*Q*

### uadratic

### Function

**QUADRATIC**

**FUNCTIONS**

Sherbrooke and the Department of Canadian Heritage.

*Production Supervisor: Jean-Paul Groleau*
*Author: Diane Vigneux*

*Content Revision: Suzie Asselin*
*Jean-Paul Groleau*
*Coordinator for the DFGA: Ronald Côté*

*Photocomposition and Layout: Multitexte Plus*

*Desktop Publishing for Updated Version: L’atelier du Mac inc.*
*English Translation: Claudia De Fulviis*

*Translation of updated Sections: Claudia de Fulviis*
*First Print: 2006*

© Société de formation à distance des commissions scolaires du Québec

All rights for translation and adaptation, in whole or in part, reserved for all countries. Any reproduction, by mechanical or electronic means, including microreproduction, is forbidden without the written permission of a duly authorized representative of the Société de formation à distance des commission scolaires du Québec (SOFAD).

**TABLE OF CONTENTS**

Introduction to the Program Flowchart ... 0.4 Program Flowchart ... 0.5 How to Use This Guide ... 0.6 General Introduction ... 0.9 Intermediate and Terminal Objectives of the Module ... 0.11 Diagnostic Test on the Prerequisites ... 0.17 Answer Key for the Diagnostic Test on the Prerequisites ... 0.21 Analysis of the Diagnostic Test Results ... 0.25 Information for Distance Education Students ... 0.27

**UNITS**

1. Determining the Maximum ... 1.1
2. Equations Involving a Maximum ... 2.1
3. Graphing an Equation of the Form *y* = *ax*2_{... 3.1}
4. Graphing an Equation of the Form *y* = *ax*2_{ + }_{c}_{... 4.1}
5. Solving a Second-Degree Equation by Factoring ... 5.1
6. Solving a Second-Degree Equation Using the Quadratic Formula ... 6.1
7. Graphing a Second-Degree Equation ... 7.1
8. Determining the Maximum or Minimum, Given a

Second-Degree Equation ... 8.1 9. Solving a Problem That Can Be Written as a

Second-Degree Equation ... 9.1 Final Review ... 10.1 Answer Key for the Final Review Exercises ... 10.4 Terminal Objectives ... 10.5 Self-Evaluation Test... 10.7 Correction Key for the Self-Evaluation Test ... 10.13 Analysis of the Self-Evaluation Test Results ... 10.17 Final Evaluation... 10.18 Answer Key for the Exercises ... 10.19 Glossary ... 10.93 List of Symbols ... 10.99 Bibliography ... 10.100 Review Activities ... 11.1

**INTRODUCTION TO THE PROGRAM FLOWCHART**

**Welcome to the World of Mathematics!**

This mathematics program has been developed for the adult students of the Adult Education Services of school boards and distance education. The learning activities have been designed for individualized learning. If you encounter difficulties, do not hesitate to consult your teacher or to telephone the resource person assigned to you. The following flowchart shows where this module fits into the overall program. It allows you to see how far you have progressed and how much you still have to do to achieve your vocational goal. There are several possible paths you can take, depending on your chosen goal.

The first path consists of modules MTH-3003-2 (MTH-314) and MTH-4104-2 (MTH-416), and leads to a Diploma of Vocational Studies (DVS).

The second path consists of modules MTH-4109-1 (MTH-426), MTH-4111-2 (MTH-436) and MTH-5104-1 (MTH-514), and leads to a Secondary School Diploma (SSD), which allows you to enroll in certain Gegep-level programs that do not call for a knowledge of advanced mathematics.

The third path consists of modules MTH-5109-1 (MTH-526) and MTH-5111-2 (MTH-536), and leads to Cegep programs that call for a solid knowledge of mathematics in addition to other abiliies.

**CEGEP**

**MTH-5110-1 ** Introduction to Vectors

**MTH-5109-1 ** Geometry IV

**MTH-5108-1 ** Trigonometric Functions and Equations

**MTH-5107-1 ** Exponential and Logarithmic Functions

and Equations

**MTH-5106-1 ** Real Functions and Equations

**MTH-5105-1 ** Conics

**MTH-5104-1 ** Optimization II

**MTH-5103-1 ** Probability II

**MTH-5102-1 ** Statistics III

**MTH-5101-1 ** Optimization I

**MTH-4110-1 ** The Four Operations on

Algebraic Fractions

**MTH-4109-1 ** Sets, Relations and Functions

**MTH-4108-1 ** Quadratic Functions

**MTH-4107-1 ** Straight Lines II

**MTH-4106-1 ** Factoring and Algebraic Functions

**MTH-4105-1 ** Exponents and Radicals

**MTH-4103-1 ** Trigonometry I
**MTH-4102-1 ** Geometry III
**MTH-536**
**MTH-526**
**MTH-514**
**MTH-436**
**MTH-426**
**MTH-416**
**MTH-314**
**MTH-216**
**MTH-116**
**MTH-3002-2 ** Geometry II

**MTH-3001-2 ** The Four Operations on Polynomials

**MAT-2008-2 ** Statistics and Probabilities I

**MTH-2007-2 ** Geometry I

**MTH-2006-2 ** Equations and Inequalities I

**MTH-1007-2 ** Decimals and Percent

**MTH-1006-2 ** The Four Operations on Fractions

**MTH-1005-2 ** The Four Operations on Integers

**MTH-5111-2 ** Complement and Synthesis II

**MTH-4111-2 ** Complement and Synthesis I

**MTH-4101-2 ** Equations and Inequalities II

**MTH-3003-2 ** Straight Lines I

**Trades**

**DVS**

**MTH-5112-1**Logic

**25 hours**= 1 credit

**50 hours**= 2 credits

**MTH-4104-2**Statistics II

**THE PROGRAM FLOWCHART**

Hi! My name is Monica and I have been asked to tell you about this math module. What’s your name?

