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NUMBER AND ALGEBRA • LINEAR AND NON-LINEAR RELATIONSHIPS
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Battleship
Game
e A bishop can move diagonally any
8
number of (unoccupied) squares at a
time. Complete the following sentence:
‘With its next move, the bishop at
b3 could capture the . . . . . . . . (name the
chess piece) at . . . . . . . . (name the
position).’
7 The game of Battleship uses alphanumeric
grids. Navigate to the Battleship weblink in
your eBookPLUS. Play the game to test your
skills with alphanumeric references.
7
6
5
4
3
2
REFLECTION
When we specify an alphanumeric grid
reference, does order matter? (That
is, does it matter whether the letter is
stated first and the number second, or
the other way around?)
15B
eBook plus
eLesson
Coordinates
and the
Cartesian
plane
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The Cartesian plane
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The Cartesian plane is named after its inventor, mathematician René Descartes. It is a visual
means of describing locations on a plane by using two numbers as coordinates (rather than a
letter and a number).
The Cartesian plane is formed by two perpendicular lines. The horizontal line is called
the x-axis, while the vertical line is referred to as the y-axis. The point where the two axes
intersect is called the origin.
Both axes must be marked (with marks being evenly spaced)
y
and numbered. The distance between each mark is one unit.
4
To locate any point on the Cartesian plane we use a pair of
numbers called Cartesian coordinates. The numbers are written
3
in a set of brackets and are separated by a comma. The first
number in brackets is called the x-coordinate of the point; this
2
shows how far across from the origin the point is located. The
second number is called the y-coordinate; this shows how far up 1
(or down) from the origin the point is.
0
If the Cartesian coordinates of the point are known, it can be
1 2 3 4 5 6 x
easily located by moving across and up (or down) from the
origin the specified number of units. For example, to find the point with coordinates (2, 3),
start from the origin and move 2 units to the right and 3 units up.
Hint: To help remember the order in which Cartesian coordinates are measured, think about
using a ladder. Remember we must always walk across with our ladder and then climb up it.
y
y
4
5
(2, 3)
3
4
2
3
1
2
0
1
2
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4
x
1
0
1
2
Chapter 15
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10 11 12 x
Coordinates and the Cartesian plane
507
NUMBER AND ALGEBRA • LINEAR AND NON-LINEAR RELATIONSHIPS
WORKED EXAMPLE 3
Draw a Cartesian plane with axes extending from 0 to 6 units. Mark the following points with a dot,
and label them.
1
a (2, 4)
b (5, 0)
c (0, 2)
d (3 2 , 1)
THINK
DRAW
1
First rule up and label the axes.
2
Mark each point.
a (2, 4) means starting at the origin, go across
2 units, and then up 4 units.
b (5, 0) means go across 5 units and up 0 units.
It lies on the x-axis.
c
d
(0, 2) means go across 0 units and up 2 units.
It lies on the y-axis.
1
y
5
(2, 4)
4
3
(3 1–2 , 1)
2 (0, 2)
1
0
(5, 0)
1
2
3
4
6 x
5
1
(3 2 , 1) means go across 3 2 units and up 1 unit.
Label each point.
WORKED EXAMPLE 4
Find the Cartesian coordinates for each of the points
A, B, C and D.
y
5
4
C
B
3
THINK
WRITE
Point A is 3 units across and 1 unit up
Point B is 1 unit across and 3 units up.
Point C is 0 units across and 4 units up.
1
Point D is 1 unit across and 1 2 units up.
A is at (3, 1)
B is at (1, 3)
C is at (0, 4)
1
D is at (1, 1 2)
2
D
A
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0
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6 x
Extending the axes
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The Cartesian axes can extend infinitely in both directions.
y
On the x-axis, the values to the left of the origin are
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negative and decreasing. Likewise, on the y-axis the
5
values below the origin are negative and decreasing.
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1st quadrant
2nd quadrant
The axes divide the Cartesian plane into four sections
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called quadrants. The quadrants are numbered in
2
an anti-clockwise direction, starting with the top right
1 Origin
corner.
x
0
-6 -5 -4 -3 -2 -1
If both x- and y-coordinates of the point are positive, it
-1 1 2 3 4 5 6
-2
will be located in the first quadrant; if the x-coordinate
-3
is negative but the y-coordinate is positive, the point
3rd quadrant
4th quadrant
-4
will be in the second quadrant. If the point is in the
-5
third quadrant, both the x- and y-coordinates of the
-6
point will be negative. Finally, if the point is in the
fourth quadrant, its x-coordinate is positive, while its
y-coordinate is negative.
