MAP4C-B
Foundations for
College Mathematics
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Authority. All rights reserved. No part of these materials may
be reproduced, in whole or in part, in any form or by any means,
electronic or mechanical, including photocopying, recording, or
stored in an information or retrieval system, without the prior
written permission of The Ontario Educational Communications
Authority.
Every reasonable care has been taken to trace and acknowledge
ownership of copyright material. The Independent Learning
Centre welcomes information that might rectify any errors or
omissions.
August 24, 2010
Foundations for College Mathematics MAP4C-B
Course Introduction, page 1
Course Description
Welcome to Foundations for College Mathematics, Grade 12,
MAP4C-B.
This course will enable you to broaden your understanding of
real-world applications of mathematics. You will analyze data
using statistical methods, solve problems involving applications
of geometry and trigonometry, solve financial problems connected
with annuities, budgets, and renting or owning accommodation,
simplify expressions, and solve equations. You will reason
mathematically and communicate your thinking as you solve
multi-step problems. This course prepares you for college
programs in areas such as business, health sciences, and human
services, and for certain skilled trades.
Materials
You will need a graphing calculator for this course. It is strongly
recommended that you use a Texas Instruments TI-83 Plus
graphing calculator, as it is demonstrated throughout this course.
A graphing calculator is not required for the Final Test.
Expectations
The expectations listed in this course describe the knowledge
and skills that you are expected to develop and demonstrate.
The overall expectations you will cover in each unit are listed on
the first page of the unit. The specific expectations are listed at
the beginning of each lesson under the heading “What You Will
Learn.”
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Course Introduction, page 2
Foundations for College Mathematics MAP4C-B
Evaluation
In each lesson, there are Support Questions and Key Questions.
Support Question
(do not send in for evaluation)
These questions will provide you with an opportunity to assess
your understanding and mastery of the ideas and skills you are
learning in the course. They will also help you to improve the
way in which you communicate your ideas. Many of the Support
Questions will prepare you to answer the Key Questions.
Do not send your answers to ILC to be marked. Suggested
Answers to Support Questions are provided so that you can
check your work.
Key Question
The Key Questions are used to evaluate your achievement of
each unit’s expectations. Your answers will show how well you
have understood the ideas and mastered the skills in the unit,
and how well you can communicate your ideas.
You must complete all of the Key Questions successfully in order
to pass each unit. When you have completed all the lessons in
a unit, submit the answers for that unit to ILC for marking.
Remember to label your answers clearly with the correct unit,
lesson, and question numbers.
Evaluation Tools
There is a Marking Guide after each Key Question that explains
how the marks are allotted for each answer. The Marking Guides
include details about what your answer must include to get full
marks and are the evaluation tools that your teacher will use to
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Foundations for College Mathematics MAP4C-B
Course Introduction, page 3
determine the marks for your answers. Ensure that you include
all of the “look-fors” that are in the Marking Guides when you
complete your Key Questions.
Submitting Your Coursework
After you have completed the work for a unit, submit it to ILC
for evaluation. Make sure that you include all the required
answers to all Key Questions in the unit. You must also complete
the Reflection.
For each unit, an ILC teacher will evaluate your work. The
teacher will write comments, giving you feedback to help you
improve your work.
Reflection
As you work through each unit, think carefully about what
you are doing. Use the Reflection to make comments, express
feelings, and give opinions about your learning and about the
course. The more you think about and reflect on what you are
doing, the better you will learn. This also provides important
feedback for the teacher. You must complete it before proceeding.
What You Must Do to Earn a
Credit
In order to receive a credit for this course, you must
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Course Introduction, page 4
Foundations for College Mathematics MAP4C-B
You will receive a mark out of 100 for each unit and for the Final
Test. Your course mark will also be out of 100.
Weighting
Coursework
Unit 1
Unit 2
Unit 3
Unit 4
Final Test
Total
Weight (%)
100%
You must receive a passing mark for each unit before starting
the next unit. If you do not receive a passing mark on any unit,
the teacher will ask you to redo and resubmit the unit. The
teacher will give you suggestions to help you pass the next time.
Final Test
Every ILC credit course has a Final Test. After you have
successfully completed the last unit of this course, you will
receive information about writing the test.
You can have two opportunities to pass the Final Test. No matter
how well you do on the unit work, if you do not pass the Final
Test, you will not get a credit for the course.
Copyright © 2009 The Ontario Educational Communications Authority. All rights reserved.
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Foundations for College Mathematics MAP4C-B
Course Introduction, page 5
Table of Contents
Unit 1: Mathematical Models
Lesson 1: Working with Exponents
Lesson 2: Exponential Equations
Lesson 3: Interpreting Graphs and Using Graphical Models
Lesson 4: Types of Graphical Models
Suggested Answers to Support Questions
Unit 2: Personal Finance
Lesson 6: Annuities
Lesson 7: Mortgages
Lesson 8: Considering an Affordable Place to Live
Lesson 9: Budgets
Lesson 10: Smart Financial Planning
Suggested Answers to Support Questions
Unit 3: Geometry and Trigonometry
Lesson 11: Measurement and Geometry
Lesson 12: Volume and Introduction to Optimal Dimensions
Lesson 13: Optimal Dimensions
Lesson 14: Introduction to Trigonometry
Suggested Answers to Support Questions
Unit 4: Data Management
Lesson 16: Sampling, Surveys, and Data Collection
Lesson 17: Lines of Best Fit
Lesson 18: Trends and Data Analysis
Lesson 19: Statistical Terms and Indices
Lesson 20: Interpreting Statistics
Suggested Answers to Support Questions
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Copyright © 2009 The Ontario Educational Communications Authority. All rights reserved.
MAP4C-B
1
Unit
Mathematical Models
Copyright © 2009 The Ontario Educational Communications
Authority. All rights reserved. No part of these materials may
be reproduced, in whole or in part, in any form or by any means,
electronic or mechanical, including photocopying, recording, or
stored in an information or retrieval system, without the prior
written permission of The Ontario Educational Communications
Authority.
Every reasonable care has been taken to trace and acknowledge
ownership of copyright material. The Independent Learning
Centre welcomes information that might rectify any errors or
omissions.
Foundations for College Mathematics MAP4C-B
Unit 1 Introduction, page 1
Table of Contents
You are here
Unit 1: Mathematical Models
Lesson 1: Working with Exponents
Lesson 2: Exponential Equations
Lesson 4: Types of Graphical Models
Suggested Answers to Support Questions
Unit 2: Personal Finance
Lesson 6: Annuities
Lesson 7: Mortgages
Lesson 8: Considering an Affordable Place to Live
Lesson 9: Budgets
Lesson 10: Smart Financial Planning
Suggested Answers to Support Questions
Unit 3: Geometry and Trigonometry
Lesson 11: Measurement and Geometry
Lesson 12: Volume Surface Area, and Introduction to Optimal
Dimensions
Lesson 14: Introduction to Trigonometry
Suggested Answers to Support Questions
Unit 4: Data Management
Lesson 16: Sampling, Surveys, and Data Collection
Lesson 17: Lines of Best Fit
Lesson 18: Trends and Data Analysis
Lesson 19: Statistical Terms and Indices
Lesson 20: Interpreting Statistics
Suggested Answers to Support Questions
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Unit 1 Introduction, page 2
Foundations for College Mathematics MAP4C-B
Introduction
Have you ever wondered if there is a way to model and predict
population growth? Do you know why it takes so long for
radioactive material to “cool down” to safe levels? Both of these
problems can be modelled by mathematical equations that use
2
.
powers. Powers are number with exponents, such as 24
In this unit, you will use the laws of exponents to simplify
expressions and solve equations in which variables are squared
or cubed. You’ll see the connections between graphical and
algebraic models and identify the best type of model to use for
a given set of data. Exponential equations will be solved using
algebra and graphs. Graphs will also be explored to identify
trends and relationships, and will be described in terms of rate
of change. College programs and careers related to mathematical
modelling will also be examined.
What You Will Learn
After completing this unit, you will be able to
involving exponential equations
modelling relationships graphically and algebraically
from real-world applications, and describe applications of
mathematical modelling in various occupations
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Foundations for College Mathematics MAP4C-B
Unit 1 Introduction, page 3
Glossary
correlation coefficient (r) a number that indicates how close to a line of
in a scatter plot are
evaluate
to “evaluate” a mathematical expression means to solve it or to
determine its final numerical value
exponent a term placed to the right of, and raised above, a base term that
indicates how many times a base term will be multiplied by itself
integer a “counting number,” including zero and any positive or negative
whole number
laws in mathematics, laws are rules or statements that always holds true
logarithmic function the inverse of the exponential function, for example,
y
if x = by then logb x
logarithmic function to solve problems in this course,
power any number multiplied by itself, indicated with an exponent, for
example, yx
radical an expression with a root sign, such as
3
5 or
6
rate of change the steepness or slope of a graph, defined as
cha nge in y y2 − y1
=
cha nge in x x2 − x1
rational number
a whole number divided by another whole number, in
other words, a fraction; any number that can be written
as a ratio of two integers
reciprocal
reciprocal of
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2
3
1
is
and the reciprocal of 2 is
2
2
3
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Unit 1 Introduction, page 4
Foundations for College Mathematics MAP4C-B
regression analysis a technique for finding a model to guess future or
past trends for a set of data; such analysis may
include finding a regression equation, fitting a line
determining the correlation coefficient, or “r-value,” of
the line/equation
regression equation given a set of data, the regression equation is a
plotted data points
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MAP4C-B
1
Working with
Exponents
Foundations for College Mathematics MAP4C-B
Lesson 1, page 1
Introduction
You see numbers with exponents in many areas of mathematics
and in everyday situations. They’re unavoidable. Exponents are
those little, small-sized numbers in powers. A power is a base
2
. Have you ever seen
number with an exponent, such as 24
is equivalent to the powers 21, 22, 2 , 24, 2 , 26, 27, and 28.
As you will see, exponents can include positive exponents,
negative exponents, and fractional exponents. There are rules
that allow you to simplify expressions that contain exponents.
This lesson will show you these rules, explain how they are used,
exponents.
Estimated Hours for Completing This Lesson
Exponent Laws
Connecting Powers and Radicals
Solving Math Problems with Scientific Calculators
1
Key Questions
1
What You Will Learn
After completing this lesson, you will be able to
powers, and for evaluating the power of a power
to simplify algebraic expressions
variety of tools and strategies
and/or rational bases
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Lesson 1, page 2
Foundations for College Mathematics MAP4C-B
Exponent Laws
Exponent
The following exponent rules,
sometimes called laws, are very
useful when trying to simplify
expressions containing powers.
The first two laws, multiplying
and dividing powers, only apply if
the bases are equal or the same.
Power
Base
Remember these terms when
working with powers.
Multiplying Powers
To illustrate the rule for multiplying powers, let’s look at an
example and a counter-example.
Example
(34) × (32)
Write each power in expanded form.
= (3 × 3 × 3 × 3) × (3 × 3) Rewrite this as a power.
= 36
Simplify by writing the original base
with a new exponent.
Conclusion
34 × 32 = 36 (Notice here that 6 = 4 + 2.)
Counter-example
(24) × (32)
Write each power in expanded form.
= (2 × 2 × 2 × 2) × (3 × 3) Rewrite this as a power.
= (2 × 2 × 2 × 2) × (3 × 3) You cannot simplify because the bases
are different.
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Foundations for College Mathematics MAP4C-B
Lesson 1, page 3
Dividing Powers
To illustrate the rule for dividing powers, let’s look at an
example.
Example
57
53
Write each power in expanded form, and divide out any
common factors from top and bottom.
5 × 5 × 5 ×5 ×5 ×5 ×5
=5 ×5 ×5 ×5 =5 4
5 ×5 ×5
Conclusion
57
=
53
4
or
7
4
xm
xn
Remember, these rules for multiplying or dividing
powers only apply if the bases are the same.
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Lesson 1, page 4
Foundations for College Mathematics MAP4C-B
Negative Exponents
To illustrate the rule for negative exponents, let’s look at an
example. Rule #2, the rule for dividing powers, can be used to
Example
4 =4
0
=
4
4
=
1
43
3
Since x0 = 1 for all values of x except when x = 0
Conclusion
4
−3
=
1
43
x− n
1
= n
x
⎛ x⎞
⎜⎝ y ⎟⎠
−n
⎛ y⎞
=⎜ ⎟
⎝ x⎠
n
Power of a Power
To illustrate the “power of a power,” which is the rule for raising
a power to a power, let’s look at an example. In some cases,
powers have exponents resulting in a power of a power. When
you say “power of a power,” you mean you are taking a power
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Foundations for College Mathematics MAP4C-B
Lesson 1, page 5
2
2 4
.
Apply the rule for multiplying exponents.
Example
2 4
2
2
2
2
Common bases, so use multiplication law
8
Conclusion
2 4
8
Now let’s see how these rules or “exponent laws” can be used to
simplify expressions.
Example
Simplify each algebraic expression using the exponent laws you
have just studied.
When no exponent
is shown, it is
assumed to be an
exponent of 1. For
example, a = a1, or
4 = 41, and so on.