I’m Andy.

Whether you are registered at an adult education center or at Formation à distance, ...

You’ll see that with this method, math is a real breeze!

... you have probably taken a placement test which tells you exactly which module you should start with.

My results on the test indicate that I should begin with this module.

Now, the module you have in hand is divided into three sections. The first section is...

... the entry activity, which contains the test on the prerequisites.

By carefully correcting this test using the corresponding answer key, and record-ing your results on the analysis sheet ...

**?**

The memo pad signals a **brief reminder **of
concepts which you have already studied.

The **calculator **symbol reminds you that
you will need to use your calculator.

The sheaf of wheat indicates a **review **designed to
reinforce what you have just learned. A row of
sheaves near the end of the module indicates the
final review, which helps you to interrelate all the
learning activities in the module.

The starting line shows where the learning activities

**begin**.

The little white question mark indicates the **questions**

for which answers are given in the text.

**?**

... you can tell if you’re well enough prepared to do all the activities in the module.

The boldface question mark
indicates **practice exercices**

which allow you to try out what you have just learned.

And if I’m not, if I need a little review before moving on, what happens then?

In that case, before you start the activities in the module, the results analysis chart refers you to a review activity near the end of the module.

In this way, I can be sure I have all the prerequisites for starting.

Exactly! The second section contains the learning activities. It’s the main part of the module.

Look closely at the box to the right. It explains the symbols used to identify the various activities.

The target precedes the

**objective** to be met.

I see!

**?**

**START**

Lastly, the finish line indicates

that it is time to go on the **self-evaluation**
**test **to verify how well you have understood
the learning activities.

A “Did you know that...”?

Later ...

For example. words in bold-face italics appear in the glossary at the end of the module...

**Great!**

... statements in boxes are important points to remember, like definitions, for-mulas and rules. I’m telling you the format makes everything much easier.

The third section contains the final re-view, which interrelates the different parts of the module.

Yes, for example, short tidbits on the history of mathematics and fun puzzles. They are in-teresting and relieve tension at the same time.

No, it’s not part of the learn-ing activity. It’s just there to give you a breather. There are also many fun things

in this module. For example, when you see the drawing of a sage, it introduces a “Did you know that...”

Must I memorize what the sage says?

It’s the same for the “mathwhiz” pages, which are designed

espe-cially for those who love math. They are so stimulating thateven if you don’t have to do them, you’ll still want to.

And the whole module has been arranged to make learning easier.

There is also a self-evaluation test and answer key. They tell you if you’re ready for the final evaluation.

Thanks, Monica, you’ve been a big help.

I’m glad! Now, I’ve got to run. See you!

This is great! I never thought that I would like mathematics as much as this!

**GENERAL INTRODUCTION**

** PARABOLAS**

In this module you will be learning about parabolas. A parabola is the graphic representation of a situation where a maximum or minimum quantity, or vertex, is reached. For example, a ball thrown up into the air rises to a maximum height before it begins to fall again. The graph representing the ball as it rises to its maximum height and falls to the ground takes the shape of a parabola.

Mathematically, the parabola is represented by an equation of the form
*y* = *ax*2_{ + }_{bx}_{ + }_{c}_{. This type of equation is called a second-degree equation because}

the variable “*x*” has the exponent “2*.*” Second-degree equations differ from
first-degree equations in that they involve a maximum or minimum value that is not
found in a first-degree equation in two variables.

All cases similar to that of a ball thrown up into the air can be expressed as second-degree equations and are represented graphically by a parabola. In business and the physical sciences, the parabola is often used to find the maximum or minimum quantity in different situations. You will be examining several such examples in this module.

You will also learn to graph a second-degree equation of the form *y* = *ax*2_{ + }_{bx}_{ + }_{c}_{, where}

*a* is a rational number other than 0 and *b* = *c* = 0, where *a* and *c* are rational
numbers other than 0 and *b* = 0, and where *a*, *b* and *c* are rational numbers other
than 0. Among other things, this involves drawing up a table of values and
determining the axis of symmetry.

You will then use factoring to solve second-degree equations of the form
*ax*2_{ + }_{bx}_{ + }_{c }_{= 0. All five factoring methods will be used. You should be familiar}

with these five methods as they were covered in a previous module entitled “Factoring.”

You will also learn a new method for solving equations algebraically. This
method involves using the** quadratic formula**, which is essential for solving

**polynomials that cannot be factored**.

Whatever method you choose, you will have to find the value or values of the
variable that make the equation equal to zero. These values are called the **roots**

of the equation.

You will also learn to use algebra to find the coordinates of the maximum point in a situation that can be expressed as a quadratic function. Lastly, given an everyday situation or a problem related to numbers or applied geometry, you will learn how to express that situation mathematically and solve the resulting second-degree equation.

