A Tour of Your Textbook
CHAPTE
6
Chapter Opener
R
Quadra
tic Equ
ations
Quadrat
Form y ic Relation
s
= ax 2
+ bx + of the
2 Exp
c
res
• This two-page spread introduces what
you will learn in the chapter.
• The specific curriculum expectations
that the chapter covers are listed.
• In the vocabulary lists are the
mathematical terms that are
introduced and defined in the
chapter.
• The chapter problem is introduced.
Questions related to the chapter
problem occur in the Connect and
Apply sections of the exercises
throughout the chapter and are
identified by a Chapter Problem
descriptor.
Many rea
s y = ax 2
l-world
+ bx +
the form
c in
situatio
quadrat
y = a(x
ns can
— h) 2 +
comple
ic relati
k by
be mode
ting the
ons. Re
quadrat
square
lled usi
lated pro
involvin
in situ
ic equa
ng
g no frac
blems
tions. Fo
tions, usin ations
and en
variety
can be
r exam
of tools.
gineerin
ga
solved
ple, de
using
g use pa
signs in
consta
2 Ske
rabolic
tch or gra
nt gravit
architect
arches.
ph a qua
y, the fl
ure
whose
skateb
Because
dratic rela
ight pa
equatio
oa
rde
tion
of
th
n
r
is given
is a parab
of a ba
form y
are use
in the
= ax 2 +
ll, a roc
ola. In
d to sol
bx + c,
ket, or
variety
business
using a
ve proble
a
of metho
minimizin
, quadrat
ms of ma
ds.
g the cos
ic
2 Det
mo
ximizin
dels
ermine
t of pro
g reven
the zer
duction
os and
maxim
In
ue
thi
or
.
the
um or min
s chap
ter, yo
imum valu
quadra
the gra
u will rel
tic rela
e of a
tion from
ph of a
ate qu
from its
its graph
quadrat
adratic
definin
by mode
or
ic relati
equatio
g equatio
lling wi
2 Solv
n.
on and
ns to
th quad
e quadrat
solve pro
ic equatio
ratic eq
real roo
blems
ns that
uations
ts, usin
have
g a vari
.
ety of met
2 Det
ermine
hods.
, through
and des
investig
cribe the
ation,
connec
between
tion
the fac
tors of
expres
a quadra
sion and
tic
the x-in
of the
tercept
graph of
s
the cor
quadra
respon
tic rela
ding
tion, exp
form y
ressed
= a(x —
in the
r)(x — s).
2 Inte
rpret rea
of quadra l and non-real
roots
tic equ
ations,
investig
through
ation usin
g graphi
techno
ng
logy, and
relate the
roots to
the x-in
tercept
corresp
s of the
onding
relation
s.
2 Exp
lore the
algebraic
of the
develo
quadra
pment
tic form
ula.
2 Sol
ve proble
Vocabula
ms aris
realistic
ry
ing from
situatio
a
complet
n repres
graph or
ing the
ented by
square
an equ
quadrat
a
ation of
ic equatio
relation
a
quadra
, with and
n
root
tic
withou
of techno
t the use
logy.
quadrat
ic
formula
Chapte
r Proble
m
Skateb
oard hal
f-pipe ram
have a
ps typica
semicir
y
cular cro
lly
manufact
ss sectio
urer dec
n. A
ides to
with a
build a
parabo
half-pipe
lic cross
section.
In this
0
chapte
ground
r, you wil
dimens
l solve
ions of
related
the new
quadra
half-pipe
tic equ
ations
.
to find
the
260
261
Get Ready
Examples and practice questions review key skills from previous
mathematics courses that are needed for success with the new
concepts of the chapter.
y
t
tercep
Lines
the y-In
Graph
pe and
the Slo
d 2: Use
Metho
.
ate
y 3
d Evalu
4 and
ute an
when x
Substit
2y 1
3x Evaluate 1
3x 2y
1
2(3)
3(4)
61
12 19
4
en a sion wh
h expres
ate eac
7
a 3b
1.
b) 2
and b
2a 3b
b3
d) 1 a) a a
1
5
a
c) 3b
f) b 2
3 b
a
2. Evalu
2
en x sion wh
h expres
ate eac
5
1. Evalu
3y 3.
b) 2x
and y 2y
x
4y
d)
a) 3x
2 1 x
y
4x y
f)
c)
e)
1 y
x
2
3
fy 3(x
Simpli
y)
2(x y)
3(x y) ) 2(x) 2(
3(y
3(x)
x 2y
3y 2
3x 5y
x
e)
4
2
ssions
fy Expre
x y).
Simpli
y) 2(
Use the
.
expand
perty to
tive pro
distribu
like
Collect
terms.
fy.
5y)
y)
4(3x
4(x x 3y)
x 3y)
a) 5(2
3y) (2
1)
pli
4. Sim
2(x b) x a 5b
2) 2(2
2b c) 3(a
fy.
y)
2(x a) 5x
4a 9b
2b y)
b) 3a
3(x 2(x y)
pli
3. Sim
c)
y
Lines
of Values
Graph
a Table
1.
d 1: Use
Metho
2x line y
the
Graph
x
y
1
0
3
1
2
3
Plot the
for
values
simple
Choose
e each
y.
x. Calculat
value for
onding
corresp
5
7
points.
ough the
line thr
Draw a
points.
r1
• Chapte
4 MHR
vi MHR • A Tour of Your Textbook
x
6
y = 2x +
4
1
2
—2 0
—2
2
x
ation
The equ
form
is in the
+ b.
2 5.
y = mx
x
ey 3
the lin
2
Graph
rise .
5).
2
,
3
e is (0,
is . So run
the lin
pe, m, 3
point on
a
,
The slo
So
.
e.
b, is 5
the lin
5).
int on
ntercept,
The y-i
xis at (0, ch another po
the y-a
rea
on
b.
to
rt
pe
Sta
slo
mx use the
form y
0.
Then,
y 2 write it in the
e 3x the lin
ation to
Graph
the equ
rrange
0
First rea 3x y 2 2
x
2.
y 3
rcept is
3 . The y-inte
rise 1
, so run
pe is 3
e.
The slo
the lin
to graph
se facts
the
e
Us
pts
Interce
12.
d 3: Use
Metho
4y line 3x
0
8
6
4
2
x
2
_ —5
y = 3x
—2
Rise + 2
—4
Run + 3
y
2
Run + 1
—2 0
—2
Rise — 3
4
2
3x — y
x
—2=0
—4
y
3x — 4y
= 12
x
the
4
t, y 0.
Graph
2
x-intercep
—2 0
At the
) 12
—2
3x 4(0 12
0).
3x
e is (4,
the lin
on
4
—4
int
x
po
is 4. A
tercept
0.
