team will show what you have learned on a stand-alone
poster.
7-1.
BEGINNING TO INVESTIGATE EXPONENTIALS
y
In Chapter 5, you graphed several exponential
functions. Some graphs, like those that modeled the
rabbit populations in problem 5-4, were increasing
exponential functions and looked similar to the two
exponential functions graphed at right.
x
y
Other graphs, such as the rebound-height graphs from
the bouncing ball activity (problem 5-20), represented
decreasing exponential functions and looked similar to
the third curve, shown in bold at right.
You already know that equations of the form
represent the family of lines, and you know what effect changing the
parameters m and b have on the graph. Today you will begin to learn
more about the exponential function family. In their simplest form, the
equations of exponential functions look like
.
By experimenting with different values of b, find three equations in
form that have graphs appearing to match the three graphs shown
above. Confirm your results using your graphing calculator and be
ready to share your results with the class.
x
7-2.
INVESTIGATING
, Part One
What types of graphs exist for equations of the form
?
Your Task: With your team, try different values of b to try to find as
many different looking graphs as possible. (Stick to small values of b,
for example, less than 10. Keep the window on your calculator set from
–10 to 10 in both the x and y direction.)
Decide as a team what different values of b to try so that you find as
many different looking graphs as possible. Be sure to keep track of
what you have tried with a sketch of the resulting graph so that you may
refer to it later. Use the questions listed in the “Discussion Points”
section below to help get you started.
What special values of b should we consider?
Are there any other values of b we should try?
How many different types of graphs can we find?
How do we know we have found all possible graphs?
y
7-3.
The graph of the function
right.
a.
is shown at
Describe what happens to y as x gets
bigger and bigger. For example, what
is y when x = 20? x = 100? x = 1000?
x = (a much larger number)?
b.
Does the graph of
have an
x-intercept? Explain how you know.
c.
approaches the x-axis.
When x is very large, the graph of
That is, as x gets larger and larger (farther to the right along the
curve), the closer the curve gets to the x-axis. In this situation, the
. You can read more
x-axis is called an asymptote of
about asymptotes in the Math Notes box at the end of this lesson.
x
Does
have a vertical asymptote? In other words, is there a
vertical line that the graph above approaches? Why or why not?
7-4.
INVESTIGATING
, Part Two
Now that you, with your class, have found all of the possible graphs
for
, your teacher will assign your team one or two of the types of
graphs to investigate further. Completely describe the graphs. Use the
“Discussion Points” section below to guide your investigation of this
graph. Look for ways to justify your summary statements using more
than one representation (equation, table, graph).
As a team, organize your graphs and summary statements into a standalone poster that clearly communicates what you learned about your set
of graphs. Be sure to include all of your observations along with
examples to demonstrate them. Anyone should be able to answer the
questions below after examining your poster. Use colors, arrows, labels,
and other tools to help explain your ideas.
How can you describe the shape of the graph?
What happens when x gets larger? What happens when x gets
smaller?
How does changing the value of b change the graph?
Which aspects of the graph do not change?
Are there any special points? Can they be explained with the
equation?
Does the graph have any symmetry? If so, where?
7-5.
7-6.
Exponential functions have some interesting characteristics. Consider
functions of the form
as you discuss the questions below.
a.
are defined only for
.
Exponential functions such as
Why do you think this is? That is, why would you not want to use
negative values of b?
b.
Can you consider
or
to be exponential functions?
Why or why not? How are they different from other exponential
function?
LEARNING LOG
Look over your work from this lesson. What questions did you ask
yourself as you were making observations and statements? How does
changing the value of b affect a graph? What questions do you still
have after this investigation? Write a Learning Log entry describing
what mathematical ideas you developed during this lesson. Title this
entry “Investigating
” and label it with today’s date.