**INTERMEDIATE AND TERMINAL OBJECTIVES OF**

**THE MODULE**

Module MTH-4108-1 consists of nine units and requires 25 hours of study, distributed as follows. The terminal objectives appear in boldface.

**Objectives** **Number of Hours*** **% (evaluation)**

1 to **6** 4 20%

**7** 10 40%

**8** 5 20%

**9** 5 20%

* One hour are allotted for the final evaluation.

1. Determining the Maximum

Given a word problem dealing with an everyday situation that can be
expressed as an equation of the form *y* = *ax*2_{ + }_{bx}_{ + }_{c}_{, find the value of the}

variables *x* and *y* that corresponds to the required maximum quantity
(maximum output, maximum profit, maximum height) by using one of the
following methods:

• filling in a partially completed table of values;

• substituting different values for *x* in a second-degree

equation. The values of *a*, *b* and *c* are rational numbers and
*a*≠ 0.

The equation and the values of *x* are given. The given values of *x* are usually
natural numbers. The answer must be written as an ordered pair (*x*, *y*) and
the steps in the solution must be shown.

2. Equations Involving a Maximum

Given a word problem dealing with an everyday situation and using a
partially completed table of values, formulate a second- degree equation of
the form * y = ax2 _{ + bx + c}*

_{, where }

_{a}_{, }

_{b}_{ and }

_{c}_{ are rational numbers and }

*≠*

_{a}_{ 0.}

3. Graphing an Equation of the Form *y* = *ax2*

Graph a second-degree equation of the form *y* = *ax2*_{, where }_{a}_{ is a rational}

number between – 5 and 5 (*a*≠ 0). The result should be the graph of a parabola
with the vertex, the axis of symmetry and the equation of the axis of
symmetry clearly indicated. The scale for each of the two axes must also be
indicated.

4. Graphing an Equation of the Form *y* = *ax2 _{+ c}*

(1) Graph a second-degree equation of the form *y* = *ax2*_{+ }_{c,}_{ where }_{a}_{ is a rational}

number between – 5 and 5 (*a*≠ 0) and where *c* is a rational number. The result
should be the graph of a parabola with the vertex, the axis of symmetry and
the equation of the axis of symmetry clearly indicated. The scale for each of
the two axes must be shown as well. (2) Indicate whether the vertex of the
parabola is a maximum or a minimum point.

5. Solving a Second-Degree Equation by Factoring

(1) Solve a second-degree equation of the form a*x*2_{ + }_{bx}_{ + }_{c}_{ = 0, where }_{a}_{, }_{b}_{ and}

*c* are rational numbers and *a* ≠ 0, using the appropriate factoring method
(removing the common factor, factoring by grouping, factoring a trinomial of
the form *ax2*_{ + }_{bx}_{ + }_{c}_{ or factoring the difference of squares) as well as the}

multiplication property of zero. The steps in the solution must be shown. (2)
Given the roots of a quadratic equation, determine the coordinates of the
points on the graph of the equation *ax*2_{ + }_{bx}_{ + }_{c}_{ = }_{y }_{ that correspond to these}

given values.

**6. Solving a Second-Degree Equation Using the Quadratic Formula**

**(1) Find the value of the discriminant **∆** = ****b****2**_{ – 4}_{ac}_{ in order to}

**determine the number of roots (0, 1 or 2) of a second-degree equation**
**of the form ****ax****2**_{ + }_{bx}_{ + }_{c}_{ = 0, where }_{a}_{, }_{b}_{ and }_{c}_{ are rational numbers and}

* a* ≠

**0. (2) If necessary, solve this equation using the quadratic**

**formula: ****x**** = –b**±± **b****2** **– 4ac**

**2a** **. The resulting values are real numbers**

**and the steps in the solution must be shown. (3) Given the graph of**

**a second-degree equation of the form ****y**** = ****ax****2**_{ + }_{bx}_{ + }_{c}_{, find the number}

**of zeros for this equation and indicate the coordinates of the points**
**corresponding to these zeros.**

**7. Graphing a Second-Degree Equation**

**Graph a second-degree equation of the form ****y**** = ****ax****2**_{ + }_{bx}_{ + }_{c}_{, where }_{a}_{,}

**b**** and ****c**** are rational numbers and *** a*≠

**0. The result should be the graph**

**of a parabola with the following information clearly indicated: the**
**vertex, the axis of symmetry and its equation, the y-intercept, the**

**point symmetric to the ****y****-intercept and, if necessary, the coordinates**

**corresponding to the zeros of this equation. The scale for each axis**
**and the calculations involved in finding each of these points must**
**also be indicated.**

**8. Determining the Maximum or Minimum, Given a Second-Degree**
**Equation**

**Determine the abscissa and the ordinate of the maximum or**
**minimum point of a parabola, given a second-degree equation of the**
**form ****y**** = ****ax****2**_{ + }_{bx}_{ + }_{c, }_{where }_{a}_{, }_{b}_{ and }_{c}_{ are rational numbers and }*_{a}*≠

_{ 0.}**The problems deal with situations related to science or business.**
**The steps in the solution must be shown.**

**9. Solving a Problem That Can Be Written as a Second-Degree Equation**

**Using factoring or the quadratic formula, solve a word problem that**
**can be written as a second-degree equation of the form**

**ax****2**_{ + }_{bx}_{ + }_{c }_{= 0, where }_{a}_{, }_{b}_{ and }_{c}_{ are rational numbers and }*_{a}*≠

_{ 0. Any}**irrelevant solutions should be disregarded. Each problem involves**
**finding a maximum of two solutions and deals with computation,**
**geometry or everyday situations. The steps in the solution must be**
**shown.**

**DIAGNOSTIC TEST ON THE PREREQUISITES**

**Instructions**

1. Answer as many questions as you can. 2. You may use a calculator.

3. Write your answers on the test paper.

4. Don’t waste any time. If you cannot answer a question, go on to the next one immediately.

5. When you have answered as many questions as you can, correct your answers using the answer key which follows the diagnostic test.