The x-in
pt, x y-interce
At the
4y 12
3(0) 12
(0, 3).
ercepts.
line is
4y g the int
on the
3
15
by findin
y
A point
3y h line
is 3.
pt
b) 5x
ph eac
rce
16
nte
7. Gra
The y-i
8y y3
or the
d) 4x
ues
a) x le of val
21
ient
3y Use a tab
x
ven
7
e.
lin
c)
h
e a con
d.
ph eac
e. Choos
3
pt metho
5. Gra
x
lin
2
rce
h
nte
b) y
ph eac
20
slope y-i
8. Gra
5y 2 6
x
2
2
d.
b)
x
metho
x
3 1
a) y 10
5
d) y x
y
1 5
a) x
d) y 4
x
60
c) y 2
ng the
3y c) 2x
t rewriti
by firs
b.
e
lin
x
h
m
ph eac
30
form y
6. Gra
y
in the
b) 2x
equation
20
0
2y y1
d) 5x
a) x 70
y
c) x
dy
Get Rea
• MHR
5
Numbered Sections
Lesson Opener
Many lessons start with a photograph and short description of a
real-world setting to which the mathematical concepts relate.
Investigate
5.3
These are step-by-step activities,
leading you to build your own
understanding of the new concepts of
the lesson. Many of these activities
can best be done by working in pairs
or small groups to share ideas.
Common Factors
During a perform
ance at a sea-life
park, a dolphin
jumps out of the
water. Its height
, h, in metres,
above the water
after t seconds can
be approximated
by the relation h
10x 5x2. This
relation can
also be written as
h 5x(2 x), becaus
e the terms
in the polynomial
10x 5x2 have
a common factor
of 5x.
Investigate A
Tools
algebra tiles
How can you use
a model to find
factors of a polyno
mial?
1. To factor 2x
4, use algebra
tiles to create a
whose length
rectang
and width represe
nt the
ular area
factors of the polyno
a) Arrange two
mial.
x-tiles and four unit
tiles to form a
area 2x
4. Then, place
rectangle with
tiles along the left
the length and width
side and top to
find
of the rectangle.
One dimension
done for you.
has been
b) Write an equatio
n for the area as
a
product of the length
and width.
2. Repeat step
1 for 6x 18. How
many
different rectang
les can you find?
3. Use algebra
tiles to find the
factors of
x2 2x. Express
the area as a produc
t
of the length and
width.
4. Use algebra
tiles to factor 2x2
4x. How many
can you find? Write
different rectang
an area statem
les
ent for each one.
5. Use algebra
tiles to factor each
expression, if possib
If it is not possib
le.
le,
a) 3x 3
d) 2x2 6x
explain why.
b) 4x 10
e) 2x 5
c) x2 4x
f) 4x2 10x
6. Reflect Explai
n how you can expres
s a polynomial as
a product of factors
.
228 MHR • Chapte
r5
ct
l a Binomial Produ
Example 1 Mode
Reasoning and Proving
Selecting Tools
Examples
2)(2x 3).
ial product (x Model the binom
Representing
Solution
Problem Solving
Reflecting
Connecting
Communicating
Method 1: Use Algebra
• Examples provide model solutions that show how
the new concepts are used.
• The examples and their solutions include several
tools to help you understand the work.
– Notes in a thought bubble help you think through
the steps.
– Sometimes different methods of solving the same
problem are shown. One may make more sense to
you than the other. Often, alternative methods,
using different technology tools, are shown.
Tiles
2
le with width x
to create a rectang
Use algebra tiles
3.
and length 2x 2 -tiles,
There are two x
six
seven x-tiles, and
unit tiles.
2
7x 6.
rectangle is 2x The area of the
2x2 7x 6
(x 2)(2x 3) Diagram
it x 2 units. Divide
length and label
segment of any
2 units long.
Draw a vertical
long and a section
a section x units
the segment into
a horizontal
l segment, draw
the top of the vertica
2x units long and
Perpendicular to
d into a section
divide
long,
units
segment, 2x 3
long.
units
a section 3
x 2 and 2x 3.
g sides opposite
drawin
by
le
ts from the section
Complete the rectang
l dashed segmen
ntal and vertica
of the four
Then, draw horizo
. Find the areas
te sides, as shown
le.
marks to the opposi
up the whole rectang
sections that make
Method 2: Use a
2x + 3
2x
x
x 2x = 2x
2
3
x 3 = 3x
x+2
2
2 2x = 4x
23=6
of the four areas.
equals the sum
t (x 2)(2x 3)
The binomial produc2
6
2 x 3x 4x (x 2)(2x 3) 2
2x 7x 6
r5
214 MHR • Chapte
of
Right Bisector
Example 1
a Chord
y
of this
the centre
tor
Verify that
the right bisec
circle lies on
d AB.
of the chor
6
4
B(4, 2)
2
—6
—4
—2
0
2
M
4
6
x
• Some examples use the four-step
problem solving model to remind
you of a very helpful way of
tackling problems.
—2
—4
A(2, —4)
—6
tes (0, 0)
if the coordina
tor of AB only
right bisec
lies on the
tor.
The centre
the right bisec
of A
equation of
coordinates
satisfy the
tor. Use the
the slope
the right bisec
, calculate
on
Then
is
AB
AB.
of
of
of the
The midpoint
and the slope
coordinates
the midpoint
slope and the
AB.
and B to find
AB. Use this
bisector of
endicular to
of the right
perp
tion
line
equa
a
of
the
to determine
AB.
of
midpoint, M,
e
and the slop
coordinates
y1
y
oint
2
midp
Find the
y2
m AB x2 x1
x1 x 2 , y1
b
2
2 (4)
M(x, y) a 2
42
2
b
2 4 4
a 2 ,
2
6
2
1)
(3,
3
slope of this
r to AB, the
cula
endi
perp
bisector is
l of m AB.
roca
Since the right
recip
the negative
bisector is
Solution
1
1
t.
3
the y-intercep
mAB
1) to find
dinates M(3,
e and the coor
Use this slop
a y-intercept
All lines with
h the
y mx b
of 0 pass throug
1
b
origin.
1 3 (3)
b
1 1
1 The coordinates (0, 0)
.
x
0b
y 3
right
bisector is
e lies on the
of the right
re of the circl
The equation
efore, the cent
equation. Ther
satisfy this
the chord PQ.
MHR 147
bisector of
s of Circles •
3.5 Propertie
A Tour of Your Textbook • MHR vii
Key Concepts
c) You can read
the maximum y-value
from
the pencil
• This feature summarizes the concepts learned in the lesson.
• You can refer to this summary when you are studying or doing
homework.
-and-paper graph
or use the
TRACE feature on
the graphing calcula
tor.
Since the maxim
um value of y is
26, the
height of each arch
is 26 m.
The x-axis represe
nts the floor of the
hallway. The width
of each arch is the
difference betwee
x-intercepts. From
n the two
the pencil-and-p
aper graph, the
appear to be about
x-intercepts
7 and 7. Use the
TRACE feature on
graphing calcula
the
tor to find that the
curve crosses the
about 6.9 and
x-axis at
6.9.