ETHODS AND MEANINGS
MATH NOTES
Graphs with Asymptotes
A mathematically clear and complete definition of an asymptote
requires some ideas from calculus, but some examples of graphs
with asymptotes might help you recognize them when they occur. In
the following examples, the dotted lines are the asymptotes, and their
equations are given. In the two lower graphs, the y-axis,
, is also
an asymptote.
y
y
y=3
y=5
x
y
x=2
y = x2
x
y
x
y=x
x
As you can see in the examples above, asymptotes can be diagonal
lines or even curves. However, in this course, asymptotes will almost
always be horizontal or vertical lines. The graph of a function has a
horizontal asymptote if, as you trace along the graph out to the left or
right (that is, as you choose x-coordinates farther and farther away
from zero, either toward infinity or toward negative infinity), the
distance between the graph of the function and the asymptote gets
closer to zero.
A graph has a vertical asymptote if, as you choose x-coordinates closer
and closer to a certain value, from either the left or right (or both), the
y-coordinate gets farther away from zero, either toward infinity or toward
negative infinity.
5.2.1
How can I undo an exponential function?
Finding the Inverse of an Exponential Function
When you first began investigating exponential functions, you
looked at how their different representations were
interconnected, as in the web at right. So far in this
Graph
chapter, you have considered how functions and their
inverses are related in different representations
including equations, x  y tables, and graphs. What would
the inverse equation for each of the parent functions you
worked with in Chapter 2 look like in each representation?
Table
Equation
Context
As you work with your team today, ask each other these questions:
What does the parent function look like in this representation?
How can that help us see the inverse relation?
Would another representation be more helpful?
How can we describe the relationship in words?
5-55.
So far, you have worked with eight different parent graphs:
i.
y  x2
ii. y  x 3
y x
v.
y x
vi.
y
1
x
iii.
yx
iv.
vii.
y  bx
viii.
x 2  y2  1
a.
For each parent, find its inverse, if possible. If you can, write the
equation of the inverse in y = form. Include a sketch of each parent
graph and its inverse. Remember that you can use the DrawInv
function on your graphing calculator to help test your ideas.
b.
Are any parent functions their own inverses? Explain how you
know.
c.
Do any parent functions have inverses that are not functions? If so,
which ones?
5-56.
THE INVERSE EXPONENTIAL FUNCTION
There are two parent functions, y  x and y  b x , that have inverses
that you do not yet know how to write in y = form. You will come back
to y  x later. Since exponential functions are so useful for modeling
situations in the world, the inverse of an exponential function is also
important. Use y  3x as an example. Even though you may not know
how to write the inverse of y  3x in y = form, you already know a lot
about it.
5-57.
a.
You know how to make an x  y table for the inverse of y  3x .
Make the table.
b.
You also know what the graph of the inverse looks like. Sketch the
graph.
c.
You also have one way to write the equation based on your
algebraic shortcut that you used in part (d) of problem 5-40. Write
an equation for the inverse, even though it may not be in y = form.
d.
If the input for the inverse function is 81, what is the output? If
you could write an equation for this function in y = form, or as a
function g(x)  , and you put in any number for x, how would you
describe the outcome?
AN ANCIENT PUZZLE
Parts (a) through (f) below are similar to a puzzle that is more than 2100
years old. Mathematicians first created the puzzle in ancient India in the
2nd century BC. More recently, about 700 years ago, Muslim
mathematicians created the first tables allowing them to find answers to
this type of puzzle quickly. Tables similar to them appeared in school
math books until recently.
Here are some clues to help you figure out how the puzzle works:
log 2 8  3
log 3 27  3
log 5 25  2
log10 10, 000  4
Use the clues to find the missing pieces of the puzzles below:
a.
log 2 16  ?
b.
log 2 32  ?
c.
log? 100  2
d.
5-58.
5-59.
log 5?  3
log? 81  4
e.
f.
How is the Ancient Puzzle related to the problem of the inverse function
for y  3x in problem 5-56? Show how you can use the idea in the
Ancient Puzzle to write an equation in y = form or as g(x)  for the
inverse function in
problem 5-56.
THE INVERSE OF ABSOLUTE VALUE
a.
Find the inverse equation and graph of y  2 x  1 .
b.
Although you know how to find the table, graph, and equation for
the inverse of absolute value, this is another function whose inverse
equation cannot easily be written in y = form. In fact, there is no
standard notation for the inverse of the absolute value function.