6. To be considered correct, your answers must be identical to those in the key. In addition, the various steps in your solution should be equivalent to those shown in the answer key.

7. Transcribe your results onto the chart which follows the answer key. This chart gives an analysis of the diagnostic test results. 8. Do only the review activities that apply to your incorrect

answers.

9. If all your answers are correct, you may begin working on this module.

1. Find the value of *y* in the equation *y* = –*x*2_{ + 5}_{x}_{ – 7 if:}

a) *x* = 3 b) *x* = –2

2. Find the value of *y* in the equation *y* = 0.25*x*2_{ – 0.3}_{x}_{ + 5 if }_{x}_{ = 4.}

3. a) Find the value of *b*2_{ – 4}_{ac}_{b)} _{Find the value of }* _{b}*2

_{ – 4}

_{ac}if *b* = –5, *a* = –3 and *c* = 0. if *a* = 1

2, *b* = 0.3 and *c* =
1
2.

4. Carry out the following operations on the polynomials below.

a) (5*x*2_{ –}_{ x }_{+ 3) + (}* _{x}*2

_{ + 2}

_{x}_{ – 5) =}

b) (*x*2_{ + 7}_{x }_{– 1) – (3}* _{x}*2

_{ + 4 – 2}

_{x}_{) =}

*y*
*x*
1
1
*y*
*x*
1
1
*x* *y*
*x* *y*

5. a) Graph the equation *y* = 5*x* – 3 and make a table of at least three related
values.

Table of values

b) Graph the equation *x* = –2 and make a table of values for this equation.

Table of values

6. Solve the following equations by showing all the steps and checking the results.

b) – 5*x*= 0

c) – 8*x*+ 6 = 0

7. Factor the following polynomials.

a) 6*x*2_{ + 3}_{x}_{b)} * _{x}*2

_{ + 5}

_{x}_{ – 2}

_{x}_{ – 10}

c) *x*2_{ + 7}_{x}_{ + 6} _{d) 3}* _{x}*2

_{ + 11}

_{x –}_{ 20}

*y*
*x*
1
1

### •

### •

*x*

*y*– 1 – 8 0 – 3 1 2 2 7

**ANSWER KEY FOR THE DIAGNOSTIC TEST**

** ON THE PREREQUISITES**

1. a) *y*= –

*x*2

_{ + 5}

_{x}_{ – 7}

_{b)}

_{y}_{ =}

_{–}

*2*

_{x}_{ + 5}

_{x}_{ – 7}

*y*= – (3)2

_{ + 5(3) – 7}

_{y }_{= – (– 2)}2

_{ + 5(–2) – 7}

*y*= – 9 + 15 – 7

*y*= – 4 – 10 – 7

*y*= – 1

*y*= – 21 2.

*y*= 0.25

*x*2

_{ – 0.3}

_{x}_{ + 5}

*y*= 0.25(4)2

_{ – 0.3(4) + 5}

*y*= 0.25(16) – 1.2 + 5

*y*= 4 – 1.2 + 5

*y*= 7.8 3. a)

*b*2

_{ – 4}

_{ac}_{ = (– 5)}2

_{ – 4(– 3)(0)}

_{b)}

*2*

_{b}_{ – 4}

_{ac}_{ = (0.3)}2

_{ – }

_{4 1}2 1 2

*b*2

_{ – 4}

_{ac}_{ = 25 – 0}

*2*

_{b}_{ – 4}

_{ac}_{ = 0.09 – 1}

*b*2

_{ – 4}

_{ac}_{ = 25}

*2*

_{b}_{ – 4}

_{ac}_{ = – 0.91}4. a) 6

*x*2

_{+}

_{ x}_{ – 2}

_{b) – 2}

*2*

_{x}_{+ 9}

_{x}_{ – 5}c) 2

*x*2

_{– }_{8}

_{x}_{ – 24}

_{d) –}

*2*

_{x}_{ + 4}

_{x }_{– }1 2 5. a)

*x* *y*
– 2 3
– 2 1
– 2 0
– 2 – 2
*y*
*x*
1
1

### •

### •

### •

### •

b) 6. a) Solution: Check: 2*x*+ 7 = 0 2

*x*+ 7 = 0 2

*x*= – 7 2(– 3.5) + 7 = 0

*x*= –7 2 or – 3.5 – 7 + 7 = 0 0 = 0 b) Solution: Check: – 5

*x*= 0 – 5

*x*= 0

*x*= 0 – 5 – 5(0) = 0

*x*= 0 0 = 0 c) Solution: Check: – 8

*x*+ 6 = 0 – 8

*x*+ 6 = 0 – 8

*x*= – 6 – 8(0.75) + 6 = 0

*x*= – 6 – 8 –6 + 6 = 0

*x*= 3 4 or 0.75 0 = 0 7. a) 6

*x*2

_{ + 3}

_{x}_{b)}

*2*

_{x}_{ + 5}

_{x}_{ – 2}

_{x}_{ – 10}3

*x*(2

*x*+ 1)