6.9 (6.9) 13.8
The width of each
arch is about 13.8
m.
I can see that the
maximum occurs
when
x = 0. From the equation
,
when x = 0, y = 26.
Technology Tip
You can get a better
approximation of
the
x-intercepts by zooming
in.
• Position the cursor
near
one of the x-interce
pts.
• Press z, select
2:Zoom In, and then
press e.
• Press u and
reposition the cursor.
Key Concepts
Communicate Your Understanding
These questions allow you to reflect on the concepts of the
section. By discussing these questions in a group, you can see
whether you understand the main points and are ready to start
the exercises.
The relation define
d by
y a x 2 bx c is a quadratic
relation.
axis of symmetry
vertex,
maximum
point
The graph of a quadra
tic relation
is called a parabo
la.
The accuracy of the
approximation improves
each time you repeat
these steps.
The vertex of a
parabola is either
the minimum point
vertex,
or the
minimum
maximum point
on the graph.
point
A parabola is symme
tric about a vertica
l line that passes
vertex. This line
through the
is the axis of symme
try.
If a relation is quadra
tic, the second differe
first differences
nces
are constant, but
are not.
the
Communicate Your
Understanding
C1
C1
El-Noor used the
following incorre
ct technique to
the relation is not
determine that
quadratic. Explai
n the flaw in his
reasoning.
x
y
First Differen
ces
—3
13
3
0
—5
1
—3
2
3
6
4
27
24
18
4
C2
Second Differen
ces
—10
—2
2
—8
2
10
In Section 4.1, Investi
gate Part A, you
found that the relatio
between thumb
length and palm
nship
area is non-linear.
quadratic? Explai
Is the relation
n.
4.2 Quadratic Relation
s • MHR 171
Exercises
Practise
• These questions provide an
opportunity to practise your
knowledge and understanding
of the new concept.
• To help you, questions are
referenced to the examples.
Communicate
Your Understan
ding
C1 How
do the graph
s of y 2x2 and
Explain the simil
y 2x2 comp
arities and the
are?
differences.
Match each graph
with the
appropriate equat
y
ion. Explain
your reasoning.
C2
a) y x 2 2
4
b) y 1 2
x
2
8. Write an
equation for
the quadratic
that results from
relation
each transforma
D
ally by a factor
of 8.
C
—4
—2 0
2
4
x
—2
2)2
—4
A
Practise
For help with
questions 1 to
4, see the
Investigate.
1. Sketch graph
s of these three
quadratic
relations
on the same
set of axes.
a) y 3x 2
c) y 1 2
x
b) y 1 x 2
4
4
2. Sketch graph
s of these three
quadratic
relations
on the same
set of axes.
a) y (x 2
9)
c) y (x 2
5)
b) y (x 2
2)
3. Sketch graph
s of these three
quadratic
relations
on the same
set of axes.
a) y x 2 8
b) y x 2 5
5. a) Make
tables of value
s for y x2,
y 2x2, y
2
x
1, and y (x 3)2.
b) Compare
the y-values
for y x2 and
y 2x2.
c) Compare
the y-values
for y x2 and
y x2 1.
d) Compare
the y-values
for y x2 and
y (x 3)2
.
Connect and
Apply
each transforma
tion.
a) The graph
of y x2 is transl
ated 6
upwa
rd.
4. Sketch the
graph of each
parabola. Label
least three point
at
s on
the parabola.
the transforma
Describe
tion from the
graph of y 2
x.
b) y 2 x 2
3
c) y x 2 5
d) y (x 2
8)
e) y 1 2
x
f) y (x 2
2
3)
g) y x 2 0.5
h) y x 2
2
a) y 4x 2
units
b) The graph
of y x2 is transl
ated 4 units
downward.
7. Write an
equation for
the quadratic
that results from
relation
each transforma
tion.
a) The graph
of y x2 is transl
ated 7
to
the left.
units
b) The graph
of y x2 is transl
ated 5 units
to the right.
c) The graph
of y x2 is transl
ated 8 units
to the left.
d) The graph
of y x2 is transl
ated 3 units
to the right.
178 MHR • Chapt
er 4
Extend
• These are more challenging and thought-provoking questions.
• Most sections conclude with a few Math Contest questions.
viii MHR • A Tour of Your Textbook
vertically by
a factor of 1
.
5
For help with
question 9, see
the Example.
9. The grass
in the backyard
of a house is
a square
with side length
10 m. A
x 10 m
square patio
is placed in
the centre. If
the side
length, in metre
s, of the patio
is x, then the
area of grass
remaining is
given by the
relation A x 2 100.
a) Graph the
relation.
b) Find the
intercepts. What
do they
represent?
c) How does
the equat
grass in the backy ion change if the
base length and
area. Use finite
differences to
determine
whether the
relation is linear
, quadratic,
or neither.
b) Determine
an equation for
the
relationshi
p between the
base length
and the area.
c) Describe
the transforma
tion
graph of y 2
x.
to graph y 2
y x 2 k both
ax and
follow what
is indicated
by the operation,
but in y (x
h)2, the
transformation
is opposite to
what the
operation seem
s to indicate.
ard of a house
square with side
is a
length 12
a) Explain
why this migh
t be so.
b) Describe
10. The relati
on l 0.04s2
can be used
calculate the
to
length
, l, in metres,
skid mark for
of the
a car travelling
at a speed, s,
in kilometres
per hour, on
dry pavement
before braking.
the result
s in part a) comp
are?
c) For what
values of s is
this
model valid?
d) How woul
d the skid mark
s and the
equation
wet?
change if the
pavement were
Did You Know
A Technic
the transforma
tion you woul
use to graph
d
y (2x)2.
13. A parab
ola y ax 2 k passes throu
the points
gh
(1, 3) and (3,
13). Find the
values of a and
k.
14. Compare
the graphs of
y (x 2)2
and y (2 2
x) . Explain any
similarities
and differences
.
15. Math Conte
st
a) Identify
the similaritie
s
and differences
in the graphs
of y (x 2) 2
5 and
x (y 2)2 5.
?
al Collision Investig
ator or Recons
specially trained
tructionist is a
police officer who
investigates serious
accidents. These
traffic
officers collect
and interpret evidenc
determine the
e to
cause of the collision
and if any charges
be laid.
should
from the
Extend
12. The transf
ormations
m?
d) For what
values of x is
each equation
valid
a) What is
the length of
the skid mark
car travelling
for a
at 50 km/h?
100 km/h?
b) How do
e has a side length
of
a) Make a table
comparing
?
6. Write an
equation for
the quadratic
that results from
relation
Connect and Apply
• These questions allow you
to use what you have
learned to solve problems
and make connections
among concepts. In answering these questions
you will be integrating your skills with many of the
math processes.