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2
4
a ×a ×a
(3 a )
2
9 a4
3
5 2 ×5 3 ×5
(5 )
3
2
⎛3 ⎞
⎜⎝ 4 ⎟⎠
−2
1 5 a2 b3 c5
3 ab4 c3
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Lesson 1, page 6
Foundations for College Mathematics MAP4C-B
Solution
a2 × a4 × a3
= a2 +4 +3 = a9
5 2 ×53 ×5
(5 )
3
=
2
5 2 +3 +1 5 6
= 6 = 5 6 −6 = 5 0 = 1
3 ×2
5
5
⎛3 ⎞
⎜⎝ 4 ⎟⎠
−2
2
42 16
⎛4 ⎞
= ⎜ ⎟ = 2 =
⎝3 ⎠
9
3
(3 a )
3
2
9a
4
=
1 5 a2 b3 c5
3 ab4 c3
3 3 (a2 )3 2 7 a2 ×3 2 7 a6
=
=
9 a4
9 a4
9 a4
a b c
a6–4
a2
a1b c2
5 ac2
or it can be expressed as
b
Example
For this example, first use exponent laws to simplify the
7
(3 8 )(3 −3 )
33
(2 )(2 )(2 )
2
3
2 −3
−5
⎛1 ⎞
⎜⎝ 4 ⎟⎠
−3
Solution
7
= 42 = 16
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Foundations for College Mathematics MAP4C-B
Lesson 1, page 7
3 8 +(−3 ) 3 5
(3 8 )(3 −3 )
= 3 = 3 5 −3 = 3 2 = 9
3
=
3
3
33
(2 )(2 )(2 ) =
2
−5
3
2 −3
⎛1 ⎞
⎜⎝ 4 ⎟⎠
−3
2 2 +3 −5
20
=
= 2 0 −(−3 ) = 2 3 = 8
−3
−3
2
2
3
⎛4 ⎞
= ⎜ ⎟ =4 3 =64
⎝1 ⎠
Support Questions
(do not send in for evaluation)
x 2 y5
x3 y
x2
x
1 5 x 4 y3 z6
5 xy4 z4
2
a2 × b4 × a
power, then evaluate.
⎛3 ⎞
⎜⎝ 5 ⎟⎠
3 2 ×3 6
(3 )
2
3
−2
There are Suggested Answers to Support Questions at the end of
this unit.
Connecting Powers and Radicals
The
or radical sign is implied to mean square root. In fact,
2 5 could actually be written as
2
2 5 , which means: “What
math courses the values of some commonly used radicals. For
example, without using a calculator, you may know some square
roots, such as:
25 =5
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9 =3
1 4 4 =1 2
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Lesson 1, page 8
Foundations for College Mathematics MAP4C-B
Square Root
used, you don’t bother writing in the 2 when writing square
roots. However, when writing other radicals, it is necessary
to include the number. For instance, the third root of eight is
written as
3
8 .
Radicals beyond the Square Root
The radical can be extended to indicate other roots. Here are
some examples:
3
8 means: “What value when cubed gives 8?” The answer is 2.
1
3
You could also write this as 8 .
4
8 1 means: “What value when raised to the exponent 4 gives
8
6 5 6 1 means: “What value when raised to the exponent 8
Example
Write each of the following using a radical sign.
256
1
8
144
1
2
177147
Solution
256
1
8
=
177147
8
1
11
144
256
=
11
1
2
1
11
= 1 4 4 or 2 1 4 4
177147
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Foundations for College Mathematics MAP4C-B
Lesson 1, page 9
Rational Exponents
A ratio is a fraction. A rational exponent is a fractional exponent.
Let’s use an example to see what happens when you apply the
power of a power rule to a rational exponent. To simplify a power
of a power, you will keep the base and multiply the exponents.
Remember that xm n = xm×n
Example
2
⎛ 1⎞
Let’s begin by proving that ⎜ 2 5 2 ⎟ = 2 5 , as follows.
⎝
⎠
2
⎛ 12 ⎞
⎜⎝ 2 5 ⎟⎠ = 2 5
1
×2
2
=251 =25
2
⎛ 12 ⎞
Therefore ⎜ 2 5 ⎟ = 2 5
⎝
⎠
This is similar to the statement
means the same as
2
(
25
)
2
=25
25
25
This allows us to conclude that 2 5
1
2
= 25
Extending this to other rational exponents, you can write
examples such as:
1
8 =2= 8
1
4
243
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1
5
This is read as the “cubed root of 8.”
= =4
=3 = 5 243
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Lesson 1, page 10
Foundations for College Mathematics MAP4C-B
Example
( )
Rewrite each of the following using the radical sign
and
evaluate without using a calculator. In other words, write the
radical as a number without a radical sign.
81
1
2
343
1
3
128
Solution
81
1
2
=
343
81 =9
1
3
=
1
7
343 =7
3
128
1
7
=
7
128 =2
Generalize the Connection
1
n
The power x represents the nth root of x, and can be written
1
using a radical sign such that x n = n x , as seen in the previous
examples.
You may have noticed that all of these rational exponents so far
1 1 1
, , ,
2 3 7
What if the exponent is a rational number where the numerator
is some value other than one? This can be generalized as follows:
x
m
n
=x
1
×m
n
⎛ 1⎞
= ⎜ xn ⎟
⎝ ⎠
=
m
( x)
n
m
Let’s extend this rule to simplify questions very similar to those
in the last example, only this time the exponents do not all have
one as the numerator.
Example
Evaluate the following without using a calculator by simplifying
first.
8
2
3
343
2
3
128
4
7
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Foundations for College Mathematics MAP4C-B
Lesson 1, page 11
Solution
2
⎛ 13 ⎞
= ⎜8 ⎟ =
⎝ ⎠
2
3
8
343
128
( )
3
8
2
2
2
3
1
⎛
⎞
= ⎜3 4 3 3 ⎟ =
⎝
⎠
4
7
1
⎛
⎞
= ⎜1 2 8 7 ⎟ =
⎝
⎠
4
=22 =4
(
3
343
)
(
7
128
)
2
4
=7 2 =49
= 2 4 =1 6
Negative Rational Exponents
The rule you just looked at also applies to negative rational
1
exponents. You’ll recall that x − n = n , so it follows that
x
−1
1
xn =
x
1
n
1
or
n
x
When dealing with negative exponents of any sort, it is
suggested that you deal with the negative first.
Example
For each of the following, first rewrite with a radical sign, then
simplify without using a calculator.
16
−3
2
⎛36 ⎞
⎜⎝ 4 9 ⎟⎠
−
1
2
Solution
16
−3
2
=
1
16
⎛36 ⎞
⎜⎝ 4 9 ⎟⎠
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−
1
2
3
2
=
1
( 16 )
3
1
=
1
1
=
43 64
1
⎛ 4 9 ⎞ 2 ⎛74 94 9⎞ 2 7
= ⎜ ⎟ =⎜ ⎟ =
⎝ 3 6 ⎠ ⎝63 63 6⎠
6
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Lesson 1, page 12
Foundations for College Mathematics MAP4C-B
Support Questions
(do not send in for evaluation)
calculator. Rewrite each using a radical sign first.
64
4.
1
2
125
1
3
81
1
4
−1
3
64
Rewrite each of the following using a radical sign, then
simplify without using a calculator.
64
2
3
16
7
4
⎛4 ⎞
⎜⎝ 9 ⎟⎠
3
2
81
−
3
4
Working with Questions that Can’t Be
Solved Mentally
In each of the previous cases, the value of the power could be
determined mentally, without a calculator. In some cases the
radical could be evaluated in its original form, and in other cases
there needed to be some manipulation of the exponent first.
But what if the exponent was not one of those that result in an
integer or “whole number” answer? In these cases, how would
you evaluate this kind of power with a rational exponent?
One option would be to interpolate from a graph. Interpolating is
the process of estimating points between those that are plotted
on a graph or between the lines on the graph to estimate the
final value of a power.
For example, you know the value of 4
same as
1
2
4
you evaluate a question like 2
13
4
? Even if you write the exponent
13
2 4 =2
to evaluate mentally. You can, however, estimate this using the
graph of y = 2x.
If you look at the following graph of y = 2x, you can see how to
use a process called interpolation.
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Foundations for College Mathematics MAP4C-B
Lesson 1, page 13
To interpolate the
G ra ph of Y = 2^x
Graph
of Y = 2^x
13
value of 2 4 , you start
at the spot on the
graph where x
35
30
25
20
y
15
draw a dotted line from
here straight up to the
curve. From here, you
draw another dotted
line straight across to
the y-axis. The y-axis value you end at will be approximately the
10
5
0
-2
-1
0
1
2
3
4
5
x
13
4
13
4
you end at approximately y = 9. This means that based on this
value of 2
x
for x =
13
graph, 2 4 = 9. If you want a more accurate estimate, you need to
“zoom in” or enlarge the graph, as you can see in the following
version of the more precise graph. Notice that this graph has
more delineations and so is more precise.
Because you can see
more clearly on this
graph exactly where
x
end at y
So, according to this
Graph of Y = 2^x
Graph of y = 2x
10
9
8
7
6
yy
5
4
graph, the value of 2
3
2
1
0
-2
-1.5
-1
-0.5
0
0.5
1
xx
1.5
2
2.5
3
3.5
4
13
4
if you were able to look
at the value of 2
13
4
This could be done for any base with any exponent. This is not
the best method to use, however. This method has the following
problems:
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Lesson 1, page 14
Foundations for College Mathematics MAP4C-B
that if you do have this technology available, then you also
have access to a scientific calculator, which is faster and more
accurate, as you’ll see in the next section.
Solving Math Problems with
Scientific Calculators
most of the instructions regarding calculators will be specific to
this calculator. There are many, many scientific calculators out
there so it’s difficult to include instructions that will work for
everyone. Here are some tips to remember when using scientific
calculators of any type:
cover many languages, but it has useful information specific to
your calculator.
3 + (2 − 5 )2
you would have
example, to answer a question like
6
to enter:
ENTER
+
(
(
2
x2
. Or, to answer a question like
enter: 10
÷
(
×
2
÷
)
6
10
, you would have to
3 ×2 2
x2
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)
ENTER
.
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Foundations for College Mathematics MAP4C-B
Lesson 1, page 15
negative, you press the ( – ) button to add a negative sign
before the number. Be sure to use the correct button for the
negative sign. Do not use the button for subtraction.
Commonly Used Buttons on the TI-83
Plus
The following table shows some common calculator functions you
will use in this lesson.
Math
Operation
TI-83 Calculator Keystrokes
ˆ
2
Notes
A common key found on
other calculators would
y
be
instead of the
2
x2
ENTER
x
ˆ
(
(–)
)
ENTER
3
(
ˆ
8
The brackets keep
the base together as a
negative number.
ˆ
÷
1
Remember that
)
3
ENTER
8
(
1
⎛2 ⎞2
⎜⎝ 5 ⎟⎠
(
1
8 = 8 3 . The brackets
are needed to keep the
exponent together.
2
1
÷
÷
)
)
2
The brackets are needed
to keep the base together
and to keep the exponent
together.
ˆ
ENTER
Example
Evaluate each of the following to three decimal places.
10
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3
4
1
7
32
⎛4 ⎞2
⎜⎝ 5 ⎟⎠
1
3
25
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Lesson 1, page 16
Foundations for College Mathematics MAP4C-B
Solution
3
4
10
7
= 10
ˆ
÷
(
32 is the same as 3 2
1
7
⎛ 4⎞ =
÷
(
4
⎜⎝ 5 ⎟⎠
(
4 ÷
or
= 0.8944271 ≈ 0.894
25
(–)
(
1
÷
7
1
÷
2
)
ˆ
1
2
3
≈
)
=1.6406707 ≈ 1.641
)
1
4
)
)
is the same as
25
1
÷
1
3
or 2 5
0
(
ˆ
1
(
ˆ
.
)
−1
3
ˆ
(
≈
)
Support Questions
(do not send in for evaluation)
decimal places.
45
5
1
3
44
2.1
1
4
16
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⎛2 ⎞
⎜⎝ 7 ⎟⎠
57
−5
3
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Foundations for College Mathematics MAP4C-B
Lesson 1, page 17
Key Questions
Save your answers to the Key Questions.
When you have completed the unit, submit them to ILC for marking.
1.
Simplify each expression. (5 marks)
a) (a )
n5
b) 3
n
d) x(x2)(x4)
e)
3 2
2.
c) (3x2)3
t 2 (t 5 )
(t 3 )2
Write as a single power, then evaluate. (6 marks)
(4 )
2
a)
3.
d)
c)
4
sign), then evaluate.
27
25
1
3
−1
2
3
5
b)
32
c)
e)
⎛64 ⎞
⎜⎝ 2 7 ⎟⎠
8
5
3
−1
3
Evaluate with a calculator. Round your answer to three
decimal places. (5 marks)
a)
250
1
4
3.7
b)
1.28
−5
5.
(22)(2–1)(23)
b)
Write each power as a root (using the
(10 marks)
a)
4.
57 ÷ 55
2
c)
⎛3 ⎞
⎜⎝ 5 ⎟⎠
−5
d)
6
28
⎛3 ⎞
For question ⎜ ⎟ , list the specific calculator keystrokes you
⎝5 ⎠
used. (1 mark)
Now go on to Lesson 2. Do not submit your coursework to ILC
until you have completed Unit 1 (Lessons 1 to 5).
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MAP4C-B
2
Exponential Equations
Foundations for College Mathematics MAP4C-B
Lesson 2, page 1
Introduction
In the first lesson you worked with exponents. In every case,
the exponents were numbers. Now you will deal with exponents
that are variables in equations. These equations are known
as exponential equations. This lesson explores the methods
of solving exponential equations while looking at real-world
applications.
Estimated Hours for Completing This Lesson
1
Applications of Exponential Equations
1
Key Questions
1
What You Will Learn
After completing this lesson, you will be able to
and error and graphs
and by using graphing technology
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Lesson 2, page 2
MAP4C-B Foundations for College Mathematics
Solving Exponential Equations
Using Trial and Error or Using a
Graph
Exponential equations can be defined as any equation where the
variable is in an exponent. In other math courses, you’ve likely
solved only two general types of equations. You may recall linear
equations such as 2x
x
2x2
so these equations can’t be classified as exponential equations.
There are many methods used to solve exponential equations.
These methods vary in difficulty and accuracy, but they also
depend on the technology you have available. You will look
at four methods in this lesson. A fifth method, using the
logarithmic function, will not be covered in this course.
Method 1: Using Trial and Error to
Estimate an Exponent
The method of trial and error will give us an estimate of the
answer, but typically takes more time than other methods, with
limited accuracy.
Example
x
= 21 accurate to two decimal places using trial and error.
Solution
2
2
= 27. This tells you that x should be
= 27 is closer to 21
is, your value for x
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Foundations for College Mathematics MAP4C-B
When x
2.7
When x
2.8
Lesson 2, page 3
When x
When x
When x
2.77
2.771
So after rounding 2.771, x is 2.77 accurate to two decimal places.
As you can see, it may take quite a few attempts before you get
to the correct answer. A lot of time can be spent on this method.
can be easier but it requires one or more graphs.