*x*(

*x*+ 5) – 2(

*x*+ 5) (

*x*+ 5)(

*x*– 2)

c) *x*2_{ + 7}_{x}_{ + 6} _{d) 3}* _{x}*2

_{ + 11}

_{x}_{ – 20}

*x*2_{ + 6}_{x}_{ + }_{x}_{ + 6} _{3}* _{x}*2

_{ + 15}

_{x}_{ – 4}

_{x}_{ – 20}

*x*(*x* + 6) + 1(*x* + 6) 3*x*(*x* + 5) – 4(*x* + 5)
(*x* + 6)(*x* + 1) (*x* + 5)(3*x* – 4)

• Trinomial of the form • Trinomial of the form
*x*2_{ + }_{bx}_{ + }_{c}* _{ax}*2

_{ + }

_{bx}_{ + }

_{c}e) 4*x*2_{ – 25}

(2*x*)2_{ – (5)}2

(2*x* + 5)(2*x* – 5)

**ANALYSIS OF THE DIAGNOSTIC TEST RESULTS**

**Question** **Answer** **Review** **Before Going**

**Correct** **Incorrect** **Section** **Page** **to Unit(s)**

1. a) 11.1 11.4 Unit 1 b) 11.1 11.4 Unit 1 2. 11.1 11.4 Unit 1 3. a) 11.1 11.4 Unit 1 b) 11.1 11.4 Unit 1 4. a) 11.2 11.11 Unit 2 b) 11.2 11.11 Unit 2 c) 11.2 11.11 Unit 2 d) 11.2 11.11 Unit 2 5. a) 11.3 11.23 Unit 3 b) 11.3 11.23 Unit 3 6. a) 11.4 11.31 Unit 5 b) 11.4 11.31 Unit 5 c) 11.4 11.31 Unit 5 7. a) 11.5 11.40 Unit 5 b) 11.5 11.40 Unit 5 c) 11.5 11.40 Unit 5 d) 11.5 11.40 Unit 5 e) 11.5 11.40 Unit 5

• If all your answers are** correct**, you may begin working on this module.

• For each **incorrect** answer, find the related section in the **Review** column
and do the review exercises for that section before starting the unit(s) listed
in the right-hand column under the heading **Before Going on to.**

**INFORMATION FOR DISTANCE**

**EDUCATION STUDENTS**

You now have the learning material for MTH-4108-1 together with the home-work assignments. Enclosed with this material is a letter of introduction from your tutor indicating the various ways in which you can communicate with him or her (e.g. by letter, telephone) as well as the times when he or she is available. Your tutor will correct your work and help you with your studies. Do not hesitate to make use of his or her services if you have any questions.

**DEVELOPING EFFECTIVE STUDY HABITS**

Distance education is a process which offers considerable flexibility, but which also requires active involvement on your part. It demands regular study and sustained effort. Efficient study habits will simplify your task. To ensure effective and continuous progress in your studies, it is strongly recommended that you:

• draw up a study timetable that takes your working habits into account and is compatible with your leisure time and other activities;

The following guidelines concerning the theory, examples, exercises and assign-ments are designed to help you succeed in this mathematics course.

**Theory**

To make sure you thoroughly grasp the theoretical concepts:

1. Read the lesson carefully and underline the important points.

2. Memorize the definitions, formulas and procedures used to solve a given problem, since this will make the lesson much easier to understand.

3. At the end of an assignment, make a note of any points that you do not understand. Your tutor will then be able to give you pertinent explanations.

4. Try to continue studying even if you run into a particular problem. However, if a major difficulty hinders your learning, ask for explanations before sending in your assignment. Contact your tutor, using the procedure outlined in his or her letter of introduction.

**Examples**

The examples given throughout the course are an application of the theory you are studying. They illustrate the steps involved in doing the exercises. Carefully study the solutions given in the examples and redo them yourself before starting the exercises.

**Exercises**

The exercises in each unit are generally modelled on the examples provided. Here are a few suggestions to help you complete these exercises.

1. Write up your solutions, using the examples in the unit as models. It is important not to refer to the answer key found on the coloured pages at the end of the module until you have completed the exercises.

2. Compare your solutions with those in the answer key only after having done
all the exercises. **Careful!** Examine the steps in your solution carefully even
if your answers are correct.

3. If you find a mistake in your answer or your solution, review the concepts that you did not understand, as well as the pertinent examples. Then, redo the exercise.

4. Make sure you have successfully completed all the exercises in a unit before moving on to the next one.

**Homework Assignments**

Module MTH-4108-1 contains three assignments. The first page of each assignment indicates the units to which the questions refer. The assignments are designed to evaluate how well you have understood the material studied. They also provide a means of communicating with your tutor.

When you have understood the material and have successfully done the perti-nent exercises, do the corresponding assignment immediately. Here are a few suggestions.

