PROBLEM SOLVING STRATEGY
GUESS AND CHECK
One way to solve a problem is to guess the answer and
then check to see whether it is correct. If it is not, you
can keep guessing and checking until you get the right
answer.
When a ball is dropped onto the ground
Falling Objects
A space shuttle depends on
gravity to return to the
Earth. Below an altitude of
about 13 700 m, the descent
of the shuttle is not powered. The shuttles
descent toward the runway is at an angle of
19 to the horizontal. This is much steeper
PROBLEM SOLVING
USING THE STRATEGIES
1. Calendar If
7. Marking points Mark six points on a
there are exactly four
Mondays in January, on what days of the
piece of paper, so that each point is 1 unit
week can January 31 not fall?
from exactly three other p
PROBLEM SOLVING STRATEGY
USE A DATA BANK
You must locate information to solve
some problems. There are many sources
of information, including the Internet,
the media, print data banks, and experts.
The brightness of planets and stars as
they appear from t
2
Specific Expectations
Quadratic Functions and
Equations
Functions
Functions &
Relations
Identify the structure of the complex number system and express complex
2.1
2
numbers in the form a + bi, where
= 1.
i
2.1
Determine the maximum or minimum value of
Motion in Space
In the Modelling Math
questions on pages 616,
634, 640, and 663, you
will solve the following
problem and other problems that involve
motion in space.
Halleys Comet orbits the sun about
every 76 years. The comet travels in an
elliptical pa
PROBLEM SOLVING
USING THE STRATEGIES
1. DrivingRashad
8. EquationsThe letters R, S, and T
started the 780-km drive
from Sault Ste. Marie to Ottawa at 09:00.
represent
His
integers. Find the possible values
average speed was 80 km/h. Three hours
of R,later
The Motion of a Pendulum
Some scientists, including Aristarchus of
Samos in the 3rd century
B.C. and
Copernicus in the 16th century
A.D.,
believed that the Earth rotates. However, no
one had been able to demonstrate this rotation scientifically.
In 1851,
Measurements of Lengths and
Areas
In the Modelling Math questions on pages
108, 118, and 132, you will solve the following
problem and other problems that involve lengths
and areas.
The ratio of the length to the width for the provincial flag of
Ontario i
GETTING STARTED
Communications Satellites
In 1972, Canada became the first country to use satellites for
communications within its own borders. The Anik A1
satellite carried radio and television programs to all parts of
the country. The satellite was in a
CHALLENGE PROBLEMS
an equation in the form
ax2 y bc c for the
quadratic function whose graph passes through (8, 2,
0), (0,
0). 8) and (
1. EquationWrite
2. RootsFind
2
the roots of
x
3. EvaluatingIf
k2 1
x 1 0.
k
2 x 2
6
0, evaluate
. x
x 2
4. Real root
CHALLENGE PROBLEMS
1. IntersectionHow
many times can an arbitrary circle intersect the
graph ofysin x?
a) at most 2 times b) at most 4 times c) at most 6 times
d) at most 8 times e) more than 16 times
2.
Sum of rootsFind
0 x 2.
3. If , then If
f
simplify
PROBLEM SOLVING STRATEGY
SOLVE RICH ESTIMATION PROBLEMS
How many words does your daily newspaper print in a year? Estimation problems like
this one are known as Fermi problems. They get their name from the physicist Enrico
Fermi, who liked to pose them to
PROBLEM SOLVING STRATEGY
USE A DIAGRAM
Diagrams provide insights that help you solve many problems.
The following problem has been adapted from a twothousand-year-old Chinese mathematics book called the Chiu
chang suan-shu.
A tree trunk has a height of 20
8
Loci and Conics
Specific Expectations
Functions
Functions &
Relations
Construct a geometric model to represent a described locus of points;
determine the properties of the geometric model; and use the properties
to interpret the locus.
8.1, 8.2
Explain
Summary of Formulas
and Equations
Note: The order of the formulas follows the order of the chapters.
Quadratic Formula
2
If ax2 bx c 0, then x b b 4ac.
2a
Transformations
Vertical Translation y f (x) k
If k > 0, translate upward k units.