Maths Quest 7 for the Australian Curriculum
NUMBER AND ALGEBRA • LINEAR AND NON-LINEAR RELATIONSHIPS
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If the point is located on the x-axis, its y-coordinate is always 0. Likewise, if the point is on
the y-axis, its x-coordinate is always 0.
WORKED EXAMPLE 5
Plot the following points on the Cartesian plane.
A(-1, 2), B(2, -4), C(0, -3), D(4, 0), E(-5, -2)
State the location of each point on the plane (that is, the quadrant, or the axis on which it sits).
THINK
WRITE
1
Draw a set of axes, ensuring that they are long enough to fit
all the values.
2
Plot the points. The first point is one unit to the left and
two units up from the origin. The second point is two units
to the right and four units down from the origin (and so on).
y
5
4
3
A 2
1
D
x
0
−5 −4 −3 −2 −1
−1 1 2 3 4 5
−2
E
−3 C
−4
B
−5
3
Look at the plane and state the location of each point.
Remember that the quadrants are numbered in an anticlockwise direction, starting at the top right. If the point
is on the axis, specify which axis it is.
Point A is in the second quadrant.
Point B is in the fourth quadrant.
Point C is on the y-axis.
Point D is on the x-axis.
Point E is in the third quadrant.
REMEMBER
1. Cartesian coordinates can be used to locate any point on a plane.
2. The Cartesian plane is formed by two perpendicular lines called axes. The horizontal
axis is called the x-axis and the vertical axis is called the y-axis. The axes intersect at
the point called the origin.
y
3. Both axes must be marked (with marks being
6
evenly spaced) and numbered. The distance
5
between each mark is one unit. The axes can
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1st quadrant
2nd quadrant
3
extend infinitely in both directions.
2
4. The location of any point on the Cartesian
1 Origin
plane is given by its Cartesian coordinates.
x
The Cartesian coordinates are a pair of
0
-6 -5 -4 -3 -2 -1
-1 1 2 3 4 5 6
numbers that are separated by a comma and
-2
are shown within brackets. The first number is
-3
3rd quadrant
4th quadrant
called the x-coordinate of the point; this shows
-4
how far across (that is, to the left or to the
-5
right) from the origin the point is located. The
-6
second number is called the y-coordinate; this
shows how far up (or down) from the origin
the point is. For example, the point (2, 3) is located 2 units to the right and 3 units up
from the origin.
Chapter 15
Coordinates and the Cartesian plane
509
NUMBER AND ALGEBRA • LINEAR AND NON-LINEAR RELATIONSHIPS
EXERCISE
15B
INDIVIDUAL
PATHWAYS
eBook plus
Activity 15-B-1
Introducing the
Cartesian plane
doc-2009
The Cartesian plane
FLUENCY
1 WE3 Draw a Cartesian plane with axes extending from 0 to 5 units. Mark the following
points with a dot, and label them.
a (4, 3)
b (1, 4)
c (3, 3)
d (2, 0)
e (0, 4)
f
(0, 0)
2 WE4 Find the Cartesian coordinates for each of the points A–L.
y
Activity 15-B-2
D
10
More of the
Cartesian plane
doc-2010
L
C
9
Activity 15-B-3
3D planes
doc-2011
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J
7
6
eBook plus
Weblink
The coordinate
plane
G
B
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K
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A
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x
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3 WE5 Plot the following points on the Cartesian plane.
A(–1, –3), B(–2, 5), C(3, –3), D(0, –4), E(–2, –2), F(–5, 0), G(3, 1), H(3, 0),
I(–4, –2), J(4, –5)
State the location of each point on the plane (that is, the quadrant, or the axis it sits on).
UNDERSTANDING
4 Each of these sets of Cartesian axes (except one) has something wrong with it. From the list
below, match the mistake in each diagram with one of the sentences.
A The units are not marked evenly.
B The y-axis is not vertical.
C The axes are labelled incorrectly.
D The units are not marked on the axes.
E There is nothing wrong.
a
b
y
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3
2
2
1
1
0
510
x
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3
4
Maths Quest 7 for the Australian Curriculum
x
0
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3
4
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NUMBER AND ALGEBRA • LINEAR AND NON-LINEAR RELATIONSHIPS
c
y
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x
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y
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eBook plus
Digital docs
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x
y
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1
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x
0
5 From the diagram at right, write down the coordinates
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3
2
1
6
2
3
2
0
1
1
2
3
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x
y
of 2 points which:
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a have the same x-coordinate
C
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b have the same y-coordinate.
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Messages can be sent in code using a grid like the one
B
D
2
drawn below, where the letter B is represented by the
coordinates (2, 1).
1 A
Use the diagram to decode the answer to the following
E
0
riddle.