1. Do a rough draft first and then, if necessary, revise your solutions before submitting a clean copy of your answer.

2. Copy out your final answers or solutions in the blank spaces of the document to be sent to your tutor. It is preferable to use a pencil.

3. Include a clear and detailed solution with the answer if the problem involves several steps.

4. Mail only one homework assignment at a time. After correcting the assign-ment, your tutor will return it to you.

In the section “Student’s Questions,” write any questions which you may wish to have answered by your tutor. He or she will give you advice and guide you in your studies, if necessary.

**In this course**

Homework Assignment 1 is based on units 1 to 7. Homework Assignment 2 is based on units 8 and 9. Homework Assignment 3 is based on units 1 to 9.

**CERTIFICATION**

When you have completed all the work, and provided you have maintained an average of at least 60%, you will be eligible to write the examination for this course.

**UNIT 1**

**DETERMINING THE MAXIMUM**

**1.1**

**SETTING THE CONTEXT**

**Counting Peaches**

Mr. Pitt owns a peach orchard with 30 trees yielding an average of 400 peaches each. He wants to plant more peach trees in order to get the maximum yield from his orchard; however, for each additional tree planted, the average yield per tree will drop by 10 peaches. How many peach trees should Mr. Pitt plant to ensure a maximum harvest?

**To achieve the objective of this unit, you should be able to solve**

**problems involving ****maximum**** yield, profit or height. The problems are**

**based on everyday situations. You will be required to fill in partially**

**completed tables of values and find the value of ****x**** or ****y**** that corresponds**

**to the required maximum. You should also be able to find the maximum**
**value for situations whose equation is given.**

The table below should help Mr. Pitt solve his problem.

Table 1.1 Projected yield of Mr. Pitt’s orchard

**Number of** **Total** **Average** **Total yield**
** peach trees** **number of** **yield per** **of orchard**

**to be added** **peach trees** **tree** **y*** x*
0 30 400 30 × 400 = 12 000
1 30 + 1 400 – 10 × 1 (30 + 1) × (400 – 10 × 1) = 12 090
2 30 + 2 400 – 10 × 2 (30 + 2) × (400 – 10 × 2) = 12 160
3 30 + 3 400 – 10 × 3 (30 + 3) × (400 – 10 × 3) = 12 210
4 30 + 4 400 – 10 × 4 (30 + 4) × (400 – 10 × 4) = 12 240

**5**

**30 + 5**

**400 – 10**×

**5**

**(30 + 5)**×

**(400 – 10**×

**5) = 12 250**6 30 + 6 400 – 10 × 6 (30 + 6) × (400 – 10 × 6) = 12 240 7 30 + 7 400 – 10 × 7 (30 + 7) × (400 – 10 × 7) = 12 210 8 30 + 8 400 – 10 × 8 (30 + 8) × (400 – 10 × 8) = 12 160 9 30 + 9 400 – 10 × 9 (30 + 9) × (400 – 10 × 9) = 12 090 10 30 + 10 400 – 10 × 10 (30 + 10) × (400 – 10 × 10) = 12 000

The first column shows the number of trees Mr. Pitt will have to add to his
orchard. The numbers range from 0 to 10 inclusive and are identified by *x*.

The second column shows the total number of peach trees obtained by adding the current number of peach trees and the additional number of trees to be planted with a view to maximizing the orchard’s yield.

The third column shows the average yield per tree. This yield is equal to the average number of peaches produced by each tree (400) less the production loss for each additional tree (10/additional tree).

The fourth column shows the orchard’s total yield after additional trees have
been planted. This yield is equal to the * product* of the total number of trees
(column 2) and the average yield per tree (column 3). This product is identified
as

*y*.

For example, the third line of Table 1.1 tells you that if two trees were added,
the total number of peach trees would be **30 + 2 = 32**, that the average yield
per tree would be **400 – 10** **x** **2** and that the orchard’s total yield would be

**(30 + 2)** **x** **(400 – 10** **x** **2) = 12 160 peaches**. If you have understood this,

you should be able to answer the following questions without any difficulty.

**?**

What is the orchard’s greatest total yield? ...
**?**

How many trees should Mr. Pitt add to his orchard in order to obtain the
greatest total yield? ...
**?**

What is the total number of peach trees required for the orchard to yield a
maximum amount of fruit? ...
**?**

What will be each tree’s average yield when the total yield will have reached
its **maximum**? ...

**?**

If Mr. Pitt adds between 0 and 5 trees, will his orchard’s total yield increase
or decrease? ...
**?**

What would happen to the orchard’s total yield if Mr. Pitt were to add
between 6 and 10 trees? ...
The greatest total yield for Mr. Pitt’s orchard is **12 250** peaches, which
corresponds to the addition of **5** peach trees. Consequently, he will need a total
of **35** trees for his orchard to yield the maximum amount of fruit. The average
yield per peach tree when the total yield has reached its maximum is equal to
400 – 10 × 5, or **350** peaches.

You will no doubt have noticed from Table 1.1 that with the addition of between
0 and 5 trees, the total yield of Mr. Pitt’s orchard **increases, **whereas with the
addition of between 6 and 10 peach trees, the total yield **decreases**.

Lastly, the ordered pair corresponding to the orchard’s maximum yield is 5 peach
trees and 12 250 peaches, that is, (**5, 12 250**). Problem solved!