The following method for solving an exponential equation is
similar to a method that was used in Lesson 1 to evaluate
powers with rational exponents.
Method 2: Using a Graph to Estimate an
Exponent
You may recall that you used graphs in Lesson 1 to evaluate
powers with rational exponents. This involved interpolating the
desired point from the graph. This process can also be used to
solve exponential equations.
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Lesson 2, page 4
MAP4C-B Foundations for College Mathematics
Example
Solve 2x = 12 using the graph of y = 2x.
70
60
50
Y
y
40
30
20
10
0
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
Xx
Solution
Yy
70
Looking at the graph of
x
60
y = 2 that appears here,
50
you must find the value
40
of x that corresponds to
30
a y-value of 12. In other
20
words, you need to find
10
an x that satisfies
x
0
12 = 2 . To do this, you
-4
-2
0
2
4
6
8
will draw a dotted line
X
x
using a straight edge,
such as a ruler. Start
at y = 12 on the vertical axis, move straight across to the graph,
then straight down to the x
can see that this value of x is between 2 and 4.
You can estimate from the graph that the solution appears to
.
6 ENTER
by entering 2 ˆ
is promising. This answer is close to your desired result of 12,
so you can continue looking for a more accurate solution by
“zooming in” on the graph of y = 2x.
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Foundations for College Mathematics MAP4C-B
Lesson 2, page 5
yY
With the graph zoomed
in, you can see that your
solution would be closer
to x
Again, this can be
verified with a scientific
calculator by entering 2
.
ENTER
.
ˆ
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
0
0.5
1
1.5
2
verifies your estimate.
2.5
3
3.5
4
Xx
As you can see, this method gives an estimate of the answer,
and relies on graphing technology to do so. In the next section,
you’ll see how graphing technology can be used faster and more
Support Questions
(do not send in for evaluation)
x in each of the following,
accurate to two decimal places. In each case, explain your
logic and show all values that were attempted.
x
x
value of x that satisfies 2x
give a rough estimate. Graph B, which is an enlarged version
of the original graph, will allow you to give a more accurate
answer.
Graph A
40
35
30
yY
25
20
15
10
5
0
0
1
2
3
4
5
6
7
8
X
x
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Lesson 2, page 6
MAP4C-B Foundations for College Mathematics
yY
Graph B
30
29
28
27
26
25
24
23
22
21
20
2
2.5
3
3.5
4
4.5
5
5.5
6
X
x
There are Suggested Answers to Support Questions at the end of
this unit.
Method 3: Solving Exponential
Equations Using a Calculator
exponential equations using graphs, only much faster and with
greater accuracy. To do so you must first understand some of the
For this exercise, you will find the point of intersection of a curve
x
×
y2
y1
×
x
.
Entering Equations into the TI-83 Plus
Y=
WINDOW
Y=
,
ZOOM
,
TRACE
, and
GRAPH
,
.
button revealing the following screen.
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Foundations for College Mathematics MAP4C-B
Lesson 2, page 7
keypad can be used to move the cursor on this
x
y
(
1
and y
.
ENTER
)
ˆ
X,T,θ,n
.
000
moves the cursor down to
. Your screen should
now look like the second screen shown above.
ENTER
Viewing Graphs with the TI-83 Plus
To view the graph of one or more equations, they must first be
entered in the Y= screen, which you have already done. Now, you
only need to press the GRAPH button. The view you will see will
totally depend on the window settings. This can be a problem
time it was used are saved in the memory. As a result, you must
always remember to change the window settings if you see no
graph or if the graph is too big or too small.
Estimating and Changing Window Settings
It can be frustrating when you push the GRAPH button and see
nothing on the graph. Don’t panic. Look at your equation and
try choosing numbers that are similar to the x- and y-values
second equation is y2
see y
of roughly y
roughly y
Determining the x
a bit easier because you can almost always start with zero. For
x
×
y1
x is zero. Let’s try a higher number like x = 20.
When x is 20, y
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Lesson 2, page 8
MAP4C-B Foundations for College Mathematics
see x-values that range from a minimum of x = 0 to a maximum
of roughly x = 20.
Pressing the WINDOW button reveals the window settings. Enter the
window settings as shown in the following sample screen.
The window screen allows you to essentially zoom in or out for
your graph. Whenever you change the window settings, you
must press GRAPH to see the changed graph. The larger the space
between the Xmin and Xmax values, the farther out you’ve
zoomed. This is also true for a large space between the Ymin and
Ymax values.
values such as 0.1, 1, or 10. Setting the Xscl to 10 will give you
x
the space between the tick marks smaller, and a larger value for
larger space.
With the two equations you have already in the Y= screen
x
GRAPH
y2
. You should see the
y1
following graph.
point of intersection of two equations. In order
for this to work, you must adjust the window
settings so the point of intersection is visible.
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Foundations for College Mathematics MAP4C-B
Lesson 2, page 9
Finding the Point of Intersection Using
the TI-83 Plus
To find the point of intersection of two equations, the following
must occur first:
intersection is visible.
2nd
TRACE
5:intersect, then
ENTER
,
ENTER
,
ENTER
.
If you perform this on the two equations you’ve
explored so far, you should get “Intersection
screen.
x
value of x
more accurate method than the previous two methods. You could
verify this result using a scientific calculator and you would get
Let’s look at another example.
Example
x
Plus.
Solution
and
previous equations.
Y=
y2
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CLEAR
ENTER
CLEAR
ENTER
y1
to erase the
x
and
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Lesson 2, page 10
MAP4C-B Foundations for College Mathematics
to see if the point of
intersection is visible. If not, use some estimating to determine
good values to use in the window screen.
GRAPH
y
at 140 and Ymax at 160 with a scale of 1.
WINDOW
and set Ymin
x
4
GRAPH
2nd
.
TRACE
5:intersect, then
ENTER
,
ENTER
,
ENTER
.
You should have the following screens:
x
This can be verified using any scientific calculator to calculate
entire value of x x
=
I think you’ll agree that if you wanted to achieve an answer
accurate to seven decimal places using any of the previous
methods, it would take far too much time. Now let’s look at using
this method to solve a real-world application of exponential
equations.
Example
x
,
annually can be modelled by the equation A
where x is the number of years invested and A is the amount of
the investment. How long would it take for this investment to be
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Foundations for College Mathematics MAP4C-B
Lesson 2, page 11
Solution
x
Option 1
Option 2
x
x
y1
and y2
1.08x
y2
y1 = 1.08x and
then find the point of intersection
point of intersection, you should
see this:
point of intersection, you should see
this:
Both methods reveal the same answer. It would take
Support Question
(do not send in for evaluation)
to three decimal places. Include your window settings and a
sketch of the graph clearly showing the location of the point
of intersection.
x
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x
= 20
x
x
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Lesson 2, page 12
MAP4C-B Foundations for College Mathematics
Method 4: Solving Exponential
Equations Using a Common Base
virtually all the time, regardless of the numbers used in the
problem. If a graphing calculator is not available, and if the
numbers work out “nicely,” you may be able to use the following
method. It’s based on the concept that if two powers with the
same base are equal, then their exponents must be equal.
Example
Solve 4x = 64
Solution
You begin by first rewriting 64 as 4 .
4x = 4
Since you have these both written using the same base, you can
set the exponents equal. Therefore, x
This can be extended to more complex questions, but you should
follow the following steps:
variable.
Example
x
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Foundations for College Mathematics MAP4C-B
Lesson 2, page 13
Solution
x
2x
.
2x = 2
Therefore x
Example
2x
Solution
2x
2x
.
2x
Set the exponents equal.
2x
2x = 4
x=2
Example
Solve 9
2x
x
Solution
9
9
9
x
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2x
x
x
x
2 2x
First, rewrite 81 as 92, then use the power of a
power rule from Lesson 1.
= 94x
x
x
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Lesson 2, page 14
MAP4C-B Foundations for College Mathematics
Support Question
(do not send in for evaluation)
10. Solve each of the following by determining a common base.
2x
= 16
x
x
2x
Applications of Exponential
Equations
Exponential equations can “model” any situation in which there
is exponential growth or decay. This is evident with compound
interest investments, which, as you know, have exponential
curves when graphed. Exponential curves are also common with
many graphs of human, animal, or bacterial population growth.
As you know, growth curves start low and end high.
When radioactive isotopes decay, or break down naturally into
another isotope, they do so according to a known half-life rate.
Half-life refers to the time needed for a quantity of the material
to decay to half of its original level. A graph of half-life decay is
an exponential curve that starts high and ends low.
In all of these cases, an exponential model is the best choice to
model the situation, and, as such, will result in an exponential
equation. The following are examples of real-world applications
of exponential equations.
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Foundations for College Mathematics MAP4C-B
Lesson 2, page 15
Financial Math
You may have seen this formula used in other math courses.
n
A=P
, where A = the amount of the investment, including
interest
P = the original principal, or amount
invested
i = the amount of interest per compounding
period, and
n = the number of interest compounding
periods.
For this lesson, you will not be expected to apply this formula.
Instead, it will be given to you as it applies to the problem. When
the unknown value is “n,” you have an exponential equation.
Example
He invests it, hoping to have $1 million by his retirement age of
n
. Will he achieve his goal?
using the equation A
Solution
x
x
= 1 000 000:
= 20.
x
y1
and y2
Find the point of intersection of these equations. Press
2nd
TRACE
5 : intersect, then
ENTER
,
ENTER
,
ENTER
.
graph are shown.
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Lesson 2, page 16
MAP4C-B Foundations for College Mathematics
The point of intersection is x = 41.42291, y = 20.
This means that it would take approximately 41.4 years to reach
The Natural World
Population growth is a common situation that can be modelled
using exponential equations. Whether it’s the growth of a
rabbit population in a forest, or bacteria in a petri dish, these
populations will typically grow at an exponential rate. This type
of growth assumes there are no limiting factors, such as habitat
or food limitations, and other limits on population such as
predators and mass migrations.
Example
A population of cells is being studied to better understand a
disease. This particular type of cell doubles every day. If a sample
of these cells is estimated to consist of 1000 cells initially, the
size of the sample after n days can be modelled with the equation
n
, where S = the sample size of cells, and n = the
S
number of days that have passed. How long would it take to
Solution
This is one of those cases where you can use a common base. You
n
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Foundations for College Mathematics MAP4C-B
Lesson 2, page 17
n
the power.
9
2n
.
2n = 29
n=9
Therefore, 9 days are needed to reach a cell population of
This last example illustrates a growth situation, but materials
decrease can be seen with the decay of radioactive isotopes
as previously mentioned. This decrease can also be seen as a
substance is naturally broken down by the human body once it is
injected.
Example
Dye is injected to test pancreas function. The mass, R grams,
of dye remaining in a healthy pancreas after t minutes is given
t
where I grams is the mass of dye
by the equation R = I
healthy pancreas, determine how much time elapses until
your solution.
Solution
t
. Since this
x
and y2
equations y1
window settings and the resulting graph.
The point of intersection is x
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y
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Lesson 2, page 18
MAP4C-B Foundations for College Mathematics
Let’s look at an example that involves a radioactive substance
into one or more “daughter isotopes.”
Example
The radioactive isotope tungsten-187 has a half-life of one day.
The decay of this substance can be modelled using the equation
n
where M
I = the
M=I
If a rock sample initially contains 41 g of tungsten-187, how long
Solution
x
y1
x
and y2
The following window setting will give the resulting graph.
The point of intersection is x
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y
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Foundations for College Mathematics MAP4C-B
Lesson 2, page 19
Support Questions
(do not send in for evaluation)
11. The company you work for has downsized and you have
and decide to invest it for retirement, earning an average
x
,
how you arrived at your answer.
12. A large piece of aluminum is gradually pressed into a thin
sheet by passing through rollers. Each pass through the
rollers results in a thinner piece of aluminum according to
x
, where A = the thickness of the
the equation A
x = the number of passes
through the rollers. How many passes are needed to end up
to solve, and include a sketch of the graph with the window
settings.
a half-life of one day, how long would this take? Show your
work, and solve without using graphing technology.
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Lesson 2, page 20
MAP4C-B Foundations for College Mathematics
Key Questions
Save your answers to the Key Questions.
When you have completed the unit, submit them to ILC for marking.
6.
Use trial and error to solve for x in the equation 5x = 75,
accurate to two decimal places. List all values that you
attempted. (2 marks: 1 for answer, 1 for showing values
attempted)
7.
Solve for x by determining a common base. Show all steps.
a) 2x = 64 (1 mark)
b) 3n+4 = 272n
(2 marks)
c) 42(x+5) – 11 = 245 (2 marks)
8.
Solve each exponential equation using the TI-83 Plus
calculator. For each equation, create blank TI-83 Plus
windows similar to those provided below to include your
window settings and a sketch of the graph. Clearly identify
the location of your solution by labelling the point of
intersection accurate to two decimal places.
a)
4x = 100
WINDOW
Xmin = 3
Xmax =
Xscl =
Ymin =
Ymax =
Yscl =
Xres = 1
(4 marks: 1 mark for appropriate window settings,
1 mark for graph, 2 marks for correctly labelled point of
intersection)
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Foundations for College Mathematics MAP4C-B
x
Lesson 2, page 21
= 0.4
WINDOW
Xmin = 1
Xmax =
Xscl =
Ymin =
Ymax =
Yscl =
Xres = 1
(4 marks: 1 mark for appropriate window settings,
1 mark for graph, 2 marks for correctly labelled point of
intersection)
9.
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The half-life of tungsten-187 is one day. How long would
it take for 64 g to decay to 4 g? Round your answer to two
decimal places. Show all work and solve without the use
of graphing technology. (3 marks: 1 mark for correct
equation, 1 mark for work shown, and 1 mark for final
answer)
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Lesson 2, page 22
MAP4C-B Foundations for College Mathematics
10. An investment can be modelled by the equation
A = 2500 (1.06)t, where A = the final amount of the
investment, and t = time in years. How many years would it
take to have a final amount equal to $10 000? Round your
answer to the nearest tenth of a year. Include a sketch of the
graph, list the functions used in the TI-83 Plus, and include
the window settings from the TI-83 Plus. (4 marks: 1 mark
for window settings, 1 mark for functions used, 1 mark
for graph, and 1 mark for answer correctly rounded)
WINDOW
Xmin = 10
Xmax =
Xscl =
Ymin =
Ymax =
Yscl =
Xres = 1
Now go on to Lesson 3. Do not submit your coursework to ILC
until you have completed Unit 1 (Lessons 1 to 5).