• There are many opportunities to use technology. If specific
tools or materials are needed, these are noted and the question
has a Use Technology descriptor.
c) y x 2 10
1 unit.
b) The graph
of y x 2 is comp
ressed
3
c) y x 2
d) y (x 11. The first
three diagrams
in a pattern are
shown. Each
squar
tion.
a) The graph
of y x 2 is stretc
hed
vertic
B
b) Solve the
second equat
ion for y.
4.3 Investigate
Transformations
of Quadratics
• MHR 179
Technology
Scientific and graphing calculators are useful for many sections.
Keystroke sequences are provided for techniques that may be
new to you.
• A TI-83 Plus or TI-84 Plus graphing
calculator is useful for some sections,
particularly for graphing relations. In the
analytic geometry chapters, alternative
methods using Cabri® Jr are shown.
Reasoning
and Proving
Representing
Selecting
Reflecting
Communicating
The
computer with
pad ®
Geometer’s Sketch
let x 0.
y 4.9(0) 2
50
50
zero
a value of
x for which
a
relation has
a value of
0
corre
sponds to
an
x-intercept
of the graph
of the relati
on
vertic
Points. Move
triangle with
Construct the
first quadrant.
and C(6, 2).
is isosceles.
verify that ABC
each side to
of
length
the
2. Measure
equal?
s of ABC are
median
Which angle
, construct the
of side AB. Then
D,
about the
ude
oint,
the midp
you concl
3. Construct
C. What can
Measure AD
from vertex C.
CD?
an
medi
the
You can also place
g Plot
vertices by choosin
Graph
Points from the
the
menu and typing
coordinates.
.
ACD and BCD
and compare
4. Measure
new location.
of ABC to a
Drag a vertex
ve the
Snap Points.
AC BC. Obser
5. Turn off
vertices until
t this process
the other
as you repea
Move one of
, and BCD
ADC, ACD
es.
measures of
ons of the vertic
tor
for various locati
the angle bisec
altitude, and
median, the
the
related to the
are
How
line segments
6. Reflect
have
d? How are these
les triangles
at vertex C relate
think all isosce
of AB? Do you
ning.
right bisector
reaso
your
Explain
these properties?
Since heigh
t
and time
cannot
be negative,
this graph
shows only
part
of a parab
ola.
I can see
that the y-inte
rcept
is 50
from the graph
The y-interc
.
ept is 50.
This repr
which the
esents the
stone was
height from
dropped,
50 m abov
To find the
e the wat
er.
x-intercep
t, or zero
operation
, of the rela
on a grap
hing calc
tion use the
• Press n
ulator.
Zero
[CALC] to
display the
and select
CALCUL
2:zero.
ATE men
• Move the
u,
cursor to
the left of
x-intercep
the
t and pres
s e.
• Move the
cursor to
the right
x-intercep
of the
t and pres
s e.
• Press e
again.
The x-inte
rcept is app
roximately
This repr
esents the
3.19.
time whe
n
the
stone hits
c) The con
the water,
stant term
3.19 s.
would be
would chan
75 instead
ge to y of 50. The
4.9x 2 equation
75.
176 MHR
• Chapter 4
• The Geometer’s Sketchpad® is used in
several sections for investigating concepts
related to analytic geometry. Alternative
steps for doing investigations using pencil
and paper are provided for those who do
not have access to this computer software.
ator
displayed,
a Graphing Calcul
axes are not
Method 3: Use
from
cation. If the
choose Axes
Cabri® Jr. appli
1. Start the
F5 menu; then,
is
Show from the
of the work area
Tools
calculator
with the
window setti
ngs show
n.
b) For the
y-intercept,
pad ®
e Snap
Geometer’s Sketch
. Then, choos
Graph menu
is in the
Grid from the
the work area
2),
1. Choose Show
that most of
es A(2, 5), B(1,
the origin so
Tip
Technology
A stone is
dropped
from the
height, y,
top of a 50-m
in metres,
cliff abov
above the
relation y
ea
water can
4.9x 2
be estimated river. Its
50, whe
re x is the
using the
a) Graph
time, in seco
the relation
nds.
.
b) Find
the intercep
ts. What
c) How
do they repr
would the
esent?
equation
from a 75-m
change if
the stone
cliff instead
were drop
of a 50-m
d) For wha
ped
cliff?
t values of
x is each
equation
valid?
Solution
a) Use a
graphing
The
Method 2: Use
Tools
Example
Falling Sto
ne
Tools
Problem Solving
Connecting
TI-84 Plus
TI-83 Plus or
tor
graphing calcula
er
118 MHR • Chapt
choose Hide/
so that most
Move the origin
the submenu.
ant.
in the first quadr
3
Technolo
gy Tip
Method 2:
Use
a Compute
r Algebra
When a solu
System (CAS
tion invo
)
lves fraction
your wor
k.
s, a CAS
is helpful
for checking
Turn on the
TI-89 calc
ulator. If
• Press n
the CAS
¡ to
does not
access the
• Select 2:Ne
start, pres
F6
menu.
HOME
wProb to
sÍ
.
• Press e
clear the
CAS.
.
Solve equ
ation for
y:
• Type in
the equatio
n
4x y 9.
• Press e
.
• Place brac
kets arou
nd the
equation.
• Type 4Í
X
. Press e
.
Substitute
9 4x in
place of y
equation
in
. Press
e. Cop
simplified
y the
form, and
Paste it into
the comman
d line. Put
around the
brackets
equation,
and add 9.
Press e
.
Copy the
new form
of the
equation,
and Paste
it into the
command
line. Put
brackets
around the
equation,
and divide
by 6. Pres
s e.
To find the
correspondin
for y:
g value
• Copy y
9 4x
and Paste
into the com
it
mand line
retype it).
(or
• Type Í
|
X
Í
=13 ÷
• Press e
6.
.
There are
several ways
to
solve linear
systems using
a CAS. The
steps show
n
here follow
the steps
used
by hand.
• Some sections show you how to use a
computer algebra system (CAS) to explore
algebraic processes or as an alternative
tool to solve algebraic problems. This
text uses the TI-89 calculator.
Technolo
gy Tip
The Í
|
key means
“such
that.” It is
used to evalu
ate
an expression
for a given
value.
The lines
intersect
at a 13, 1
6 3b .
Note that
the solution
to Exampl
fractions.
e 3 involves
This is an
example
solved accu
that cann
rately by
ot be
graphing,
calculator
unless a grap
is used.
hing
24 MHR •
Chapter 1
Technology Tip
• This margin feature points out helpful hints or alternative
strategies for working with graphing calculators or The
Geometer’s Sketchpad®.
Technology Appendix
• The Technology Appendix provides detailed help with all
the functions of the graphing calculators and The Geometer’s
Sketchpad® that are used in this course.
A Tour of Your Textbook • MHR ix
Assessment
Key Concepts
The midpoint
of a line segm
ent can
be found by
adding half of
the run
and half of the
rise to the coord
inates
of the first endp
oint.
Each
coordinate of
the midpoint
a line segment
of
is the mean of
the
corresponding
coordinates of
the
endpoints.