If k < 0, transla
Investigate & Apply
Modelling Double Helixes
1. Spiral staircasesThe
up and down handrails of
two spiral staircases inside a building are in the shape of
a double helix. The staircases are 25 m high and there are
12 turns (half periods) of the handrails f
Investigate & Apply
Confocal Conics
Confocal conics are conic sections that share a common focus. For
example, the elliptical orbits of the planets are confocal. They all have the
sun as one of their foci.
Write an equation for a circle, an ellipse, a hyp
8.2
Equations of Loci
A locus is a set of points defined by a rule or condition. For
example, if a dog is attached by a 10-m leash to a post in the
middle of a large yard, then the locus of the farthest points that
the dog can reach is a circle with a rad
CHALLENGE PROBLEMS
even function satisfies
f(f(x)
x) for all values of x
in its domain. An odd function satisfies
x) f(x)
f( for all values of x in
its domain. Classify each of the following as even, odd, or neither.
2
a) f(x) 2x 1
b) f(x) 3x 4
c) f(x) x
PROBLEM SOLVING
USING THE STRATEGIES
1. Natural numbers There
5. Numbers The sum of two numbers
are three
natural numbers that give the same result
is 3. The product of the numbers is 2.
a) Find the numbers.
when they are added and when they are
b)Find th
5
TRIGONOMETRIC FUNCTIONS
Specific Expectations
Functions
Functions & Relations
Determine the sine, cosine, and tangent of angles greater than 90, using a 5.2
suitable technique, and determine two angles that correspond to a given single
trigonometric fun
GETTING STARTED
Human Physiology
The Heart
1. For the average person at rest, the heart
pumps blood at a rate of about 5 L/min.
Therefore, the volume, V litres, pumped over
a period of time, t minutes, is given by the
equation V5t.
a) Let V y and
t x, co
GLOSSARY
angle of elevation
The angle, measured upward,
between the horizontal and the line of sight from an
The distance of a number
from to an object.
observer
A
absolute value
zero on a real number line.
acute angle
Object
An angle whose measure is
. l
PROBLEM SOLVING STRATEGY
USE LOGIC
The ability to think logically is an important skill that you
will use in any profession you choose and in everyday life.
This skill can be improved with practice.
Most counterfeit coin problems include a balance
scale w
PROBLEM SOLVING
USING THE STRATEGIES
1. MeasurementThe
7. Tiling Suppose that a
perimeters of a
regular hexagon and an equilateral triangle
domino covers two squares
are equal. What fraction of the area of on
thea checkerboard. If two
hexagon is the area
7.3
Investigation: Compound Interest, Geometric
Sequences, and Exponential Growth
Banks and financial institutions offer a
variety of accounts and investments. When
you invest or borrow money, the interest
rate can greatly affect the amount of
interest yo
Investigate & Apply
Interpreting a
Mathematical Model
There are several steps in the inquiry process.
They do not necessarily occur in a given
order and they may often be revisited in
completing an investigation. These steps
include the following.
Formul
Investigate & Apply
Modelling Restrictions Graphically
There are several steps in the inquiry process. They do not necessarily occur
in a given order and they may often be revisited in completing an
investigation. These steps include the following.
Formul
ANSWERS
12 c)1 d) 0 15. a)noneb) all 16. a)2 b)4 c)3 d) 5
65
17. all values except
0x
Chapter 1
b)
Getting Started, p. 2
1. a)250
f 21 000b) 760 f 15202. a)yes
b)no 3. a)nob)yes4. grasshopper
5. a)1000 f 20 000
1 6. a)the soprano
b)
b) probably not
11
Re
GETTING STARTED
Store Profits
A company is hiring staff for its new
mega-bookstore. If there are too few
staff members, they will not be able
to run the store effectively. If there are
too many staff members, their salary
costs will be too high.
A consult
A Tour of Your Textbook
To understand the textbooks structure, begin by taking a brief tour.
CHAPTER INTRODUCTION
SPECIFIC EXPECTATIONS
The specific expectations listed on the first page of each chapter describe the
concepts and skills that you are expec
REVIEW OF KEY CONCEPTS
4.1 Reviewing the Trigonometry of Right Triangles
Refer to the Key Concepts on page 271.
Solve each triangle. Round each side length to the nearest tenth of a
unit, and each angle to the nearest tenth of a degree.
1.
a)
b)
A
9.7 cm