1 2 3 4 5 x
Q Where did they put the man who was run over by a
steamroller?
y
U V W X Y
A (4, 2)(4, 3)(3, 2)(5, 3)(4, 4)(1, 4)(4, 2)(5, 4)(1, 1)
5
(2, 3)(4, 2)(4, 3)(3, 5)(1, 1)(3, 4)(4, 1)(4, 4)(4, 4)(4, 2)
P Q R S T
4
(4, 5) (4, 4)(5, 1)(2, 5)(5, 1)(4, 3)(5, 1)(4, 2)(2, 2)(3, 2)
K L M N O
3
(5, 4)(1, 1)(4, 3)(4, 1)(4, 3)(4, 2)(4, 3)(5, 1)
F G H I J
Rule up a Cartesian plane with both axes extending from 0 to
2
A B C D E
10 units. Plot the following points and join them in the order
1
given to make a geometric figure. Name each shape.
0
a (2, 2)–(5, 2)–(2, 6)–(2, 2)
1 2 3 4 5 x
b (4, 4)–(8, 4)–(6, 8)–(4, 4)
c (1, 1)–(10, 1)–(8, 9)–(2, 9)–(1, 1)
d (0, 0)–(8, 0)–(10, 10)–(2, 10)–(0, 0)
Here is an exercise which may require care and concentration. On graph paper or in your
exercise book rule up a pair of Cartesian axes. The x-axis must go from 0 to 26 and the y-axis
from 0 to 24. Plot the following points and join them in the order given.
(0, 15)–(4, 17)–(9, 22)–(10, 21)–(12, 24)–(16, 22)–(15, 21)– (18, 19)–(20, 24)–
(22, 18)–(26, 12)–(26, 10)–(23, 4)–(20, 3)–(18, 4)–(14, 7)–(11, 7)–(4, 6)–(2, 7)–
(212 , 8)–(0, 15)
N
School
Complete the picture by joining (19, 2)–(21, 2)–(20, 0)–(19, 2).
What is the area of a rectangle formed by connecting the points
(2, 1), (7, 1), (7, 4) and (2, 4) on a Cartesian plane?
Spreadsheet
Plotting points
doc-0002
Home
Chapter 15
Coordinates and the Cartesian plane
511
NUMBER AND ALGEBRA • LINEAR AND NON-LINEAR RELATIONSHIPS
10 Consider the following set of points: A(2, 5) B(–4, –12) C(3, –7) D(0, –2) E(–10, 0) F(0, 0)
G(–8, 15) H(–9, –24) I(18, –18) J(24, 0).
Which of the following statements is true?
a Points A and J are in the first quadrant.
c Only point I is in the fourth quadrant.
e Point F is at the origin.
g Point D is two units to the left of point F.
b Points B and H are in the third quadrant.
d Only one point is in the second quadrant.
f Point J is not on the same axis as point E.
y
3
2
1
11 Consider the triangle ABC at right.
a State the coordinates of the vertices of the triangle
ABC.
b Find the area of the triangle.
c Reflect the triangle in the x-axis. (You need to
copy it into your workbook first.) What are the new
coordinates of the vertices?
d Now reflect the triangle you have obtained in part c
into the y-axis, and state the new coordinates of the
vertices.
0
−5 −4 −3 −2 −1
−1
A
C
1 2 3 4 5 x
−2
−3
−4
REFLECTION
eBook plus
Why must the x-coordinate
always be written first and the
y-coordinate second?
Digital docs
WorkSHEET 15.1
doc-2004
15C
B
Plotting simple linear relationships
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When a set of points is plotted on the Cartesian plane, a pattern may be formed. If a pattern
forms a straight line, we call it a linear pattern.
The coordinates of the points that form a pattern can be presented as a set, or in a table. If
shown in a table (similar to the one shown below), the coordinates of each point should be
read ‘in columns’; that is, the top number in each column gives the x-coordinate and the
bottom number gives the corresponding y-coordinate of the point. Consider, for example, the
table of values and the set of points below. Both show the same information.
x
0
1
2
3
y
8
6
4
2
(0, 8) (1, 6) (2, 4) (3, 2)
The Cartesian coordinates of the points are ordered pairs. That is, the first number always
represents the x-coordinate, and the second always represents the y-coordinate of a point. A
set of ordered pairs forms a relation between x and y.
If the points form a linear pattern when plotted, we say that the relation between x and y is
linear.
WORKED EXAMPLE 6
Plot the following set of points on the Cartesian plane, and comment on any pattern formed.
(1, 3) (2, 4) (3, 5) (4, 6) (5, 7)
THINK
1
512
Look at the coordinates of the points in the set: the
x-values range between 1 and 5, while the y-values
range between 3 and 7. Draw a set of axes, ensuring they
are long enough to fit all the values.
Maths Quest 7 for the Australian Curriculum
WRITE