Communicate Your Understanding
B (x , y )
2 2
x1 + x
y +y
2 _______
M _______
2
, 1
2
2
(
)
A (x , y )
1 1
rise
___
2
rise
___
2
run
___
2
run
___
2
The midpoint
of a line segm
ent with endp
oints ( x , y )
has coordinates x1 x2 y1 y
1
1 and (x2 , y )
2
a
2
,
b.
2
2
To find an equat
ion for the medi
coordinates of
an of a triang
le, first find
the midpoint
the
of the side oppo
coordinates of
site the vertex
the midpoint
. Use the
and the vertex
the median.
to calculate the
Then, substitute
slope of
the slope and
point into y the coordinates
mx b to solve
of either
for the median’s
To
y-intercept.
find an equat
ion for the right
the slope and
bisector of a
line segment,
midpoint of
the segment.
first find
to calculate the
Use the line
slope of a perpe
segm
ent’s
slope
ndicular line.
slope and the
Then, substitute
coordinates of
the midpoint
this
for the right
into y m x
bisector’s y-inte
b to solve
rcept.
• These questions provide an opportunity to assess your
understanding of the concepts before proceeding to use your
skills in the Practise, Connect and Apply, and Extend questions.
• Through this discussion you can identify any concepts you need
to study further.
Communicate
Your Understan
ding
C1 Descr
y
ibe two meth
ods for findin
midpoint of
g the
this line segm
ent.
C2
P (1, 3)
3
2
1
0
Describe how
to determine
an equation
for the median
from vertex A
of ABC.
Q (5, 1)
1 2 3 4 5
6
A(—2, 3)
3
2
B(—1, 1) 1
C(3, 0)
1 2 3 x
—1 —2 —10
C3
Describe how
to determine
an equation
for the right
bisector of line
segment PQ.
P(—4, 2)
x
y
y
6
4
2
—6 —4 —2 0
Q(6, 5)
2 4 6
x
2.1 Midpoint of
a Line Segment
• MHR 65
18. Chapter
Problem
Here are
stages of
the first thre
a Koch snow
e
flake, nam
Swedish
mathematic
ed after the
ian
von Koch
(18701924 Niels Fabian Helg
e
). This snow
starts with
an equilate
flake
ral triangle.
stage, the
middle thir
At each
d of each
replaced
side is
by two segm
ents, each
length to
equal in
the segment
they replace.
a) Find
the coordina
tes of the
vertices on
three new
the bottom
in the seco
of the snow
nd stage.
flake
b) Are Koc
h snowflak
es fractals?
Explain.
Special Connect and Apply Questions
• Some questions are related to the Chapter Problem.
• Achievement Check: The last Connect and Apply
question of some sections provides an opportunity to
demonstrate your knowledge and understanding and
your ability to apply, think about, and communicate
what you have learned. Achievement Check questions
occur every two or three sections and are designed to
assess learning of the key concepts in those sections.
Extend
20. The
point A(x
, 1) is 5 unit
point (2,
s from the
6).
a) Find
a possible
valu
e for x.
b) Is this
value the
only solu
tion? Exp
21. a) On
lain.
a grid, draw
line segm
lengths
ents
with the
listed
method. How below. Describe you
r
do you kno
line segm
w that each
ent is the
correct leng
th?
ii) 25
iii) 213
iv) 241
the line segm
those draw
ents you
drew to
n by a clas
smate. Veri
both sets
of line segm
fy that
ents have
correct leng
the
ths.
i) 22
b) Compare
(0, 0)
(3, 0)
Go to www
.mcgrawhil
l.ca/links/p
the links to
rinciples10
learn more
and follow
about Koch
snowflake
s.
Achievemen
t Check
19. Ligh
tning star
ts
a forest fire
Reasoning
and Proving
Representing
at a
point with
Selecting
Tools
map
coordinates
Problem
Solving
Connecting
F(23, 25).
The
Reflecting
nearest tow
Commu
ns
nicating
coordinates have
A(15, 19)
and B(23
a) Which
, 17).
town is at
greater risk
the fire?
from
b) Describ
e
any assumpt
ions you
for your answ
made
er to part
a).
und midway
two towns,
between the
a camper
learns abou
from a new
t the
scast. The
camper estim fire
that he is
about 9 km
ates
from the
camper corr
fire. Is the
ect? Justify
your resp
onse.
c) At a cam
pgro
22. List
the points
that satis
and have
integer coor fy each condition
dinates.
a) 5 unit
s from the
origin
b) 5 unit
s from the
point (2,
c) 10 unit
1)
s from the
poin
t (5, 2)
23. Sally
has hidden
her brother’
present
s
birthday
somewhere
in the back
writing inst
yard. Whe
ructions for
n
she used
finding the
a coordina
present,
te syst
unit on the
grid represen em with each
positive y-ax
ting 1 m.
The
is
The instruct of this grid points
north.
ions read
“Start at
walk half
the origin,
way to (8,
6), turn 90°
then walk
left, and
twice as far.”
present?
Where is
How far is
the
it from the
origin?
24. Math
Contest The
number of
of five lette
arrangem
rs from the
ents
contain the
word mag
netic that
word net
is
A 60
B 100
C 360
D 630
25. Math
E 720
Contest Sho
w that the
larger sem
area
icircle is
equal to the of the
areas of the
sum of the
two smaller
semicircles.
2.2 Length
Chapter Problem Wrap-Up
This summary problem occurs at the end of
the Chapter Review. The Chapter Problem
may be assigned as a project.
of a Line Segm
ent • MHR
79
Chapter 3 Rev
iew
3.1 Investigate
Properties of
Triangles,
pages 110—116
1. a) Define
a median.
b) List two
additional prope
rties of the
medians of a
triangle.
c) Outline
how you could
use geometry
software to show
that the medi
ans of
all triangles have
these properties.
2. a) Use congr
uent triangles
A
to show how
the
lengths of the
medians
to the two equal
sides
M
N
of an isosceles
triangle
are related.
b) Describe
how you can
B
C
use geometry
software to demo
nstrate
that this relati
onship appli
es for all
isosceles triang
les.
5. a) Verify
that the altitu
de from vertex
bisects side KL
J
in the triangle
with
vertices J(5,
4), K(1,
8), and L(1
, 2).
b) Classify
JKL. Explain
your reasoning.
3.3 Investigate
Properties of
Quadrilaterals,
pages 128—136
6. List two
properties of
the diagonals
of each geom
etric shape.
a) square
b) parallelogr
am
c) kite
7. a) Show
that the line
segment joinin
midpoints of
g the
opposite sides
of any
parallelogram
bisects the area
of the
parallelogram.
A
d with the coord
inate axes.
Then, draw the
right bisectors
of all
three sides.
b) Show that
the point of inters
ection
the
of
right bisectors
of the sides of
right triangle
any
lies on the hypo
tenuse.
c) Describe
how you can
use geometry
software to answ
er part b).