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MAP4C-B
3
Interpreting Graphs and Using
Graphical Models
Foundations for College Mathematics MAP4C-B
Lesson 3, page 1
Introduction
Graphs can help you see connections between things, such as the
amount of caffeine you drink each day compared to the number
of hours you sleep each night. Or you could study a group of
people whose names start with “B” to see if they are generally
taller than average. In some cases, it is clear that there is no
connection. In other cases, there is definitely a relationship, or
trend. This lesson will explore these types of graphs and examine
how the trends can be used to make predictions or justify
decisions. The graphs will also be described according to their
rate of change using proper units.
Estimated Hours for Completing This Lesson
Graphs of Mathematic Relationships
Formula for Determining Rates of Change
Comparing Rates of Change
2
Key Questions
1
What You Will Learn
After completing this lesson, you will be able to
proper units
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Lesson 3, page 2
MAP4C-B Foundations for College Mathematics
Graphs of Mathematic
Relationships
The tools of mathematics are often used to describe relationships
or trends when variables are compared. Some data sets reveal
a trend, and are related to each other, while some do not reveal
a trend. One problem is that when you are only looking at the
data as a table of numbers, these trends don’t always appear so
readily.
Is There a Trend?
Look at this table of values comparing a soft drink company’s
sales volume to the number of paid advertisements promoting
their drinks. These ads are played on traditional radio stations
as well as Internet radio and podcasts.
Number of
Ads per Week
Sales
($1000s)
2
2
4
8
7
8
6
10
9
12
10
9
7
11
11
7
12
9
14
12
16
11.8
17
12.1
18
11.9
As you might expect, if the company
runs more radio ads in a week, they’ll
have more sales. This is confirmed as
you look down the table of values.
What may not be so obvious is that
these sales volumes will level off as
you reach the bottom seven entries in
the lower part of this table.
So, if the soft drink company knows
at what point sales stop increasing, it
would not buy more than that number
of ads. By understanding the trend,
the company can avoid wasting money
on ads that play so often they are
ignored or become annoying to the
listeners.
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Foundations for College Mathematics MAP4C-B
Lesson 3, page 3
The trend in this data would be seen more clearly in a graph, as
shown here:
Sales from Radio Ads
($1000s)
SalesSales
(Thousands
of Dollars)
14
12
10
8
6
4
2
0
0
5
10
15
20
Number of Radio Ads per Week
off or reach a “plateau” at approximately $12 000 in sales. This
graph, like many others, only compares two variables. There are
many other factors that are not shown here. Let’s consider a few
of those factors.
The sales of a soft drink will be affected by the advertising, but
they will also be affected by other factors, such as the following:
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Lesson 3, page 4
MAP4C-B Foundations for College Mathematics
Using the Trend
In general, a graph will provide a visual image of the
relationship between sets of data. It will make it easier to see
what type of trend, if any, is present. Later in the unit you will
see how you can use algebra to interpret these relationships
in greater detail, but for the time being you will only examine
relationships in general terms.
Example
The Boston Marathon is a well-known 42 km race that has
been run since 1897. The race is always run on an official
holiday on the third Monday in April, called Patriots’ Day, a
day when libraries, schools, and many businesses are closed in
Massachusetts. The graph that follows shows the winning times
answer the questions that follow.
Winning Times (min)
Boston
Marathon
WinningTimes
Times, -1927–2020
Boston
Marathon
Winning
1927 - 1994
180
160
140
120
100
80
60
40
20
0
1920
1940
1960
1980
2000
2020
Year
in the year 2020.
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Lesson 3, page 5
Solution
running shoes, aerodynamics, diet, and exercise. The more
gradual changes you see in recent times reflect the fact that
the time can’t continue to decrease at the same rate over
time. The human body can only run a marathon so fast and
we are quickly reaching the limits of the body’s endurance.
estimate the marathon time in 2020, as follows:
Winning Times (min)
Boston
Marathon
Winning Times
Times, 1927–2020
Boston
Marathon
Winning
- 1927 - 1994
180
160
140
120
100
80
60
40
20
0
1920
1940
1960
1980
2000
2020
Year
If the trend were to continue, the winning time may be
approximately 120 minutes.
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Lesson 3, page 6
MAP4C-B Foundations for College Mathematics
Support Question
(do not send in for evaluation)
14. The discus throw is a sport in which competitors try to throw
winning discus throws from the Olympics to answer the
questions that follow.
Olympic
DiscusResults,
Results - 1928-2004
Olympic
Discus
1928–2020
80
Winning
Distance
(m)
Winning
Distance
(m)
70
60
50
40
30
20
10
0
1920
1940
1960
1980
2000
2020
Year
throw in the 2020 Olympics.
There are Suggested Answers to Support Questions at the end of
this unit.
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Foundations for College Mathematics MAP4C-B
Lesson 3, page 7
Formula for Determining Rates of
Change
Earlier in this lesson, you only described the relationships
visible in a graph in general terms. A more specific description
would include the rate of change. Basically, this refers to the
steepness of the graph. You may have also heard of it referred to
as the slope of the graph.
Keep in mind that this formula is only valid for straight lines,
of change is calculated by taking two points on the graph and
comparing the differences in the x and y values. If these two
points are shown as ordered pairs such as
change in y y2 − y1
x2, y2
x1, y1
=
change in x x2 − x1
In the following example, you will see that for convenience, the
two compared points you select should be “easy” whole numbers,
will sometimes need to estimate values from the graphs.
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MAP4C-B Foundations for College Mathematics
Example
Calculate the rate of change for each of the following graphs:
120
100
yy
80
60
40
20
0
0
5
10
15
20
25
30
x
x
120
100
yy
80
60
40
20
0
0
2
4
6
8
10
12
14
xx
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Lesson 3, page 9
45
40
35
30
y
y
25
20
15
10
5
0
0
5
10
15
20
25
x
Solution
Let’s look at some ways you could use to determine the rates of
change for these three graphs.
always easy to find points on a graph that are clear and easy
to read, but care should be taken to select points accurately.
Now apply the formula for rate of change:
Rate of change =
8 0 −20 60
=
=4
2 0 −5
15
Therefore, the rate of change for Graph A is 4.
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MAP4C-B Foundations for College Mathematics
120
100
y
y
80
60
40
20
0
0
5
10
15
20
25
30
xx
would be
2 0 − 1 0 0 −8 0
=
= −8 . Therefore, the rate of change
1 0 −0
10
this example:
that go down to the right.
x and y values, always start with the
values from the same point.
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Lesson 3, page 11
120
100
yy
80
60
40
20
0
0
2
4
6
8
10
12
14
x
part in the following graph you examine. Let’s look at each
section:
Section A: This is a straight line, so you can determine the
The rate of change would be
15 − 5 10
=
=2
5−0
5
Section B: This is also a straight line, and the rate of change
rate of change would be
16 − 16 0
= =0
10 − 6 4
Section C: This section is curved, so it’s hard to calculate the
rate of change because it’s changing. It is less steep near the
beginning of the section, then steeper as you move right.
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MAP4C-B Foundations for College Mathematics
45
40
35
30
25
yy
C
B
20
15
A
10
5
0
0
5
10
15
20
25
x
Units for Rate of Change
When measuring rate of change on a graph, the units will be
determined by the axes of the graph. Since rate of change
change in y
units for y .
=
then the units will be
change in x
units for x
Let’s take another look at the previous examples once you put
the data into context by including the units on the axes.
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Foundations for College Mathematics MAP4C-B
Lesson 3, page 13
120
Distance
(metres)
DIstance
(metres)
100
80
60
40
20
0
0
5
10
15
20
25
30
Time (seconds)
This rate of change was calculated to be 4, but now that we see
the units included, we should be more specific. The rate for this
graph would actually be 4 metres per second, or 4 m/s.
Fuel Consumption (litres)
120
100
80
60
40
20
0
0
2
4
6
8
10
12
14
Time (hours)
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Lesson 3, page 14
MAP4C-B Foundations for College Mathematics
Example
Determine the rate of change, including units, for each of the
following graphs. Show your work.
Money vs. Time
300
Money
Earned
($) ($)
Money
Earned
250
200
150
100
50
0
0
5
10
15
20
Time
(hours)
Time
(hours)
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Lesson 3, page 15
Temperature vs. Time
Temperature
(C°) Celsius)
Temperature
(degrees
120
100
80
60
40
20
0
0
5
10
15
20
25
30
TimeTime
(min)(min)
Banquet
BanquetHall
HallCosts
Costs
14000
Total
Costs
Total
Cost($)
($)
12000
10000
8000
6000
4000
2000
0
0
100
200
300
400
500
Number
of People
Attending
Number
of People
Attending
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MAP4C-B Foundations for College Mathematics
Solution
Answers will resemble the solutions in the following table
however, different points may be used.
Graph
Points that May
be Used
Substitution into Formula
A
150 −0 150
=
=1 5
10 −0
10
B
6 0 − 1 0 0 −4 0
=
= −4
10 −0
10
C
10 000 −4000
6000
=
=30
200
3 2 0 −1 2 0
Estimated Rate of Change
Let’s look at a further example that involves money. The
following comparison illustrates a practical example of rate of
change.
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Foundations for College Mathematics MAP4C-B
Lesson 3, page 17
Example
This graph compares two simple interest investments. One is
answer the questions that follow.
Total
Amount
ofof
Investment
Total
Amount
Investment($)
Comparing Simple Interest Investments
3500
3000
2500
2000
6% Investment
1500
9% Investment
1000
500
0
0
5
10
15
20
25
30
Time Invested (years)
greater rate of change. Give two reasons for your answer.
units. Show your work.
Solution
of change for the following reasons:
change.
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Lesson 3, page 18
Interest
Rate
MAP4C-B Foundations for College Mathematics
Points that May
be Used
Apply Formula
2 0 0 0 −1 0 0 0 1 0 0 0
=
= 9 0 .9 1
11
11 −0
Estimated Rate of
Change
$90.91/year
1 7 5 0 −1 0 0 0
750
=
= 5 7 .6 9
13
13 −0
provided by the graph are limited by your estimation skills and
the accuracy of the graph. If the values of the points are not
perfectly clear, then the rate of change may be off a bit. For the
if you look at the table of values that was used to create these
graphs.
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Foundations for College Mathematics MAP4C-B
table to recalculate the rate
of change using exact points
rather than points estimated
from the graph.
Lesson 3, page 19
Time
Invested
(years)
1 7 8 0 −1 0 0 0 7 8 0
=
= $ 6 0 /yea r
1 3 −0
13
1000
1000
1
1060
1090
2
1120
1180
1180
1270
1240
6
7
1420
8
1480
9
Rate of change =
1 9 9 0 −1 0 0 0 9 9 0
=
= $ 9 0 /yea r
1 1 −0
11
Total
Value of
Investment
(9%)
0
4
Rate of change =
Total
Value of
Investment
(6%)
1720
1810
10
1600
1900
11
1660
1990
12
1720
2080
1780
2170
1840
2260
14
1900
Now you can see that estimates from graphs are generally less
accurate than using the table of values that was used to generate
the graph.
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MAP4C-B Foundations for College Mathematics
Support Questions
(do not send in for evaluation)
naturally break into three segments. Describe the general
help describe this trend.
Canada's
Population
1861 - 1996
Canada’s
Population,- 1861–1996
Population (millions)
35
30
25
20
15
10
5
0
1850
1900
1950
2000
2050
Year
16. This graph shows the commission earnings for two
They each have a different base salary, which is paid to them
the following questions.
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Foundations for College Mathematics MAP4C-B
Lesson 3, page 21
Commission Earnings
Earnings
Commission
1400
Earnings
($)
Earnings
1200
1000
800
Bob’s
Earnings
Bob's Earnings
600
Joanne’s
JoAnne's Earnings
400
200
0
0
1000
2000
3000
Sales ($)
Sales
units.
reasons for your answer.
Comparing Rates of Change
examine the graph of a relationship. For example, look at the
following graph showing population growth for three towns.
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MAP4C-B Foundations for College Mathematics
Population (thousands)
Population Growth
Growth for
Towns
Population
forThree
Three
Towns
60
50
40
Happyville
30
Bustlingtown
20
Dimsville
10
0
0
5
10
15
Year
Here are some facts you can infer from this graph:
fairly constant, but the population for Bustlingtown is
graph appears to be more curved, becoming steeper as time
goes on.
Here are some general facts to keep in mind when looking at
rates of change:
increases to the right, you have a positive rate
of change.
decreases to the right, you have a negative rate
of change.
There is no change in y for any change in x.
straight line, the rate of change is constant.
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Foundations for College Mathematics MAP4C-B
Lesson 3, page 23
curved, the rate of change is changing.
Example
Tariq’s family has asked him to buy some items at the store. This
graph shows the relationship between his distance from home
and time.
Distance
Travelled
from
Home
Distance
Travelled
from
Home
(m)
(metres)
Tariq's Trip to the Store
1200
C
1000
800
D
B
600
400
A
200
E
0
0
20
40
60
80
100
Time
Time(minutes)
(min)
answers.
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MAP4C-B Foundations for College Mathematics
Solution
E. The graph is a straight line. You could also consider
segment C, which is constant with a zero rate of change.
horizontal line so there is no change in the vertical
y
D since it is the steepest. The final part of segment
B would be steepest, but in terms of segments, D is
steepest.
the graph is curved.
400 − 0 400
Rate of change =
=
= 20 metres/minute, or 20 m/
20
−
0
20
min.
In some cases, it is useful to see multiple graphs together on one
set of axes. This makes it easier for comparisons to be made.
Example
Two different liquid solutions are brought to the boiling point
using two different hot plates, then placed in a freezer to cool
down. These liquids heat at different rates, and the graph of
your answer.