With your team, invent a symbol to represent the inverse, and give
examples to show how your symbol works. Be sure to explain how
your symbol handles that fact that the inverse of y  x is not a
function or explain why it is difficult to come up with a reasonable
notation.
5-60.
In problem 5-56, you looked at the inverse of y  3x . Finish
investigating this function.
5-61.
Consider the function f (x) 
5-62.
log100 10  ?
2
7 x
.
a.
What is f (7) ?
b.
What is the domain of f (x) ?
c.
If g(x)  2x  5 , what is g(3) ?
d.
Now use the output of g(3) as the input for f to calculate f (g(3)) .
Amanda wants to showcase her favorite function: f (x)  1  x  5 . She
3.1.3 Can two transformations be equivalent?
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Stretching Exponential Functions (and your brain)
Atthispointinyourmath‐learningcareeryouhaveadvancedtothecoursesthat
studentsgenerallyonlytakeiftheyintendtogoontocollege.Ifyouareinterested
inmathematics,thesciences,orengineering,thetopicscoveredinthisclasswillbe
essential.But,thereisagoodreasonforstudyinghardinthisclassevenifyou
intendtoneverseeanothermathcourseinyourlife:Studyingmathematicsisan
excellentwaytolearntothinkwell,andeveryprofessionwantspeoplewhocanthink
carefullyandproblem‐solvewhentheyneedto.
Inthelastfifteenyears,muchresearchhasbeendoneontwotopicsthatare
especiallyrelevanttostudyingmath.First,neurologistshavedemonstrated
(contrarytowhatwasthoughtevenafewyearsago)thatyourbrainwillgrownew
neuralconnectionsatanyage,ifyouuseitinnewwaystolearnnewwaysofthinking.
Itissomewhatlikeliftingweightstodevelopmusclesthatcanbeusedeitherwhen
playingtennisorfootball,orsimplywhileworkingatajob.Theanalogyisnotexact,
musclesdevelopdifferentlythanabraindoes,buttheideaisthesame:ifyou
exerciseit,itwillimprove.
Second,andthisisthemostimportantfact,
mathabilityisnotsomehowmagicallyfixed
atbirth:youcanalwaysdobetterifyoutry.
Thereisnolistwrittenintheskythatsays
thatMaryis,andalwayswillbe,betteratall
maththanJamie.JustasMarycanget
betteratshootingbasketsifshetries,so
canJamie.AndJamiecangetbetteratmath
thanMary.Itisallintheeffort.Some
peoplelearnthingsmorequicklythan
others,ofcourse,buteveryoneatthislevel
canlearnthismath.
Whatdoesexercisingyourbrainmean?Basically,itmeanstryingtolearninaway
thatmakesconnections.Yourbrainphysicallygrowsbothnewneuronsandnew
connectionsbetweenneurons.Atthesametime,youneedtotrytomake
connectionsbetweenthenewideasyouareseeingandtheideasyoualready
understand.Whenyoustudyproportions,youcanconnectitto,say,linear
equationsandnoticethataproportionisaspecialkindoflinearequation.Making
thisintellectualconnectionhelpswithbuildingthesebrainconnectionsandmore
importantly,itreduceshowmuchyouneedtoremember.Numerousstudieshave
shownthatpeoplewhohavelotsofconnectionsintheirbrainonaspecifictopicuse
lessoftheirbrainwhenworkingonaprobleminthatspecificarea.Theinferenceis
thatyoucan“chunk”informationintolargerpiecesandthengototherightpiece
whenyouneedtoselectone.
Wewilltalkmoreaboutdevelopingyourbrainduringthiscourse,butforrightnow
rememberthatmakinganefforttounderstandwillhelpyounotonlyinthisclass,but
ineveryfutureclassyoutakeandinlifeaswell.
InLesson3.1.1,youinvestigatedandcomparedhorizontalandverticalstretches.
Youcannowshiftandstretchgraphsbothhorizontallyandvertically.Inthislesson
youwillseehowtwodifferenttransformationsofexponentialfunctionscanbe
equivalent.
3-31.