4. a) Verify
that DEF is
a right triang
le.
y
B
C
b) Describe
another way
to answer part
a).
8. Verify that
quadrilateral
trapezoid.
E(—2, 2) 2
—2
F(3, 1)
0
2
4
x
b) Describe
another meth
od that you could
use to answer
part a).
152 MHR • Chapt
er
3
4
JKLM is a
M(3, 4)
J(—3, 2)
2
L(7, 1)
—4
—2
of another point
,
is a right triang
le.
12. Divers from
scuba clubs in
St. Catharines,
Hamilton,
and Oakville
are working
together to pract
ise rescues in
deep water.
Describe how
the clubs could
find a
practice site
that is equid
istant from the
three towns.
11. a) Verify
that points P(5,
7) and Q(7, 5)
on the circle
lie
with equation 2
x y2 74.
b) Verify that
Oakville
the right bisec
tor of the
chord PQ passe
s through the
centre
of the circle.
Lake Ontario
Hamilton
3.4 Verify Prope
rties of Quad
rilaterals,
pages 137—144
K(—2, —2)
D(1, 4)
4
C, on the circle
.
c) Show that
ABC
F
y
3.2 Verify Prope
rties of Trian
gles,
pages 117—127
of Circles, pages
are
oints of a diam
eter of the circle
defined by x2
y 2 169.
b) State the
coordinates
D
E
3. a) Draw
a right triang
le with the short
sides aligne
er
3.5 Properties
145—151
10. a) Show
that A(12, 5)
and B(12, 5)
endp
0
2
4
6
—2
9. a) Classify
the quadrilater
al
8
x
with vertices
T(2, 4), U(8,
2), V(7, 1),
and W(1, 1).
Justify this classi
fication.
b) Verify a
property of the
diagonals of
TUVW.
St. Catharines
Chapter Probl
em Wrap-Up
The golden ratio
is also know
n as the
divine propo
rtion, the golde
n mean, and
the golden sectio
n. Many peopl
e find that
objects with
proportions in
the golden
ratio are pleas
ing to the eye.
For this
reason, artist
s and architects
sometimes
use these propo
rtions in their
work.
a) Research
the golden ratio
at a library
or on the Intern
et. Is the golde
n ratio
a rational numb
er? Explain.
b) Give an exam
ple of a regul
ar
geometric shape
that involves
the
golden ratio.
Describe how
the
golden ratio
appears in this
shape.
c) Give an exam
ple of a natur
al object that
ratio. Explain
may contain
why researcher
the golden
s disagree on
occurs frequently
whether the
in nature.
golden ratio
d) Some resea
rchers argue
that designers
golden ratio
in ancient times
in buildings
used the
such as the Parth
and the Great
enon in Athen
Pyramid at Giza,
s, Greece,
Egypt. Do you
are valid? Justif
Go to
think these claim
y your answer.
s
e) Measure
www.mcgrawh
a textbook. Are
ill.ca/
links/principles
any of the dime
10 and follow
golden ratio?
nsions relate
Explain.
the links to learn
d by the
more
about the golden
ratio.
Chapter 3 Review
• MHR
x MHR • A Tour of Your Textbook
153
Practice Test
• Each chapter ends with a
practice test.
Chapter 8 Prac
tice
Test
1. Find the
length of the
indicated side,
to the
nearest centim
etre.
C
25 cm
b=?
45°
B
74°
A
2. Find the
length of the
indicated side,
to the
nearest tenth
of a metre.
K
3.6 m
R
3. Find the
measure of the
indicated angle
, to the
nearest degre
e.
7. Solve XYZ
. Round the side
length to the
nearest tenth
of a metre
and the
measures to
the nearest degre angle
e.
11. Tess was
flying from Toron
to to
Hamilton
Y
at night. She
noticed that
heading indic
her
ator was malfu
nctioning
and decided
to check her
position.
She called Hami
lton Tower and
St. Catharines
Radio, know
ing that the
stations were
50 km apart.
Hami
reported that
lton
the position
of her plane
formed an angle
of 65° with the
joining the two
line
stations. St. Catha
reported that
rines
her position
made
of 48°. How
far was the plane an angle
from
Hamilton, to
the nearest kilom
etre?
12. Scuba diver
s count fin stroke
s to estimate
the distance
they have travel
led under
water. Felipe
uses this meth
od and a
compass to swim
300 m north
from the
dive boat. He
then turns right
120° and
swims 400 m.
How far from
the boat is he,
to the nearest
metre? In what
direction
must Felipe
swim to return
to the boat?
13. While stand
ing exactly halfw
ay between
two tall trees,
Nikki spots a
blue jay at the
top of one tree
and a cardinal
at the top of
the other, as
shown.
Z
8.3 m
r=?
36°
4.3 m
68°
T
7.2 m
X
Y
8. In acute
NTW, N 54°, n 2.3
and w 1.8
km,
km.
?
16 km
a) Sketch this
triangle and
label the given
information.
D
4. Find the
measure of the
indicated angle
, to the
nearest degre
e.
18 mm
H ?
13 km
65°
b) Solve this
triangle. Roun
d the side
length to
the nearest tenth
of a metre
and the angle
measures to
the nearest
degree.
21 mm
24 mm
5. Two wires
are
N
E
S
supporting a
80°
tent pole,
as shown.
2.2 m
2.8 m
a) How far
apart are
the wires fixed
in the ground,
to the
nearest tenth
of a metre?
b) Find the
angle each wire
makes with
the ground, to
the nearest degre
e. State
any assumption
s you make.
6. Cheryl is
trying to hit
her golf ball
between
two trees. She
estimates the
distances show
n.
20 m
Within what
angle
must Cheryl
35 m
make her
shot, in order
to pass
between the
trees?
Round to the
nearest
tenth of a degre
e.
9. Students
of Fowler’s Aeron
autica
have
l School
this crest on
their pilot’s cap.
the total length
Find
of gold trim
needed to
make a crest,
to the nearest
tenth of a
centimetre. State
any assumption
make.
s you
16 cm
46°
6.2 m
join the town
Port, Sackville,
s of West
and Jonestown
, as shown.
West Port
82°
17 km
12 km
Sackville
The fire statio
n serves a total
area of
126 km2. Find
the number of
minutes it
would take for
a fire truck to
drive the
perimeter of
this region, assum
ing an
average speed
of 80 km/h.
15. From his
cockpit, the
pilot of a flying
J
12 km
plane
observes a jet
P
flying
directly above
an airport,
as shown. From
the
16 km
plane, the angle
of
elevation to the
jet is 14°
A
and the angle
of depression
to the airport
is 38°. Find the
altitude of both
aircraft, to
the nearest kilom
etre.
Achievement
Check
C
measurements
given to find
the
height of the
Peace Tower
in
Ottawa, to the
nearest metre
.