Many other problems involve finding a maximum value. Below is an example taken from the printing industry.

*N.B.* You may use a calculator for all calculations in the module, unless otherwise
indicated.

**Example 1**

The XYZ publishing company prints 36 books a day, which are sold for $40 each. The company can print 50 books a day. However, for each additional book printed, the selling price per unit decreases by $1. How many additional books must the company print in order to maximize its daily sales? What is this maximum sales figure?

**?**

To solve this problem, complete the table below by following the example of
the lines that are already completed.
Table 1.2 Number of books sold by the XYZ publishing company

**Additional** **Total number** **Selling price** **Daily sales**
**books** **of books** **per book**

**x*** y*
0 36 $40 36 × $40 = $1 440
1 36 + 1 = 37 $40 – $1 = $39 37 × $39 = ...
2 36 + 2 = 38 $40 – $2 = ... 38 × .... = ...
3 36 + .... = ... $40 – ... = ... .... × .... = ...
4 .... + .... = ... ... – ... = ... .... × .... = ...

**?**

What ordered pair corresponds to the maximum sales figure? ...
**?**

How many additional books must the company print in order to maximize its
daily sales? ...
**?**

What is the maximum sales figure?
The completed table looks like this:
Table 1.2 Number of books sold by the XYZ publishing company

**Additional** **Total number** **Selling price** **Daily sales**
**books** **of books** **per book**

**x*** y*
0 36 $40 36 × $40 = $1 440
1 36 + 1 = 37 $40 – $1 = $39 37 × $39 =

**$1 443**2 36 + 2 = 38 $40 – $2 = $

**38**38 ×

**$38**=

**$1 444**3 36 +

**3**=

**39**$40 –

**$3**=

**$37**

**39**×

**$37**=

**$1 443**4

**36**+

**4**=

**40**

**$40**–

**$4**=

**$36**

**40**×

**$36**=

**$1 440**

The ordered pair corresponding to the maximum sales figure is 2 additional
books and $1 444 per day, or (**2, 1 444**). To reach the maximum sales figure of

**$1 444**, the company must print **2** additional books each day.

**Procedure for finding the value of ****x**** or ****y**** that**

**corresponds to a maximum, using a table of values:**

1. Read the problem carefully.

2. Complete the table of values that represents the situation.
3. Find the ordered pair that corresponds to the maximum.
4. Depending on the question asked, find the value of *x* or *y*
in the ordered pair that corresponds to the required
maximum.

*N.B.* The ordered pair corresponding to the maximum is the * vertex *of a

* parabola*, which we will study in a later unit. Now it’s time for you to put what
you have learned into practice.

**Exercise 1.1**

1. Mickey Mouse Electronics can sell 300 VCRs at a * profit *of $60 per unit.
Seeking to increase its profits, the company decides to take its VCRs off the
market to increase the demand. It calculates that its sales will increase by
50 units for each week that the VCRs are off the market, but that its profit
will drop by $5 per unit owing to storage costs. Complete the following table
based on this information.

Table 1.3 Profits on VCR sales

**Number of weeks** **Number** **Profit on** **Total profit**
**off the market** **of units** **each unit**

**x****sold** * y*
0 300 $60 $60 × 300 = $18 000
1 300 + (1 × 50) = 350 $60 – ($5 × 1) = $55 $55 × 350 = $9 250
2
3
4
5
6

a) What ordered pair corresponds to the maximum profit generated by VCR sales?...

b) How many weeks should the company keep its product off the market in order to earn a maximum profit? ...

*Did you know that...*

…the idea of withholding an item in order to raise its market value is not new? Indeed, one of the basic principles of economics is that “scarcity creates demand.” This means that a certain clientele is ready to pay dearly and sometimes even very dearly for a product that is not easily found. The purchase of a luxury car is a good example. Some people wait for over two years for their Ferrari Testarossa, which sells for approximately $250 000, service and taxes not included!

**Example 2**

The height reached by a ball *t* seconds after it is thrown up into the air is
represented by the * functionh* = – 5

*t*2

_{ + 30}

_{t}_{. How many seconds will it take}

for the ball to reach its maximum height?

Comment

Throughout this module, we will use the term **function** to designate

* equations *relating to problems involving a maximum or

*value. We use the term*

**minimum****function**because, in this type of situation, we find the values of a

**variable****as a function**of the values given to the other variable. You will have the opportunity to study this concept in more depth in a subsequent module.

**?**

To solve this problem, complete the following *, where*

**table of values***t*represents the time and

*h*the height.

Equation: *h *= – 5*t*2_{ + 30}_{t}

Table 1.4 Calculating the height reached by a ball

*t* *h* Breakdown of calculations for the table of values
0 0 If *t* = 0, then *h* = – 5(0)2_{ + 30(0) = 0.}

1 25 If *t* = 1, then *h* = – 5(1)2_{ + 30(1) = 25.}

2 40 If *t* = 2, then *h* = – 5(2)2_{ + 30(2) = 40.}

3 45 If *t* = 3, then *h* = – 5(3)2_{ + 30(3) = 45.}

To complete Table 1.4, replace *t* by its given value on each line. The result is the
following table.