Can two different transformations give the same result? Consider the parent
graph f (x)  x 2 .
a.
f (2x)  (2x)2 . Describe the transformation from f (x) .
b.
Find a value of a so that f (2x)  ax 2 .
c.
Describe the transformation of g(x)  ax 2 from f (x) .
d.
Does f (2x)  g(x)  4 f (x) ? Using algebra, justify your answer. It is
not enough to say they “look” the same.
e.
Can two different transformations give the same result? Do you think
this equivalence of transformations will be true for all functions f (x) ?
Inworkingwithexponentialfunctions,itisnotsurprisingthatwewillneedto
useexponentlaws.Ifyoucan’tfillinthefollowingblanks,youshouldreview
themnow. x ab =__.__.
3-32.
Now we want to consider the graphs of exponential functions. Let
g(x)  3 2 x .
3-33.
a.
Describe g(x) as a transformation of y  2 x .
b.
Sketch the graphs of y  2 x and g(x)
on the same axes. Label each graph
with its equation.
c.
Find the y-intercept of g(x) .
Consider h(x)  3 2 x  1 .
a.
Describe h(x) as a transformation of y  2 x .
b.
Sketch the graph of h(x) .
c.
What is the equation for the horizontal asymptote?
d.
What is the y-intercept?
e.
What would be the y-intercept for y  A  2 x  B ?
3-34.
Using the laws of exponents to show that 2  3x1  6  3x .
3-35.
Consider k(x)  3  2(x2) .
3-36.
a.
Describe the transformation from y  2 x to y  k(x) .
b.
What is the y-intercept of y  k(x) ?
c.
Find an exponential function in the form m(x)  A  2 x with the same
y-intercept as part (b) above.
d.
Is m(x)  k(x) ? Justify your answer.
Let f (x)  3  4 x . Use the laws of exponents to explain why each of the
following equations is true.
a.
b.
16 f (x)  f (x  2)
f (x)
4
 f (x – 1)
3-37.
Using the laws of exponents to solve for A: 6  4 x2  A  4 x .
3-38.
One way to write a general equation for exponentials is
xh
y  a b
 k . This form can be converted to remove the “– h” from the
exponent, as you will prove in a moment. As you saw in problem 3-35, for
an exponential function, every horizontal shift is equivalent to a vertical
stretch. To see what we mean, look at the diagrams below. Discuss them
with your team to be sure you see what is going on.
y
Hereisa
typical
increasing
exponential
function‐‐‐‐‐‐‐>
x
y
Thesamecurve
canbeobtained‐‐‐‐>
bySTRETCHING
theoriginalgraph
vertically.
y
Thiscurvecanbe <‐‐‐‐‐‐‐‐‐‐‐
obtainedby‐‐‐‐‐‐‐‐>
<‐‐‐‐‐‐‐‐‐‐‐
movingthe
originalgraph
<‐‐‐‐‐‐‐‐‐‐‐
totheLEFT.
<‐‐‐‐‐‐‐‐‐‐‐
y  8  2x
<‐‐‐‐‐‐‐‐‐‐‐
x
The diagrams help us to understand, but they do not prove the two are
equivalent. Let’s look at the algebraic proof.
Consider a typical exponential function y  b x . We can move it horizontally
“h” units to the left by writing y  b xh . We can stretch it vertically by
writing y  a(b x ) . We need to use algebra to show that we can start with an
x
equation in y  b xh form and turn it into another one that is in y  a(b x )
form. Here is the proof:
b xh  b x  b h Why is this a true statement?
Now b h is just a number. Let’s call it a for short.
Then b x  b h  b x  a (or a  b x ), which is what we wanted to get.
3-39.
3-40.
Now try following the steps in problem 3-38 to rewrite an equation.
a.
Start with y  7 x 3 and rewrite that equation in y  a(b x ) form.
b.
Try this method on a more complicated example: Let y  12(5 x2 )  7
(three transformations of y  5 x ). Rewrite the equation in the simpler
form y  a(b x )  k (only two transformations of y  5 x ).
c.
Record the fact that y  a(b xh )  k (three
transformations of y  b x ) is equivalent to y  A(b x )  k
(two transformations of y  b x ) into your Tool Kit.