AB 50 m
XAY 43°
XAB 60°
ABX 82°
16. A small
lake has
Y
40 m
From Nikki’s
point of view,
the angles
of elevation to
the blue jay and
to the
cardinal are
60° and 50°,
respectively.
The trees are
40 m apart.
a) If Nikki
is 1.5 m tall,
find the heigh
of each tree,
t
to the nearest
tenth of
a metre.
X
B
Reasoning and
Proving
Representing
Selecting Tools
three approximat
ely
Problem Solving
straight shore
lines
Connecting
so it forms a
Reflecting
Communicating
triangle, LAK
.
L
a) List all of
the
possible
k
a
measurements
that could be
A
K
l
determined for
this lake.
b) If you were
a surveyor stand
ing at point
L, which of these
measurements
you measure
would
and which ones
would
you calculate?
Justify your choic
e.
c) Could you
have made a
different choic
in part b)? Is
e
this choice better
than
your previous
one? Explain.
N
8
b) How far
apart are the
two birds, to
the nearest tenth
of a metre?
Chapter 8 Practi
ce Test
Tasks
Jonestown
Fire Station
B
10. Use the
A
432 MHR • Chapt
er
14. A fire statio
n serves an area
bordered by
three highways
that
• MHR 433
Tasks
• Tasks are presented at the end of Chapters 3, 6 and 8.
• These are more involved problems that require you to use
several concepts from the preceding chapters. Each task has
multi-part questions and may take about 20 min to complete.
• Some tasks may be assigned as individual or group projects.
Cari Sports
Centre
Some of the
costs of runn
of people
ing Cari
using the
facility incr Sports Centre decr
staff need
ease
eases. Oth
ed to serv
er costs, such as the number
e the add
itional peo
as the sala
As part of
ple, grow
ries of
a budget
.
review, city
from the
cou
Cari Sports
Centre. City ncil would like to
and develop
improve
council surv
s a formula
the profit
• the pric
eys people
e of admissio by examining
using the
centre
• the num
n to the cent
ber of tick
re
ets sold
• the con
stant dail
y expense
s in runn
The centre
ing the cent
manager,
re
Emily, was
of admissio
given the
n tickets,
formula that
the price
Emily lost
of a ticket,
the
and the dail related the number
already calc paper on which the
y profit. Unf
formula was
ulated the
ortunately,
following
written, but
data list.
Ticket Price
she had
($)
Daily Profi
t ($)
8
56
10
200
12
296
14
344
a) Copy the
table
you conclud and find the first
and second
e about the
differenc
formula?
b) Make
es. What
a scatter
can
plot of the
price will
data. Draw
give the grea
a curve of
test daily
c) What are
best fit. Wha
profit?
the constan
t ticket
t daily exp
d) What rang
enses?
e of admissio
least $326?
n prices will
ensure that
the daily
profit is at
Chapter Review
322 MHR
• Chapter 6
• This feature appears at the end of each chapter.
• By working through these questions, you will identify
areas where you may need more review or study before
doing the Practice Test.
Chapter 4
Review
4.1 Investig
ate Non-Lin
ear Relation
pages 164
s,
—167
1. Which
scatter
using a curv plot(s) could be mod
elled
e instead
of a line of
Explain.
best fit?
a) y
Cumulative Review
Montréal?
c) What
is the max
imu
x
0
x
tested the
strength of
wood beam
s
by securing
beams of
various leng
ths and
placing a
500-kg mas
s at
the end of
each beam
.
The table
shows the
mean defl
ections, in
centimetres
.
a) Make
a scatter
plot of
the data.
Draw a curv
e
of best fit.
b) Describ
e
Length Defle
4.3 Investig
ction
ate Transfor
(m)
mations
pages 174
(cm)
of Quadrat
—179, and
ics,
1.0
4.4 Graph
0.33
y = a(x — 2
h) + k, pag
1.5
1.48
5. Sketch
es 180 —18
the graph
8
2.0
of each para
the transfor
2.51
bola
. Describe
mation from
2.5
the graph
4.22
a) y x 2
of
y
6
x 2.
3.0
6.11
b) y 0. 2
5x
3.5
8.17
c) y (
x 2) 2
4.0
10.52
d) y 2 2
x
4.5
12.98
6. Copy
5.0
and complet
16.72
e
the
relationship
between
the length
beam and
of
the table
parabola.
for each
Replace the
heading for
second colu
the
mn with
the equatio
parabola.
n for the
Then, sket
ch each para
bola.
Property
the deflecti
the
on.
c) Use you
r curve of
best fit to
deflection
predict the
of a
1
11
2
18
27
4
5
• A Course Review follows the tasks at the end of Chapter 8.
This comprehensive selection of questions will help you
to determine if you are ready for the final examination.
Vertex
4.2 Quadrat
ic Relation
s, pages
3. Use finit
168 —173
e differenc
es to dete
each rela
rmine whe
tion is line
ther
ar, quadrat
a)
ic, or neit
x
her.
y
b)
3
38
51
202 MHR
• Chapter 4
x
y
—2
—10
—1
—2
0
0
1
2
2
10
fly from Toro
nto
m height
aircraft? At
of the
what time
does the
reach this
aircraft
height?
0
6.0-m-long
beam.
Course Review
to
modelled
h 2.5t 2
by the rela
200t, whe
tion
minutes,
and h is the re t is the time, in
height, in
a) Graph
metres.
the relation
.
b) How
long does
it take to
to
b) y
2. A scie
ntist
• A cumulative review occurs at the end of Chapters 3, 6, and
8. These questions allow you to review concepts you have
learned in the chapters since the last cumulative review.
They also help to prepare you for the tasks that follow.
4. The fligh
t of an airc
raft from
Montréal
Toronto
can be
c)
x
—2
y
—9
—1
—6
0
—3
1
0
2
3
y = a(x —
h) 2 + k
Axis of symm
etry
Stretch or
compressio
n
factor relati
ve to y = 2
x
Direction
of opening
Values x may
take
Values y may
take
a) y (
x 1) 2 4
b) y 2(
x 3) 2 1
c)
y 1 (x
2
4 5) 1
d) y (
x 2) 2 6
A Tour of Your Textbook • MHR xi
Other Features
The Mathematical Process
These seven mathematical processes are integral to learning
mathematics:
• problem solving
• reasoning and proving
• reflecting
• selecting tools and computational strategies
• connecting
• representing
• communicating
Reasoning and Proving
Representing
Selecting Tools
Problem Solving
Connecting
Reflecting
Communicating
These processes are interconnected and are used throughout the
course. Some examples and exercises are flagged with a math
processes graphic to show or remind you which of the processes
are involved in solving the problem.
Making Connections
rdo of Pisa
fraction is a
st A continued
1
22. Math Conte
form a fraction of the
b
1
c
st In 1202, Leona
acci,
23. Math Conte
known as Fibon
better
i. In it,
(11751250),
called Liber Abac
published a book
ers now
sequence of numb 1, 2, 3, 5,
he showed a
1,
acci sequence:
called the Fibon
8, 13, 21, ….
a
and
next three terms
a) Find the
nce.
for this seque
formation rule
successive
the ratios of
b) Examine
nce
Fibonacci seque
terms of the
ble
(n 1)th term . Conjecture a possi
nth term
1000th term .
ratio 999th term
value for the
.