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Foundations for College Mathematics MAP4C-B
Lesson 3, page 25
Heating
Coolingof
of Two
Heating
and and
Cooling
TwoLiquids
Liquids
Temperature (°C)
Temperature
(°C)
120
100
80
liquid A
liquid B
60
40
20
0
0
10
20
30
40
50
Time(minutes)
(minutes)
Time
rate?
Solution
starting time, which is zero minutes.
is shown by the highest temperature reached before it is
cooled.
This is shown by the single straight line as the temperature
increases.
segment of the graph.
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Lesson 3, page 26
MAP4C-B Foundations for College Mathematics
Support Questions
(do not send in for evaluation)
17. In many cases, a worker’s hourly wage depends on the
number of hours worked. If a certain number of hours is
reached, the pay rate beyond that point increases. When
working beyond a certain number of hours, some employees
remaining hours. This type of arrangement is shown in the
following graph of pay vs. time.
Total Weekly Pay ($)
Pay vs. Time
worker’s pay structure in terms of hourly pay, and when the
hourly pay changes.
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Lesson 3, page 27
18. Two brothers, Charlie and Ed, decide to invest for their
retirement. They both expect to retire in 40 years. The
following graph shows their investment.
The investments
Charlie and
The investments
ofofChang
andEdEng
Total
Amount
Invetsment
($)
Total
Amount
of of
Investment
(dollars)
80000
70000
60000
50000
40000
30000
20000
10000
0
0
10
20
30
40
50
Time (years)
Charlie
Chang
Ed
Eng
like increasing, constant, or zero rate of change.
all of the phrases by placing the phrase
each phrase only once to describe each brother’s investment.
Include reasons for matching the phrase to that brother.
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MAP4C-B Foundations for College Mathematics
interest
Stored his money under his mattress for two years, then
invested it
Charlie
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Ed
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Lesson 3, page 29
Key Questions
Save your answers to the Key Questions.
When you have completed the unit, submit them to ILC for marking.
11. Use the graph below to answer the questions that follow.
Fuel Consumption (L/100 km)
Driving Speed and Fuel Consumption
Speed (km/h)
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a)
Describe the trend that is present in this graph. Include
specific references to the graph with your description.
(2 marks for describing trend, 1 mark for specific
references, for a total of 3 marks)
b)
What range of driving speeds produces the lowest fuel
consumption? (1 mark)
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Lesson 3, page 30
MAP4C-B Foundations for College Mathematics
12. A company has tracked its monthly profits as compared to its
sales volume and the resulting graph is shown below.
Monthly Profits
Profit ($1000's)
(1000s)
Profit
25
20
15
Profit per month
10
5
0
0
50
100
150
200
Sales Volume (units per month)
the sales volume that would result in the highest profit.
(2 marks for describing trend, 1 mark for maximum
point)
peak sales volume. (1 mark)
(1 mark)
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Lesson 3, page 31
temperature vs. time is shown below.
Temperature (C°)
Temperature of Solution
Time (min)
Answer the following questions using the letters A through D on
the graph above. More than one letter may be required for each
question.
(1 mark)
(1 mark)
(1 mark)
(1 mark)
(2 marks)
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MAP4C-B Foundations for College Mathematics
needs to do a few errands along the way, and Dave wants to
drive straight to the cottage. The distance/time graph of their
respective trips is shown below.
Time (hours)
thinner line? Explain your reasoning. (2 marks)
each brother was travelling? Include an example. (2 marks)
hours.
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Lesson 3, page 33
Time (min)
Which line belongs to whom? Explain why one line has a steady
rate of change and the other starts out curved. Include specific
information to explain your reasoning with references to the
second graph. (3 marks)
Now go on to Lesson 4. Do not submit your coursework to ILC
until you have completed Unit 1 (Lessons 1 to 5).
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MAP4C-B
4
Types of Graphical Models
Foundations for College Mathematics MAP4C-B
Lesson 4, page 1
Introduction
When you study a mathematical relationship you need to know
what type of relationship it is. Is it linear? Is it quadratic? Is it
exponential? In this lesson, you will look at all three of these
types of relationships. You will identify them by their graph
or data, and you will compare their graphs by describing their
initial conditions and rates of change. You will also learn how to
equation based on a set of data.
Estimated Hours for Completing This Lesson
Mathematical Relations and Difference Tables
1
1
Comparing Graphs of Pairs of Relations
1
Representing Data Algebraically
1
Key Questions
1
What You Will Learn
After completing this lesson, you will be able to
conditions and rates of change
the data and/or graph
related problems
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MAP4C-B Foundations for College Mathematics
Mathematical Relations and
Difference Tables
relationships or trends when two data sets are analyzed. If
you describe a relationship between two variables using math
symbols, it can be called a mathematical “relation.” As you know,
when you are examining data as a table of numbers, it is difficult
learn a way to detect the type of relation present between two
variables without graphing the data.
What Type of Relationship Is It?
When you examine data for trends, it can be useful to first know
what type of data you’re dealing with. In past math courses
you’ve examined linear and quadratic relationships and been
able to classify them. These concepts will now be reviewed and
expanded to include exponential relationships. By the end of
the lesson, you should have the tools necessary to develop an
appropriate algebraic or graphical model for a given relationship.
Determining the Type of Relationship
from the Data
To determine the type of relationship, the data will be set up in
what is called a table of finite differences, or a difference table.
This is simply a table of values with the following properties:
x-values appear in sequential order.
x-values increase in equal intervals.
yvalues.
The following examples will illustrate how the difference table
works and how to distinguish between linear, quadratic, or
exponential relationships. As you will see in the following tables,
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Foundations for College Mathematics MAP4C-B
Lesson 4, page 3
the first difference is calculated by subtracting each y-value
from the y-value that comes after it. The second differences
are calculated by subtracting the first differences in a similar
manner.
Linear Relationship
Look at the following difference table showing a linear
relationship. Take note of any patterns you see.
x
y
First
Difference
Second
Difference
0
2
4
0
6000
6
8
0
0
7000
The first differences are calculated by subtracting all adjacent
y
Summary
Since the first and second differences are constant,
the relationship is linear.
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MAP4C-B Foundations for College Mathematics
Quadratic Relationship
The following is a difference table showing a quadratic
relationship. Take note of any patterns you see.
x
y
First
Difference
Second
Difference
1
4
2
–4
0
–4
–4
4
–4
–8
Again, the adjacent y-values are subtracted. Since the first
differences are unequal, you know it’s not a linear relationship.
The process continues. You see that the second differences are
–
–
–
Summary
Since the second differences are constant and the first
differences are not constant, the relationship is quadratic.
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Lesson 4, page 5
Exponential Relationship
Again, look at the following difference table, which shows an
exponential relationship. Take note of any patterns you see.
x
y
First
Difference
Second
Difference
0
200
1 400
6
1 600
9 800
11 200
9
12 800
78 400
89 600
12
102 400
You see here that neither the first or second differences are
equal, so the relationship is neither linear nor quadratic. If you
examine the adjacent y-values a bit differently, you will see that
there is a common ratio. The ratio is calculated from the y-values
1600 ÷ 200 = 8
12 800 ÷ 1600 = 8
102 400 ÷ 12 800 = 8
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Summary
Since the ratio between y-values
is constant, the relationship is
exponential.
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Lesson 4, page 6
MAP4C-B Foundations for College Mathematics
Support Question
(do not send in for evaluation)
19. For each set of data, determine if the relationship is linear,
reasoning.
x
y
6
First
Difference
Second
Difference
First
Difference
Second
Difference
–21
–
20
–91
27
–126
–161
x
y
0
2
24
4
6
6 144
8
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Foundations for College Mathematics MAP4C-B
x
Lesson 4, page 7
y
0
First
Difference
Second
Difference
First
Difference
Second
Difference
0
6
9
12
x
y
4
8
109
12
16
469
20
There are Suggested Answers to Support Questions at the end of
this unit.
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Lesson 4, page 8
MAP4C-B Foundations for College Mathematics
Using Graphs to Determine the
Type of Relationship
Depending on what type of relationship you’re dealing with, its
graph will have a recognizable shape. The three relationships’
characteristics are summarized in the following table.
Type of
Relationship
Patterns in
the Data
Appearance of Graph
Linear
First and
second
differences
are constant
Straight lines
Quadratic
Only the
second
differences
are constant
Parabolas
or
Exponential
Constant
ratio between
y-values
Curves
that either
increase to
the right, or
decrease to
the right
or
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Foundations for College Mathematics MAP4C-B
Lesson 4, page 9
There are many technologies and software applications that
can graph a relationship. In most cases, though, these rely on
you entering an equation. In this case, you wish to see a graph
come in handy.
Graphing from Data Using the TI-83 Plus
data, you will revisit the first example in this lesson. Note that
your calculator uses an upper case X and Y instead of lowercase
x and y. The data is shown in the following table.
x
y
0
2
4
6000
STAT
and
ENTER
to select the 1:Edit
option. You should see columns labelled
6
8
7000
and
values.
CLEAR
To begin the process, you will enter the x
and y
the following steps.
ENTER
This is where the data will be placed. If data
is already present, you should clear it. To
clear the L1 column, first move the cursor to
highlight the top of a column. In this case,
key to highlight L1, and press
push the
. This can be done to any columns that have
,
5:SetUpEditor and press
should reset the columns to the factory settings.
STAT
ENTER
. This
Begin entering all of the x-values, pressing ENTER after each
entry. Repeat this process by moving to the first spot in the
L2 column and entering the y-values in the same manner. The
completed screens from each of the two steps of the data entry
process are shown here.
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Lesson 4, page 10
MAP4C-B Foundations for College Mathematics
basically have to tell the calculator where to get the points to
Y=
ENTER
to see the STAT PLOT menu
graph. Press 2nd
shown here. Following this you move the cursor to the settings
highlighted in black on the following screen and press ENTER to
activate them.
ENTER
.
scatter plot.
x-coordinates are
taken from the L1 column of data.
y-coordinates are taken from the L2
column of data.
your personal preference.
Important:
to ensure that
there are no equations entered. If there are, delete them from
the Y= screen by pressing CLEAR .
Y=
button. Based on the data you’ve entered, a sample window setting and graph are shown here.
GRAPH
Clearly the graph of this data forms a straight line. If your graph
window does not look like this, press WINDOW to enter the settings
above and then press the
GRAPH
button.
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Foundations for College Mathematics MAP4C-B
Lesson 4, page 11
The very first difference table in the lesson found this
relationship to be linear, and this graph reinforces that finding
since all graphs of linear relations are straight lines.
Example
the lesson.
x
y
1
2
4
Solution
Begin by pressing STAT ENTER and clear out data in all columns.
Refer to the previous table to get the x and y data you will need:
x-values into column L1 and y-values into column L2.
Y=
to ensure no equations are entered here.
to ensure that Plot 1 is on, select
scatter plot, and the “mark” style of your choice.
2nd
Y=
ENTER
The following screens show the data entered, an appropriate
window, and the corresponding graph.
Example
remember from earlier in the lesson.
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Lesson 4, page 12
MAP4C-B Foundations for College Mathematics
x
y
0
200
6
1600
9
12 800
12
102 400
Solution
Follow the steps as outlined in the previous two examples and
Note: Since the vertical, or y-scale, is so large, the lower points
don’t appear on this graph.
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Foundations for College Mathematics MAP4C-B
Lesson 4, page 13
Support Question
(do not send in for evaluation)
from the previous Support Question. In each case, include
an appropriate set of window settings, and then sketch the
accompanying scatter plot graph.
x
y
6
20
27
x
y
0
2
24
4
6
6144
8
x
y
0
0
6
9
12
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Lesson 4, page 14
MAP4C-B Foundations for College Mathematics
x
y
4
8
109
12
16
469
20
Comparing Graphs of Pairs of
Relations
It is often useful to graph a pair of relations on the same set
of axes. Not only will the graph help determine the type of
relationship, but seeing the graphs together will allow you to
draw comparisons between them. After looking at a graph of
two relations, you may find that a relation is linear. Recall from
change in y y2 − y1
. You will need to use this formula in the
=
change in x x2 − x1
following example involving the growth of two investments.
as
Example
The values of two investments are shown on the following graph.
Both investments start at the same time, but one is calculated
using simple interest, and the other is calculated using
compound interest. Your bank has said that compound interest
investments are better than their simple interest counterparts,
but this fact becomes crystal clear through the use of a graph.
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Foundations for College Mathematics MAP4C-B
Lesson 4, page 15
Value of Investments
90000
80000
Investment Value ($)
70000
60000
50000
Simple Interest
Compound Interest
40000
30000
20000
10000
0
0
10
20
30
40
50
Time (years)
investment, including units.
investment, including units.
interest investments in general.
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MAP4C-B Foundations for College Mathematics
Solution
simple interest investment starts at twice as much, $20 000.
cha nge in y y2 − y1
, to find
=
cha nge in x x2 − x1
the rate for the simple interest line.
Graph
Simple
Interest
Points that May
Be Used
Substitution into Formula
50 000 −20 000
30 000
=
=1200
25
25 −0
Estimated Rate of
Change
$1200/year
interest graph is curved, the rate of change is changing. It
appears that it initially has a lower rate of change, then as
time progresses, the rate of change gets steeper.
investment was half the initial simple interest investment.
This graph clearly shows that even with half as much
interest. This is good information for someone saving for
compound interest investment had a lower annual rate of
interest, which shows the power of compounding.
Comparing the Growth of Bacteria
Bacteria are tiny, one-celled organisms that are often grown in
glass dishes in a laboratory for research. A group of bacteria is
called a colony. In this second example, you’ll use your knowledge
growth of two bacteria colonies.
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Foundations for College Mathematics MAP4C-B
Lesson 4, page 17
Bacteria Example: Part A
In two colonies of bacteria, the population doubles every hour.
The initial population of Colony A is twice that of Colony B. If
x for time and y for number of bacteria.
generate a graph of the data on one set of axes. Include
the window settings used.
Solution
follows.
0
Time (hours)
Colony A
1
2
1000
2000
4000
8000
16 000
1000
2000
4000
8000
Colony B
4
x for time and y for number of bacteria, we can divide
each successive y-value to find the common ratio. The
common ratio is 2 for both colonies. Look at this table to see
how an expression can be developed.