If the exponent is a linear function of x, the expression can be rewritten with
just x as the exponent by using the properties of exponents.
Example: Write 5(2)3x2 in the form A(b) x .
Solution: 5(2)3x2  5(2)3x (2)2 
5
4
(2 3 ) x  54 (8)x
Write in the form A(b) x .
a.
3-41.
5 2x
b.
32 x 3
Using the laws of exponents, solve for A in each of the following equations.
a.
5  3 x2   A  3x
b.
1
25
c.
16  2 x 4   A  2 x
d.
1  3(x2)
3
 5(x4)  A  5 x
 A  3x
6.1.1
What is e?
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Exponential Functions
6-1.
6-2.
6-3.
6-4.
Suppose you deposit $1 at 100% annual interest (hard to get these days!).
To the nearest cent, how much will you have at the end of one year if the
interest is compounded:
a.
Once a year?
b.
Once a month?
c.
Once a day?
d.
n times a year?
Yong Li is looking for a really good bank in which to deposit $1.
Her friend, Loutfi, just heard some wonderful news. For a limited
time only, Megabux Bank is offering 100% annual interest,
compounded continuously!
a.
Yong Li knows that for any given interest rate, there is a
maximum amount of money that can be earned in one year,
no matter how often the interest is compounded. Use your
calculator to determine this maximum amount, to the
nearest cent.
b.
Without rounding, the answer to part (a) is a special number that has its own symbol: e.
The first person to use the symbol “e” was Leonhard Euler (1706-1783) (pronounced
“oiler”). Using n for the number of times per year that the interest is compounded,
express e using limit notation.
c.
Evaluate this limit to as many decimal places as you can.
ALL ABOUT e
a.
Since e is so important, your calculator has a key for it. Obtain the value of e using this
key and compare this with your answer to part (c) of the previous problem. The number
e is frequently used as the base of an exponential function. Use your calculator to
evaluate e2.
b.
What is lim 1 +
(
n→∞
)
1 nx ?
n
At this point, it is natural to ask how y = ex changes. In fact, the derivative of y = ex is very
special.
d
dx
(e x ) using the definition of a derivative as a limit.
a.
Rewrite
b.
Use your calculator to approximate the derivative.
c.
Why is this derivative so special?
Chapter 6: More Tools and Theorems
267
6-5.
As you know, e is a frequently used base of exponential functions. What about exponential
functions with other bases? What is the shape of the graphs of their derivatives? Using the
definition of a derivative as a limit, we can investigate derivatives of exponential functions
further.
a.
Use the definition of a derivative as a limit and your calculator to approximate a
derivative function for y = 2x. How does the shape of the graph of y = 2x relate to the
shape of its derivative graph?
b.
Use this technique to estimate
related?
c.
d (2 x )
d (3x )
d (e x )
Graph y = dx
, y = dx
, and y = dx
on the same set of axes. Using your
d (e x )
results from parts (a) and (b), explain why the graph of y = dx
lies in between the
other two curves.
d
dx
(3x ) . How are the graphs of y = 3x and y =
d
dx
(3x )
6-6.
d (b x ) = (some constant) ⋅ b x
In the previous problem you saw that dx
, for b = 2, 3, …. When
b = 2, the constant is approximately 0.693; when b = 3, the constant is approximately 1.099.
For what value of b is the constant 1?
6-7.
Although you may have a conjecture from the previous problem, you can analytically determine
this value of b.
d
dx
(b x ) using the definition of a derivative as a limit.
a.
Write an expression for
b.
Factor out bx. Let
c.
Explain why b = e when h → 0. You may wish to refer to the following Math Notes box.
b h −1
h
= 1 , our constant, and solve for b.
M ATH N OTES
STOP
All About “e” and ex
(
n→∞
The constant e is defined as a limit: e = lim 1 +
)
1 n
n
or e = lim (1 + n)1/n
n→0
The constant e arises frequently in mathematics. In fact, it is most often used as the base of an
exponential function, y = ex, which is y ≈ (2.71828)x. In calculus, the reason for the importance of e
d ex = ex .
is that dx
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Calculus