1
d p
fraction
the continued
a) Evaluate
2
1
3
.
1
1
5
4
nued
the infinite conti
b) Evaluate
1
fraction 1 1
1
.
This feature points out some of the connections between
topics in the course, or to topics learned previously.
system of
st Solve the
24. Math Conte
1
1 p
1
equations.
0
abc
1
acd
0
a 2b 2d
bd3
Literacy Connections
This margin feature provides
tips to help you read and
interpret items in math.
in
Mak
about
ts you have learned
onnections
many of the concep
version for
picture of how
here. Make a larger
provide a mental
One is started
A mind map can
are connected.
that help you.
rs 4, 5, and 6
any other details
quadratics in Chapte
ions
parts and add
Quadratic Equat
te the shaded
Comple
f.
yoursel
0
ax2 + bx + c =
ons
Quadratic Relati
A y = ax + bx
a > 0 means
2
:
expression factors
If the quadratic
s) = 0
• a(x — r)(x —
2 zeros are at
y
+c
a < 0 means
c is the
s
0
x
t
of squares
= 0 difference
try is
axis of symme
(h, k) is the
zeros↔
x-intercepts↔
roots
are found:
If no easy factors
form B
square to get
• complete the
tic formula
• use the quadra
la
exist, then parabo
if no real roots
r and s are the
atic Formula •
6.4 The Quadr
MHR 303
It is a good idea to read a
word problem three times.
Read it a second time for
understanding. Express the
problem in your own words.
Read it a third time to plan
how to solve the problem.
Internet Links
This logo is shown beside questions in which it is suggested
that you use the Internet to help solve the problem or to
research or collect information. Some direct links are provided
on our Web site www.mcgrawhill.ca/links/principles10.
Did You Know?
This feature provides interesting facts
related to the topics in the examples
or the exercises.
xii MHR • A Tour of Your Textbook
onnections
Read it the first time to get
the general idea.
perfect square
p)
• a(x — p)(x +
roots are
c
2 +k
B y = a(x — h)
as in form A
a means the same
— s)
C y = a(x — r)(x
as in form A
a means the same
• a(x — t) = 0
1 zero at t
2
r
Literac
Did You Know ?
Household white vinegar is
5% acetic acid. A 5% acetic
acid solution means that 5%
pure acid is mixed with 95%
water. For example, 1 L of
white vinegar contains
50 mL of pure acetic acid
and 950 mL of water.
Appendices
Challeng
e Problem
s Append
ix
1. Write
an equatio
n
that form
so that the
s a system
system has
of equatio
ns with x
a) no solu
y 4,
tion
b) infinitely
many solu
tions
c) one solu
tion
Challenge Problems Appendix
2. Sketch
a graph to
represen
solu
t a system
tion, so that
of two equ
the two line
ations with
s have
one
a) different
x-inte
A varied selection of more difficult problems is presented on
pages 448–457. Some are directly related to the content of this
course, while others are more general “puzzler” problems.
These problems will provide new challenges and enrichment.
They will help if you are planning to take the grade 11
university course to prepare for more difficult problems.
rcepts and
b) the sam
different
y-intercepts
e x-intercep
ts but diffe
c) different
rent y-interc
x-intercep
epts
ts but the
d) the sam
sam
e y-interc
ept
e x-intercep
t and
the same
3. Find the
y-intercept
point of inte
rsection of
a) 2(x each
4) y 6
3x 2(y
3) 13
linear syst
em.
b) 2(x 1) 3(y 3) 0
3(x 2)
(y 7)
20
c) 2(3x 1) (y 4) 7 4(1
2x) 0
3(3 y) 12
d) 2(x 0
1) 4(2y
1) x 3(3y
1
2) 2
4. In the
figure, the
side of each
Determin
small squa
e the area
re represen
of the shad
ts 1 cm.
ed part.
5. Solve
the followin
g linear syst
em for x
a 1 and
and y by
b 1.
letting
x
y
1
3
3
x
y
4
2
5
x
y
12
3
Prerequisite Skills Appendix
If you need help with any of the topics in the Get Ready
for each chapter, refer to this appendix on pages 458–475.
Examples and practice questions are provided. The topics
are arranged in alphabetical order.
448 MHR
• Challeng
e Problems
Appendix
Prerequi
site Skill
s
Appendix
Adding Pol
ynomials
To add 2x
5y 4,
add the like
3x 2y terms.
____
_______5
2x 5y 4
3x 2y ____
_______5
5x 3y 9
1. Add.
a) 3x 2y 5
4 x 3y ____
_______7
c)
Technology Appendix
b)
3x 4y 5
6x 2y ____
7
_______
3a 2 a 2
2
2a
4a 1
______
_____
f) 6a
2b 4
4a 3b ___
________7
x 6x 4
2x 2 4 x
___
8
_______
_
2
d)
e) 2y 2 y3
3y 2 y ____
_______2
The Technology Appendix, on pages 476–503, provides detailed
help for some basic functions of the TI-83 Plus or TI-84 Plus
graphing calculator, the computer algebra system on the TI-89
graphing calculator, and The Geometer’s Sketchpad®. These
pages will be particularly helpful if you have not used these
tools before.
Angle Pro
perties
To find the
measure
of x, reca
interior angl
ll that the
es of a trian
sum of the
x 72° gle is 180
°.
67° 180
°
x 139°
180°
x 180°
139°
x 41°
When a tran
sversal inte
a) the alte
rsects two
72°
rnate angl
parallel line
67°
es
s:
are equal
b) the corr
espondin
g angles
are equal
c)
the co-inter
ior angles
are supplem
entary
x
y
To find the
measures
of x, y, and
fact that,
since AC
z, use the
is parallel
alternate
to EF, the
angles are
equal.
BEF ABE (alte
rnate angl
x 54°
es)
BFE CBF (alte
rnat
e angles)
458 MHR
• Prerequisite
Skills Appendi
x
x + y = 180°
A
54°
x
E
B
y 73°
C
z
F
Other Back Matter
Glossary
A complete illustrated glossary is included. All key terms of the text, as
well as other mathematical terms, are listed on pages 570–581. This is a
good resource if you want to check the exact meaning of a term.
Answers
Complete illustrated answers are provided for all questions in each
Get Ready, numbered section, Chapter Review, and Practice Test. Refer
to pages 504–569. Answers for the Achievement Check questions, the
Chapter Problem Wrap-Up, the Investigate questions, and Communicate
Your Understanding questions are provided in Principles of Mathematics 10
Teacher’s Resource.
Index
A general index is included on pages 582–585.
A Tour of Your Textbook • MHR 1