0
Time (hours)
Colony A
Colony B
1
2
4
× 20
× 21
× 22
×2
× 24
× 20
× 21
× 22
×2
× 24
16 000 =
×2
If you follow the patterns in this table, you see the
expressions will become:
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Colony A: y
x
Colony B: y
x
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Lesson 4, page 18
MAP4C-B Foundations for College Mathematics
c)
The equations from b) should be
entered for Y1 and Y2 into the Y=
screen as shown:
To determine the appropriate
window settings, look at the range of minimum and
maximum values for x and y in the data you have been
given. For Xmin and Xmax, the x-values range from 0 to
5 hours. For Ymin and Ymax, the y-values could range
from nearly 0 to 16 000 bacteria after 5 hours. Two sets
of appropriate window settings and graphs are shown
below.
Since these graphs have the same rate of change, the
curves will never meet as x increases.
Bacteria Example: Part B
In this example, you will keep all conditions the same as
in Part A, except have Colony B triple every hour. You will
answer the same questions, as follows. The initial population
of Colony A is twice that of Colony B. If Colony A starts with
500 cells:
a)
Create a table of values for both colonies, using time =
0–5 hours, in 1 hour increments.
b)
Create an expression for the number of bacteria for each
colony. Use x for time and y for number of bacteria.
c)
Use the expression from part b) and the TI-83 Plus to
generate a graph of the data on one set of axes. Include
the window settings used.
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Foundations for College Mathematics MAP4C-B
Lesson 4, page 19
Solution
follows.
Time
(hours)
0
Colony A
1
2
1000
2000
4
4000
8000
16 000
Colony B
x for time and y for number of bacteria, we can divide
each successive y-value to find the common ratio. The
seen the expression for Colony A determined above in Part
A. The new expression for Colony B is shown below:
Time
(hours)
Colony B
0
1
0
2
1
4
2
4
Therefore, the new expressions for these colonies will be as
follows:
Colony A: y
x
Colony B: y
x
the Y= screen as shown in the following:
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1
and Y2 into
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Lesson 4, page 20
MAP4C-B Foundations for College Mathematics
To determine the appropriate window settings, look at what
the population of Colony B has surpassed the population of
Colony A. For Xmin and Xmax, the x-values range from 0 to
4 hours, so that we can see what happens during the hour
y-values could be
One possible set of window settings is shown here with the
graph.
To distinguish between graphs on the same
axes, press Y= and move the cursor left
of Y1. Keep pressing
until the slash
ENTER
press the GRAPH button. This will produce a thicker line, as
shown here.
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Foundations for College Mathematics MAP4C-B
Lesson 4, page 21
Support Question
(do not send in for evaluation)
21. The following graph illustrates the costs of taking a cab for
two different taxicab companies.
TaxiCab
Taxicab Comparison
Comparison
conditions of each
graph.
40
of change of each
graph, including
units.
Cost ($)
32
24
Company A
Company B
16
8
the costs equal for
each company?
0
0
10
20
30
40
50
60
Distance Travelled (km)
conditions when one
company should be used over the other company.
Representing Data Algebraically
As you have seen in some of this lesson’s examples, graphs can
be somewhat inaccurate or imprecise. If an algebraic expression
can be determined from a set of data, it becomes a mathematical
model. This model can be used to generate more accurate
estimates. Accuracy is important when making decisions about
future events or trends. For example, it would be useful to
accurately model human population growth so that adequate
farmland is preserved to grow enough food for the world in the
year 2020.
The problem here, however, is that in most cases, the data
collected in the real world may not fit your model perfectly. For
instance, a line may look like it has a slope of 2.00, when, upon
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Lesson 4, page 22
MAP4C-B Foundations for College Mathematics
closer inspection or with the addition of more data points, it
actually has a slope of 1.98. When researchers collect data, they
which will help you generate algebraic models for the data, and
also help you determine the best type of model to use.
Determining the Regression Equation
Using the TI-83 Plus
The term regression analysis refers to a technique used to
find an algebraic model that “best fits” a set of data. The process
simplifies things a great deal.
The first step is to determine the type of relationship. You
could use difference tables as you saw earlier in the lesson, but
when you use real data it is unlikely that the finite differences,
or common ratios, will work out as nicely as you saw earlier.
Statisticians assign a number to indicate how close to a line of
best fit the points in a scatter plot are. This number is called the
correlation coefficient, or the r-value. In this case, you will
r-value.
When you calculate the squared value of r, the result is called
r value of
the coefficient of determination or r2. An r2
strong correlation. In other words, the closer your r-value gets to
Let’s look at the original examples again to determine the
algebraic expressions.
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Foundations for College Mathematics MAP4C-B
Lesson 4, page 23
Example: Linear Relationship
Here you see the table of values from the linear relationship
x
0
6000
6
8
STAT
ENTER
2
4
x
y
7000
then
then enter all the values.
Once the values are entered, press STAT
to view CALC,
and scroll over with
then 4 for 4:LinReg(ax+b), which will
perform a linear regression and deliver
the result in the form ax b.
Press ENTER and you should see the following screen:
Note: If the r-value is not appearing on your
screen, press 2nd 0 to get the Catalog, and
until you see
scroll down using
DiagnosticOn. Then press ENTER
the above steps and your r-values should appear.
ENTER
. Retry
In the LinReg screen notice that
a is the slope, or rate of change if the data were graphed
b is the initial value of the graph
r2 is 1, which indicates a prefect fit; since r
is a positive correlation
So, the regression equation that represents this data is y
x
On the CALC menu screen, you may also notice 5:QuadReg
0:ExpReg
regression tools in the next two examples.
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Lesson 4, page 24
MAP4C-B Foundations for College Mathematics
Example: Quadratic Relationships
Here you see the table of values from the quadratic relationship
equation.
x
y
1
x-values in for L1,
and the y-values in for L2.
2
STAT
4
5:QuadReg ENTER
should look like:
to view CALC
. This is what your screen
For this data, the regression equation that represents the data is
x
y
x2
Sample Problem: Exponential Relationships
exponential data.
x
y
0
200
6
1600
9
12 800
12
102 400
Solution
The screens below show the data entered, and the regression
equation.
The regression equation for
x
. But, as
this data is y
you read earlier in this lesson,
the real-world data sets aren’t
always so nice. Let’s look at
some real data now.
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Foundations for College Mathematics MAP4C-B
Lesson 4, page 25
Sample Problem: Find the Orbit of the Missing Planet
This table shows data related to the planets of our solar system.
x
y
takes the planets beyond Earth a much longer time.
Planet
Average Distance from the
Sun (millions of km)
Orbit Time (days)
Mercury
88
Venus
108.2
Earth
149.6
Mars
227.9
687
Saturn
1429
10 760
2871
Neptune
60 190
There is a ring of rocks called the asteroid belt that orbit the Sun
think that the asteroid belt was once a planetary body, which
broke into many pieces during the formation of our solar system.
Determine the regression equation for this data and use it to
the Sun.
Solution
x
y
r2
Planet Data
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Linear
Quadratic
Exponential
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Lesson 4, page 26
MAP4C-B Foundations for College Mathematics
Although the linear equation has an r-value close to 1, the
quadratic is better with an r2
This means that by substituting values a, b, and c, the best
expression for us to use here would be y
x2
x
x
y
x2
represents x
y
2
y
Therefore, it would take approximately 2684 days for the
planetary body to orbit the Sun.
Support Question
(do not send in for evaluation)
22. In ancient times, it was harder to survive and life expectancy
was much lower. Due to disease, drought, and famine, most
humans did not even survive childhood. Scientists believe
that 74 000 years ago, only about 10 000 people existed. This
theory is based on recent studies of the human genome. By
.
The following table shows
estimates of the world’s
Year
Population (billions)
2.781
1960
and determine the
linear, quadratic,
and exponential
regression equations,
including r-values.
x and y
represent in these
equations?
1970
4.084
1980
4.447
4.844
1990
2000
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6.072
6.449
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Foundations for College Mathematics MAP4C-B
Lesson 4, page 27
c)
Which regression equation is best to use in this case? (Hint:
Consider what you know about populations, and do not
necessarily use the equation with the highest r-value.)
d)
Use the equation to predict the world’s population in the
year 2020.
Key Questions
Save your answers to the Key Questions.
When you have completed the unit, submit them to ILC for marking.
15. Create tables of difference like the following to determine if
each relationship is linear, quadratic, exponential, or neither.
Justify your answer. (1 mark for each answer and 1 mark
for each justification, for a total of 8 marks)
a)
x
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y
2
29
4
53
6
77
8
101
10
125
First
Difference
Second
Difference
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Lesson 4, page 28
MAP4C-B Foundations for College Mathematics
x
y
First
Difference
Second
Difference
First
Difference
Second
Difference
1
27
7
2187
9
x
y
2
7
4
6
8
109
10
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Foundations for College Mathematics MAP4C-B
x
Lesson 4, page 29
First
Difference
y
Second
Difference
1
7
9
16. Two banquet halls are being considered for a wedding
reception. Hall A charges $40 per person, and Hall B charges
shown below.
Cost ($)
Banquet Hall Costs
18000
16000
14000
12000
10000
8000
6000
4000
2000
0
Hall A
Hall B
0
100
200
300
400
500
Number of Guests
(2 marks)
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Lesson 4, page 30
MAP4C-B Foundations for College Mathematics
same amount. Describe the conditions under which one hall
would be cheaper than the other. Include references to the
initial conditions of the graph and the rate of change of the
graph. (4 marks: 1 mark for identifying approximate
point, 1 mark for selecting proper hall for each
range of conditions, 2 marks for using correct initial
conditions and rate of change to justify)
17. Two different people invest money for a 40-year period, both
interest compounded annually. The graph of their investment
amounts is shown below.
Compound
Simple
amount for each investor. (2 marks)
for an investment, simple or compound interest. (2 marks)
rate of change is constant, and explain your reasoning. (3
marks: 2 marks for correct type of interest and
1 mark for correct reasoning)
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Foundations for College Mathematics MAP4C-B
18. a)
Lesson 4, page 31
Enter the following two lists of data into the TI-83
Plus and graph them. What values would you enter
for the Xmin, Xmax, Ymin, and Ymax of your window
settings? Sketch your scatter plot graph using
technology or graph paper. (4 marks)
x
1
3
5
7
9
11
13
15
y
50
14
2
14
50
50
194
302
b)
State the type of mathematical relationship that exists. Once
you’ve recognized the type of relationship (linear, quadratic,
or exponential), determine the regression equation for
this relationship. Explain your reasoning for the type of
relationship. (3 marks: 1 mark for type of relationship,
1 mark for regression equation, and 1 mark for logical
reasoning)
c)
Use the equation to determine the value of y if x is 8.2.
(1 mark)
Now go on to Lesson 5. Do not submit your coursework to ILC
until you have completed Unit 1 (Lessons 1 to 5).
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MAP4C-B
5
Working with Formulas
Foundations for College Mathematics MAP4C-B
Lesson 5, page 1
Introduction
Formulas are used in all types of mathematics. They are used
to solve for something that is unknown. To do this effectively
you need to know how to work with formulas. Some people may
find it easier to substitute what you know first, then solve for
the unknown. Others may find it easier to isolate the unknown
first. It is also useful to determine the type of equation you’re
dealing with to decide on the best strategy. In this lesson, you
will explore all of these topics, along with careers and college
programs that utilize mathematical modelling.
Estimated Hours for Completing This Lesson
Problem Solving with Formulas
2
Solving Multi-Step Problems in Real-World Applications
1
Researching Careers and College Programs in Mathematical Modelling
1
Key Questions
1
What You Will Learn
After completing this lesson, you will be able to
xn = a using rational exponents
by first substituting known values and by first isolating the
variable
applications
modelling in occupations and college programs
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MAP4C-B Foundations for College Mathematics
Problem Solving with Formulas
Formulas help you to solve for an unknown or “missing” value.
There are many different formulas that allow you to solve for
missing values. The following table shows some examples of
formulas used in geometry.
Formula
A=
πr
Calculates
the area of a circle
2
P = 2l
w
πd
V = πr2h
the perimeter of a rectangle
the circumference of a circle
C=
the volume of a cylinder
At first glance, it would appear that these four formulas allow
you to solve for A, P, C, and V. In fact, they allow you to solve
for many more variables, depending on which variables of
the equation are known. Consider the formula for finding the
capacity of a cylinder:
Volume of a Cylinder (V =
r
πr2h)
h
If the known values are…
r
h
…you are left with the
following unknown…
...allowing you to
calculate…
V
Volume of the cylinder
V
h
r
Radius of the cylinder
V
r
h
Height of the cylinder
To be able to effectively work with formulas, you’ll need good
algebraic skills. The examples in this lesson will review some
of these skills. The lesson will also look at the two different
approaches to working with formulas:
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Foundations for College Mathematics MAP4C-B
Lesson 5, page 3
unknown, or
values
Before you begin solving formulas, you’ll need to learn how to
categorize some commonly used formulas after substituting
the known values. You will make connections between three
categories of equations: linear, quadratic, and exponential. In this
lesson, the term equation will be used rather than function. A
function is a specific type of equation in which one value x
y
x and y
are typically used is to make it easier to graph the equation on
the xy-plane. In other math courses you may be familiar with,
the output value in functions is often represented as f x
than y.
What Type of Equation Is It?
The type of equation you will be left to solve depends on what
the known values in your formula are. If you know what type
of equation you are dealing with, you can choose the proper
method to solve it. This section of the lesson will always use y as
the output variable. Let’s first review the types of equations you
might see in this lesson:
exponent of 1. In other words, when there is no visible
exponent on the variable, the equation “has a degree of 1.”
Examples include 2x = y, y
d, y
t, and y = x.
exponent of 2. Examples include y = πr2, 2x2
y = x 2.
an exponent. Examples include y
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n
x
x
y = 0, and
= y, and y
x
.
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Lesson 5, page 4
MAP4C-B Foundations for College Mathematics
Example
Consider the formula V = πr2h. Into which variables could
following?
Keep in mind that π is already a constant with an approximate
π.