*t* *h* Breakdown of calculations for the table of values
0 0 If *t* = 0, then *h* = – 5(0)2_{ + 30(0) = 0.}
1 25 If *t* = 1, then *h* = – 5(1)2_{ + 30(1) = 25.}
2 40 If *t* = 2, then *h* = – 5(2)2_{ + 30(2) = 40.}
3 45 If *t* = 3, then *h* = – 5(3)2_{ + 30(3) = 45.}
4 40 If *t* = 4, then *h* = – 5(4)2_{ + }_{30(4)}_{ = }_{40}_{.}
**5** 25 If *t* = 5, then *h* = **– 5(5)2**_{ + }_{30(5)}_{ = }_{25}_{.}
6 **0** If *t* **6,** then *h* = **– 5(6)2**_{ + }_{30(6)}_{ = }_{0}_{.}

The ordered pair (3, 45) corresponds to the maximum point of the equation
*h* = – 5*t*2_{ + 30}_{t}_{; thus, it will take 3 seconds for the ball to reach its maximum}

height.

**Procedure for finding the value of ****x****or ** **y**** that**

**corresponds to a maximum, given an equation:**

1. Determine what *x *and *y* represent.

2. Complete a table of values for the given values of *x*.
3. Find the ordered pair that corresponds to the maximum.
4. Depending on the question asked, find the value of *x* or *y*
in the ordered pair that corresponds to the required
maximum.

The following exercise will test your ability to find the value of *x* or *y* that
corresponds to the required maximum.

**Exercise 1.2**

1. Consider the function *h *= – 5*t*2_{ + 20}_{t}_{ + 30 representing the height in metres}

attained by an object *t* seconds after it is thrown up into the air. Find the time
required for this object to reach its maximum height. What is its maximum
height?

Let *t* = 0, 1, 2, 3 and 4 and complete the four steps for solving a problem that
involves finding a maximum value.

1. ... .

2. Table of values:

Table 1.5 Height reached by an object after it is thrown into the air

**t****h****Breakdown of calculations for the table of**
**values for the equation ****h**** = – 5****t****2 _{ + 20}**

_{t}

_{ + 30}0 1 2 3 4 3. ... 4. ... ...

The theory presented in this unit can be summarized by the two procedures shown in the boxes on pages 1.6 and 1.9. Review them before moving on to the

**?**

**1.2**

**PRACTICE EXERCISES**

1. A survey conducted during an amateur theatre festival in Sherbrooke indicated that 100 people would attend the festival if the tickets were sold for $4.50 apiece. For each $0.30-decrease in the price of a ticket, 20 more people would attend the festival.

a) What ticket price would bring in the most revenue? b) What would be the maximum revenue?

Complete the following table in order to find the answers to these questions.

Table 1.6 Revenue for the Sherbrooke theatre festival

** Number of** **Total number** **Ticket price** ** Revenue**
** extra people** **of people** **y**

** *** x*×

**20**0 100 $4.50 $4.50 × 100 = $450 1 × 20 100 + 20 = 120 $4.50 – ($0.30 × 1) $4.20 × 120 = $504 2 × 20 3 × 20 4 × 20 5 × 20 6 × 20 7 × 20 a) ... . b) ... .

2. The function *h* = 40*t* – 5*t*2_{ represents the height of an object }_{t}_{ seconds after it}

is thrown up into the air. Draw up a table of values for *t* = 0, 1, 2, 3, 4, 5, 6,
7 and 8. How many seconds will it take for the object to reach its maximum
height? To answer this question, complete the four steps for finding the
maximum value. Write your answers in the spaces provided below.

1. ... .

2.

Table 1.7 Height of an object *t* seconds after being thrown

3. ... .

**1.3**

**REVIEW EXERCISES**

1. Complete the following sentences.

a) Using a table of values representing a situation or using an ... ,
we can find the value of *x* or *y* that corresponds to a ... .

b) Using a table of values, we can find the value of *x* or *y* that corresponds to
a maximum by using the following procedure:

1. Read the ... carefully.

2. ... the table of values that represents the situation.
3. Find the ordered pair that corresponds to the ... .
4. Depending on the question asked, find the value of ... or *y* in the
ordered pair that corresponds to the required ... .

2. Given an equation, we can use a four-step procedure to find the value of *x* or
*y *corresponding to a maximum. Describe the procedure.

1. ... ... . 2. ... ... . 3. ... ... . 4. ... ... .

**1.4**

**THE MATH WHIZ PAGE**

**A Well-Hidden Maximum!**

A little challenge for the more daring among you!

At present, Chip Electronics could sell 100 calculators at a profit of $5.00 per unit. The company decides to withdraw its calculators from the market in order to boost demand. The company estimates that for each week that the calculators are off the market, it can sell 20 more calculators. Furthermore, for each week that the calculators are off the market, the profit per calculator goes down by $0.25 owing to the suspension of sales and storage costs. How long should Chip Electronics wait if it wishes to earn maximum profits?

To solve this problem, complete the table of values on the following page.

Table 1.8 Profit on calculator sales

**Number of** **Number of** **Profit on** **Total**
**weeks off** **calculators** **each** **profit**
**the market** **sold** **calculator**

**x*** y*
0 100 $5 100 × $5 = $500
1 100 + 1 × 20 = 120 $5 – 1 × $0.25 = $4.75 120 × $4.75 = $570
2
3
4
5
6
7
8
9