Solution
Basically, you need to change V = πr2h by identifying which
variables to make constant and which variable to leave “as is.”
r to a
constant and you are left with h and V, for instance
V = π 2 h. You could use any number in place of r. The
important thing is that the variables “left alone,” in this case
h and V, have an exponent of 1, which means the equation
will be a line when graphed on the xy-plane.
You could set either V or h as constants, leaving you with r2
V is
V = πr2
the output variable.
If you were to write the previous solution as a function, you
would get y = 100πx2. You could say that you have “constrained”
the variable h by setting it equal to 100, which has resulted in a
quadratic equation.
Let’s look at an equation that also has three variables but does
not include π.
Example
A population of bacteria is modelled by the equation P = I
where
n
,
P is final population
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Foundations for College Mathematics MAP4C-B
Lesson 5, page 5
I is initial population
n is the number of 4-hour periods
What variables would you need to set to a constant to generate
an exponential equation?
Solution
Recall that an exponential equation is an equation with the
variable as an exponent, so the only way to do this here is
to leave the variable n as a variable, and set either P or I as
constants.
Substitute then Isolate
In cases where a single calculation is needed, it may be useful to
first substitute all known values, then solve for the unknown. By
doing this, there is a chance the equation can be simplified first.
To illustrate this, the compound interest formula
A=P
in
will also illustrate connections between formulas and linear,
quadratic, and exponential equations. In this formula
A is total amount
P
i is annual interest rate
n is number of compounding periods
Example
A=P
i = 0.06, n = 10, and A
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i
n
P
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Lesson 5, page 6
MAP4C-B Foundations for College Mathematics
Solution
First, substitute what is
10
P
10
P
P(1 .0 6)1 0
2500
=
(1 .0 6)1 0
(1 .0 6)1 0
2500
=P
(1 .0 6)1 0
P
P
Note: This is a linear equation
since the variable is of degree 1.
Now simplify.
10
to
isolate the variable P.
10
n
to isolate P, then calculate. Be careful to follow
the proper order of operations on your calculator. Try it to ensure
that you get the same answer. You will need your calculator to do
the following problems.
Example
A=P
i n, calculate i if A
P=
n = 2. Round your answer to two decimal places.
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Foundations for College Mathematics MAP4C-B
Lesson 5, page 7
Solution
First substitute what is
known.
i
Note: This is a quadratic equation
i
in the base of a power with degree
2.
2
5675
= (1 + i)2
5000
1 .1 3 5 = (1 + i)2
isolate term with the variable i.
Divide the fraction to simplify it.
1 .1 3 5 = (1 + i)
Take the of both sides
1 .1 3 5 − 1 = i
i = 0 .0 6 5 3 6 3 7 8 7 6 , or
Subtract 1 from both sides to
i = 0 .0 6 5 4
isolate the variable and evaluate
= 6 .5 4 %
using a calculator.
Example
A=P1
A = $20 000, P
i n, calculate n
i = 0.07.
Solution
First, substitute what is
known.
Note: This is an exponential
n in
n
20 000
= (1 .0 7 ) n
5000
4 = (1 .0 7 ) n
the term with the variable.
Now you’re left with an exponential
equation.
following these steps:
Press
Y=
.
Enter Y1= 4 and Y2 = 1.07
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Press
WINDOW
Press
2nd
ˆ
X,T,θ,n
in the Y= screen
and select appropriate window settings.
TRACE
5:intersect then
ENTER
ENTER
ENTER
.
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Lesson 5, page 8
MAP4C-B Foundations for College Mathematics
The results and a possible set of window settings are illustrated
in the following three screens.
Therefore, in this case the value of n that satisfies the equation
n
is approximately 20.49.
In the previous example, the
equation was of the form ax = b,
or an exponential equation. A
similar-looking type of equation,
xa = b, requires a different type of
strategy.
Remember to turn off
the STAT PLOT from
the previous lesson.
Example
. With the cursor
on OFF, press ENTER .
Press
2nd
Y=
ENTER
For each of the following, solve for
x accurate to three decimal places.
x4
x9
x6
Solution
In each case, the approach is to transform the equation to the
form x1 = a number, or x = a number. This will be your solution.
To do this, use the reciprocal of the exponent on x as follows:
x4 = 20
( )
x4
1
4
= 20
1
4
1
x1 = 20 4
x = 2.114742527
x ≅ 2.115
x9 = 87
( )
x9
1
9
1
= 87 9
1
x1 = 87 9
x = 1 .64248775
x ≅ 1 .642
Copyright © 2009 The Ontario Educational Communications Authority. All rights reserved.
x6
( )
x6
1
6
1
= 2500 6
1
x1 = 2500 6
x = 3.684031499
x ≅ 3.684
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Foundations for College Mathematics MAP4C-B
Lesson 5, page 9
A few things to remember:
1
1
÷
20 4 on a calculator, use the keystrokes 20
)
ENTER
4
.
(
ˆ
doing it accurately.
Isolate then Substitute
In some cases, it may be beneficial to isolate the desired
variable first then substitute. This is true in a case where
many calculations would need to be made, as illustrated in this
example.
Example
Four cylinders need to be produced, all with a height of 1.2m,
this table.
Cylinder A
Required
volume
V, in m
Cylinder B
1.0
Cylinder C
Cylinder D
2.0
Solution
If you substituted the known values first, then solved for the
unknown, you’d have to do this four times. It is more efficient to
r
values.
Volume of a cylinder is given by the formula V = πr2h. Isolate r
after substituting for h. You know that h = 1.2 m in this case, so
substitute this in for h.
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MAP4C-B Foundations for College Mathematics
V = π r2 (1 .2 ) Substitute in 1.2 for
V = 1 .2 π r2 h then simplify.
V
= r2
1 .2 π
V
= r2
1 .2 π
r=
Divide both sides by
1.2π to isolate r2.
Take the square root
V
1 .2 π
1.2π is treated as a
constant since π is
approximately 3.14.
When calculating, use
the 2nd
ˆ button
on your calculator for
π to achieve greater
accuracy.
V
to calculate the radius of the
1 .2 π
four cylinders. Be careful when entering the keystrokes on your
calculator. You must follow proper order of operations. One
You can use the formula r=
possible set of keystrokes to use for entering
V
1 .2 π
(
Required Volume ÷
1.2
)
)
×
2nd
ENTER
.
ˆ
Try calculating the following four radii on your calculator to
make sure that you get the same values.
Plus would be:
x2
2nd
Cylinder A
Cylinder B
Cylinder C
1.0
Required
volume
V, in m
Radius
r
0 .5
1 .2 π
≈
2.0
1
1 .2 π
≈
Cylinder D
1 .5
1 .2 π
≈
Copyright © 2009 The Ontario Educational Communications Authority. All rights reserved.
2
1 .2 π
≈
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Foundations for College Mathematics MAP4C-B
Lesson 5, page 11
Support Questions
(do not send in for evaluation)
K
v, measured
m
K=
1
mv2.
2
K if m = 20 kg and v = 18 m/s.
m if K = 486 and v = 9 m/s.
v if K
m = 7 kg.
answer.
24. Solve for x in each equation accurate to three decimal places.
1
x 3 = 14
x9
x
SA = 2πr2
πrh, where r = radius and h = height. Complete
h
cylinder with the given surface area and radius, accurate to
two decimal places.
Cylinder A
Cylinder B
Cylinder C
Surface area,
2
SA
Radius, r
8
6
Height, h
There are Suggested Answers to Support Questions at the end of
this unit.
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Lesson 5, page 12
MAP4C-B Foundations for College Mathematics
Solving Multi-Step Problems in
Real-World Applications
In mathematics, there is often more than one way to solve
able to manipulate formulas allows you to solve problems more
efficiently. These examples illustrate how to work with formulas
to solve multi-step problems.
Example
A cylindrical tank is to be covered with two coats of paint,
a radius of 2.4 m. If the surface area of a cylinder is 2πr2
determine:
πrh,
2
Solution
two ways:
Method 1
Method 2
For one tank at a time:
For two tanks, the
adjusted formula would
be:
SA = 2π
2
π
= 88.9699
≈ 88.97 m2
SA
π
For two tanks:
SA
SA = 88.97 × 2 = 177.94 m2
= 177.94 m2
The amount of paint required =
approximately 10 L of paint.
2
π
1 7 7 .9 4
= 9.886, or
18
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Foundations for College Mathematics MAP4C-B
Lesson 5, page 13
While you are thinking about the formula for the volume of a
cylinder, let’s use this formula again to solve a different problem.
Example
A new road is being constructed and a large pipe is needed to
allow a stream to pass under the road. The following cylindrical
rtotal
h
rrinner
determine the volume of concrete you will need to construct this
tube.
Solution
To solve this problem, think of the concrete tube as two cylinders:
an outer “total volume” cylinder and an empty “inner volume”
cylinder. To get the volume of the concrete, you must subtract the
empty cylinder from the total volume.
Vcylinder = πr2h
Vconcrete = Vtotal
Vinner
To use the formula for the volume of a cylinder, you need to know
the height and the radius of both cylinders.
Height of both cylinders = 12 m
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MAP4C-B Foundations for College Mathematics
Method 1
Method 2
Vtotal = π
2
× 12
Vconcrete = πr2totalh – πr2innerh
= hπ(r2total – r2inner
Vinner = π
2
= 12π
× 12
= 49.48 m
Vconcrete = Vtotal
Vinner
Vconcrete
= 49.48 m
In step 2
of Method
2, πh is
factored out.
By factoring
this out, you
can isolate
the squares
of the two
radius values
and subtract
them.
Therefore, the volume of concrete needed to construct the culvert
tube is 49.48 m .
Support Question
(do not send in for evaluation)
26. The following gumball is to be created with an outer layer of
candy and an inner core of bubble gum.
rtotal
rinner
the complete gumball is to have a radius of 1.2 cm, determine the
volume of candy required to produce 10 000 of these gumballs.
4
The formula for volume of a sphere is V = π r3 .
3
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Foundations for College Mathematics MAP4C-B
Lesson 5, page 15
Researching Careers and College
Programs in Mathematical
Modelling
In the past, there were two main ways to do research for a
report. One method was to search in books, magazines, and
newspapers in a library. The other method was to search in a set
encyclopedias? Encyclopedias are a book or set of books that give
information on many subjects. They are often expensive, with
many volumes, totalling thousands and thousands of pages of
Today, with technology at nearly everyone’s fingertips, the first
place many people turn is the Internet. The Internet has made it
much easier to research information, but it is far from perfect.
This section will help you prepare for one of this lesson’s Key
Questions. In the question, you will research college programs
and careers that use mathematical modelling. It is suggested
that you speak to people in different careers that may utilize
mathematical modelling, but you are not required to do so. Many
students will choose to use the Internet to conduct most of the
research. Here are a few points to remember when conducting
research to generate a report:
paste” from a website, you should only use the information as
a source for your report.
formation you use in your report. If you need to use a word-forspecify the source and date of the quote.
conducted with a knowledgeable/qualified person, provide
the date of the interview, the person’s name, and his or her
job title. Be sure to ask the person’s permission to use what
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Lesson 5, page 16
MAP4C-B Foundations for College Mathematics
he or she says in your report.
title, date of publication, headline/article title, author, page
number.
carefully to narrow down your search.
Canadian, Canada, Ontario, and/or a city in your search
criteria.
websites may be biased, so make sure you use a variety of
sources.
based information is. Make sure you know something about
the company or organization that is the source of the
information you are using.
Researching Colleges
If you plan on attending an Ontario college, you will apply
can be found at the following website: www.ontariocolleges.ca
To apply for out-of-province colleges, you will need to contact the
college directly.
The OCAS website is very thorough and informative. When
you visit the site, you will see the following four options: Plan,
Find, Apply, and Confirm. If you click on the Plan section and
then choose “About the Colleges,” you may select a college from
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Foundations for College Mathematics MAP4C-B
Lesson 5, page 17
easily browse the websites of the different colleges and to look
through their programs. Clicking on the Find section allows
you to search based on different criteria. If you select “Advanced
Search,” you get more choices, including “Program Categories.”
This will likely be the most helpful, as you can select from many
different categories of programs.
Hint: Choose “Back to Search Criteria” rather than the Back
button of your browser in order to navigate back to where you
were.
Apply and Confirm
applying to college so they are not useful for the report you
are doing. If you require further information on a particular
program, the colleges will most likely be happy to send you
materials. Also, in many cases, names and contact numbers are
provided for people responsible for a specific program.
Researching Career Opportunities
There are many websites that provide information on careers.
provide a great starting point:
colleges, and universities.”
government agency that oversees apprenticeships. It provides
a very thorough list of links to a wide variety of occupations.
Social Development.”
The Human Resources and Social Development Canada
website allows you to search for occupations in a variety of categories. Each job category is given a unique NOC code
These websites provide some of the best ways to research
college programs and careers that use mathematical modelling.
However, you could also try using a search engine by entering
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Lesson 5, page 18
MAP4C-B Foundations for College Mathematics
search terms such as “mathematical modelling” and “Canada”
and “career” and then including the name of a specific college
or career. Remember: If you keep the quotation marks around
“mathematical modelling” and include several good search terms,
you will get fewer (and probably better) results.
Key Questions
Save your answers to the Key Questions.
When you have completed the unit, submit them to ILC for marking.
19. Use the compound interest formula A = P(1 + i)n to answer
the following questions.
a)
Isolate the variable P. (1 mark)
b)
Isolate the variable i. (1 mark)
c)
What type of equation is present if n is a constant equal to
1, rather than a variable? (1 mark)
d)
What type of equation is present if i, A, or P are constants
rather than variables? (1 mark)
e)
Explain your reasoning for parts c) and d) in one or two
sentences. (1 mark)
4
20. The volume of a sphere is given by the formula V = π r3 ,
3
where r is the radius of the sphere. If the volume of the
sphere is 800 cm3, calculate the radius, accurate to two
decimal places. Show your work. (2 marks)
21. Solve for x in x4 = 36, accurate to two decimal places. Show
your work. (2 marks)
22. The following two cylinders of equal size are to be constructed
so that the combined total volume is 20 m3.
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Foundations for College Mathematics MAP4C-B
r
Lesson 5, page 19
r
Cylinder 1
h
Cylinder 2
h
If the volume of a cylinder is V = πr2h, determine the height if
the radius is to be 1.2 m across. Show your work. (3 marks)
money at the same time for an 8-year period. Sachi invests
8
. Nari is shrewder, and was able to
formula A
invest only $8000 and end up with the same amount at the
end. His final amount can be modelled using the equation
A
i 8.
Showing all of your work, determine the value of i for Nari
and the yearly interest rate his money earned to allow this to
happen. (4 marks)
24. In this question, you will research an occupation and a
available to you, complete a table that gives information
about a college program and an occupation that involve
an application of mathematical modelling. Make sure you
choose an occupation that is not strongly related to the
“accounting certificate” program at George Brown College
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Lesson 5, page 20
MAP4C-B Foundations for College Mathematics
one occupation that involves applications of
mathematical modelling. Include the following:
required
modelling relates to the job
information, you may also interview someone about his
or her career if the occupation is related to mathematical
(4 marks)
one college program that involves applications of
mathematical modelling. Include the following:
entry into this program
program
of mathematical modelling
(4 marks)
Organize your results in a table. Include the sources of all
information. You must choose an occupation that is not directly
related to your chosen college program. Your table should
resemble the one that follows and have the same row headings,
with your answers in the right-hand column. (1 mark)
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Foundations for College Mathematics MAP4C-B
Lesson 5, page 21
and the Education it
Requires
Occupation
Employment
Outlook
Salary Range or
Average
Name and Location
Courses Required
for Entry
College Program
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Career Possibilities
How the Program
Explores an
Application of
Mathematical
Modelling
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Lesson 5, page 22
MAP4C-B Foundations for College Mathematics
If you have difficulty thinking of an occupation to research, you
may wish to consider one of the following:
Accountant
Business planner
Architect
Financial officer (bank)
Computer systems analyst
Computer technician
Engineer
Forest ecologist
Dentist
Electronics technician
Land surveyor
Small business owner
Biologist
Computer programmer
Financial planner
Market researcher
Nuclear medicine
technician
Urban planner
This is the last lesson in Unit 1. When you are finished, do the
Reflection for Unit 1. Follow any other instructions you have
received from ILC about submitting your coursework, then send it
to ILC. A teacher will mark your work, and ILC will return it to you
as soon as possible.
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Foundations for College Mathematics MAP4C-B
Unit 1
Suggested Answers to Support Questions
page 1
Unit 1
Lesson 1
x 2 y5
y4
-1 4
=
x
y
=
x
y
or
x
x3 y
2
2
2
6
x
x = 9x = 9x
3 x 3 z2
15 x 4 y3 z6
4 −1 3 − 4 6 − 4
3 −1 2
= 3 x y z = 3 x y z or
y
5 xy4 z4
a2 × b4 × a = a
× b4 = a b4
=2 =8
32 × 36
=
(3 )
2 3
⎛3 ⎞
⎜⎝ 5 ⎟⎠
64
−2
1
2
125
81
64
64
16
1
3
1
4
−1
3
2
3
2
25
⎛5 ⎞
= ⎜ ⎟ =
⎝3 ⎠
9
=
64 =8
3
=
125 = 5
81 = 3
1
1
=
=
3
64 4
=
=
7
4
32 + 6 38
= 6 = 32 = 9
2× 3
3
3
=
4
( 64 ) = 4 = 16
( 16 ) = 2 = 128
2
3
2
7
4
3
7
3
3
⎛ 4 ⎞
8
⎛2 ⎞
⎛4 ⎞2
⎜⎝ 9 ⎟⎠ = ⎜ 9 ⎟ = ⎜⎝ 3 ⎟⎠ = 2 7
⎝
⎠
81
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−
3
4
=
(
1
4
81
)
3
=
1
1
=
3
27
3
Copyright © 2008 The Ontario Educational Communications Authority. All rights reserved.
Suggested Answers to Support Questions
x2 = x6
Suggested Answers to Support Questions
page 2
45
Foundations for College Mathematics MAP4C-B
Unit 1
1
3
2.1
⎛2 ⎞
⎜⎝ 7 ⎟⎠
5
44
1
4
16
57
6.
−5
3
=
(
57
)
3
=57
3
2
= 4 3 0 .3 4 1
1
^
You can calculate: 45 3
You can calculate:
÷
1
(–)
4
1
4
16
)
1
=
16
1 ÷
or 1 6
1
4
−1
4
ENTER
= 16
ˆ
(
ENTER
Lesson 2
7.
Your attempted values will vary, but the correct answers to
8.
A good estimate from Graph A would be x = 4.6, a good
estimate from the Graph B would be x
Graph A
40
35
30
yY
25
20
15
10
5
0
0
1
2
3
4
5
6
7
8
X
x
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Foundations for College Mathematics MAP4C-B
Unit 1
Suggested Answers to Support Questions
page 3
yY
Graph B
2
2.5
3
3.5
4
4.5
5
5.5
6
Xx
9.
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Window settings and sketches of graphs will vary. The
x, y
follows:
Copyright © 2008 The Ontario Educational Communications Authority. All rights reserved.
Suggested Answers to Support Questions
30
29
28
27
26
25
24
23
22
21
20
Suggested Answers to Support Questions
page 4
Foundations for College Mathematics MAP4C-B
Unit 1
4 2 x = 1 6 3 x−1
4 2 x = (4 2 )3 x−1
4 2 x = 4 6 x− 2
2 x=6 x−2
2 =4 x
x=
2
1
or
4
2
x
2x = 1024
2x = 210
x = 10
3 2 x−1 + 7 = 8 8
3 2 x−1 = 8 1
3 2 x−1 = 3 4
2 x −1 = 4
2 x=5
x=
5
or 2 .5
2
x
must first simplify:
x
x
Enter the equations y1
Plus.
x
= 100
and y2
Choose appropriate window settings so that the point of
Solution is x
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Foundations for College Mathematics MAP4C-B
Unit 1
Suggested Answers to Support Questions
page 5
pass could be done with decreased pressure.
⎛1 ⎞
0 .2 5 = 2 5 6 ⎜ ⎟
⎝2 ⎠
x
1
= 2 5 6 (2 −1 ) x
4
1
= 2 −x
1024
1
= 2 −x
10
2
2 −1 0 = 2 − x
−1 0 = − x
10 =x
Lesson 3
distances increasing over time, then at approximately
the year 1980, the distances level off, and actually
did not attend the 1984 Olympics due to political
being linear, then the graph curves to form a plateau.
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Copyright © 2008 The Ontario Educational Communications Authority. All rights reserved.
Suggested Answers to Support Questions
The solution is at x
Suggested Answers to Support Questions
page 6
Foundations for College Mathematics MAP4C-B
Unit 1
improved training, techniques, and technology that
enabled athletes to reach peak athletic performance.
Although performance-enhancing drugs are now
forbidden, perhaps drugs used during training played
a part as well. However, you would expect to see a
levelling off because of fewer discus participants,
the limits of the human body, and possibly tighter
restrictions on drug use among athletes.
winning distance in 2020 can be estimated as follows.
Ol ympic Dis cus Re s ults - 1 9 28 -2004
Olympic Discus Results, 1928 2020
80
Winning Distance (m)
70
60
50
40
30
20
10
0
1920
1940
1960
1980
2000
2020
YYear
e ar
If the trend continues, the winning distance in the year
the following calculation demonstrates.
Between 1860 and 1900,
change in y 2 million
=
= approximately 5 0 0 0 0 people/year
change in x 4 0 years
change in y 8 million
=
= approximately 1 6 0 0 0 0 people/year
change in x 5 0 years
cha nge in y 1 6 million
=
= a pproxima tely 3 4 7 8 2 6 people/yea r
cha nge in x
4 6 yea rs
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Foundations for College Mathematics MAP4C-B
Unit 1
Suggested Answers to Support Questions
page 7
16.
Salesperson
Points that May be
Used
Bob
Rate of Change
1 2 0 0 −1 0 0 0
200
=
= 0 .1
2000 −0
2000
$0.10 per $1 in
sales, or 10 cents
for every dollar in
sales
900 −700
200
=
= 0 .2
1000 −0
1000
$0.20 per $1 in
sales, or 20 cents
for every dollar in
sales
Bob, because his pay starts at the $1000 value on the
y
the y
Section of the
Graph
Points that May be
Used
Less than 40
hours
Left side
Hours beyond
40 hours
Right Side
Apply Formula
Rate of
Change
1200 −0
1200
=
=30
40
40 −0
2 1 0 0 −1 2 0 0
900
=
=45
20
60 −40
worked, then he or she earns a time-and-a-half wage
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Suggested Answers to Support Questions
Apply Formula
Suggested Answers to Support Questions
page 8
Foundations for College Mathematics MAP4C-B
Unit 1
until 20 years, then the rate increases to a higher
rate, but is still constant. Ed’s graph starts horizontal,
or with a zero rate of change. It then grows at an
increasing rate, which means that the rate of change is
not constant.
Charlie
Ed
y
y
Stored his money under his
mattress for 2 years, then invested
After 20 years the investment
for 2 years; no rate of change;
corresponds to money under
20 years rate of change increases;
Lesson 4
19.
x
y
First
Difference
Second
Difference
6
0
20
0
27
0
Since the first differences are the same, this
relationship is linear.
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Foundations for College Mathematics MAP4C-B
Unit 1
x
y
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page 9
First
Difference
Second
Difference
0
2
24
6
6 144
86 400
92 160
8
The first and second differences aren’t equal, but there
is a common ratio of 16 when the y-values are divided.
The relationship is exponential.
x
y
0
0
First
Difference
6
Second
Difference
648
1026
9
972
1998
12
Neither the first nor second differences are equal, and
when the y-values are divided, there is no common ratio.
For these reasons, there is no relationship.
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4
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page 10
x
y
Foundations for College Mathematics MAP4C-B
Unit 1
First
Difference
Second
Difference
4
84
8
109
64
148
12
64
212
16
469
64
276
20
Since the second differences are equal and not zero, the
relationship is quadratic.
20. Your answers should resemble the following screens.
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Foundations for College Mathematics MAP4C-B
Unit 1
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page 11
$12.00.
change in y y2 − y1
, to
=
change in x x2 − x1
find the rate.
Graph
Points that May
be Used
Substitution into Formula
20 −8
12
=
= 0 .4 8
25 −0
25
Company B
2 4 −1 2
12
=
= 0 .3
40 −0
40
$0.48/km
companies. The cost is $18.00.
km, it is better to go with Company B. You pay more at
Plus.
Linear
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Quadratic
Exponential
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Company A
Estimated Rate
of Change
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Foundations for College Mathematics MAP4C-B
Unit 1
Note regarding scientific notation or “E notation”:
decimal left four places. Similarly, for the value
scientific notation form. To do so after entering a
2nd
,
(–)
x represents the year and y represents the
r-value was with the quadratic equation.
This should not be selected because if you were to look
would imply that the population graph would appear
parabolic, like this:
It does not make sense that humans could have lived in
great numbers before the advent of modern medical and
agricultural techniques.
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Foundations for College Mathematics MAP4C-B
Unit 1
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page 13
In reality, the world’s population started small and is
increasing rapidly. If you looked far before the year
x = 2020 into the equation:
2020
= 8.7 billion people
y
Lesson 5
K=
K
1
2
2
1
m(9 )2
2
= 40 .5 m
40 .5 m
=
40 .5
=m
486 =
486
486
40 .5
12
1
(7 )(v2 )
2
3 4 5 = 3 .5 v2
345 =
345
=v
3 .5
9 .9 2 8 3 1 4 4 8 8 = v
v ≈ 9 .9 2 8
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Therefore the equation you should use is
x
y
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page 14
Foundations for College Mathematics MAP4C-B
Unit 1
the variable has degree = 2.
x
1
5
(x )
5
=345
1
5
x = 3 .2 1 7 8 3 5 3 5 5
x ≈ 3 .2 1 8
x9 = 1000
(x )
9
1
9
=1 0 0 0
1
9
x = 2 .1 5 4 4 3 4 6 9
x ≈ 2 .1 5 4
1
3
x = 14
3
⎛ 13 ⎞
⎜⎝ x ⎟⎠ = 1 4
3
x=2744
h in the formula:
SA = 2 r2 + 2 rh
SA 2 r2 = 2 rh
SA 2 r2
=h
2 r
This formula can be used for all three calculations, by
substituting in values for SA and r.
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Foundations for College Mathematics MAP4C-B
Unit 1
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page 15
Cylinder A
Cylinder B
Cylinder C
Surface area,
2
SA
Radius, r
Height, h
8
h=
6
2 5 0 0 2 (8 )2
2 (8 )
π
5 0 0 0 2 (6 )2
2 (6 )
h=
4 2 7 5 2 (1 5 )2
2 (1 5 )
h = 1 2 6 .6 2 9 4
h = 3 0 .3 5 9 6
h 1 2 6 .6 3 cm
h
3 0 .3 6 cm
π × 82
h
h ≈ 41.74 cm
26. There are two methods you could use to solve this problem.
Method 1:
4
π (1 .2 )3 = 7 .2 3 8 2 3 cm 3
3
4
= π (0 .5 )3 = 0 .5 2 3 5 9 9 cm 3
3
= 7 .2 3 8 2 3 − 0 .5 2 3 5 9 9
Vtotal =
Vinner
Vcandy
Vcandy = 6 .7 1 4 6 3 1 cm 3 for one gumb a ll
Note: You don’t need to round this answer for one gumball
because you will be multiplying by 10 000 at the end. Any errors
due to rounding would be magnified, so it is advised to keep the
extra decimal places for greater accuracy.
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Note that this can be entered
into a calculator as:
h=
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page 16
Foundations for College Mathematics MAP4C-B
Unit 1
Method 2:
Vcandy = Vtotal − Vinner
4
4
3
3
π rtotal
− π rinner
3
3
4
3
3
= π (rtotal
− rinner
)
3
4
= π (1 .2 3 − 0 .5 3 )
3
4
= π (1 .6 0 3 )
3
= 6 .7 1 4 6 3 1 cm 3 for one gumb a ll
=
. Therefore,
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