Studenttext
Studenttext
Studenttext
Content:
27.1 Positive Coordinates
27.2 Coordinates
27.3 Plotting Straight Lines
27.4 Plotting Curves
27.5 Mid-Points of Line Segments
27.6 Three-Dimensional Coordinates
MEP Jamaica: STRAND H UNIT 27 Coordinates: Student Text
27 Coordinates
27.1 Positive Coordinates
Coordinates are pairs of numbers that uniquely describe a position on a rectangular grid.
The system was invented by the 17th century French mathematician, René Descartes.
These numbers are sometimes referred to as Cartesian coordinates.
The first number refers to the horizontal (x-axis) and the second the vertical (y-axis).
The coordinates (4, 3) describe a point that is 4 units across and 3 units up on a grid from
the origin (0, 0).
y
5
3 (4, 3)
0
0 1 2 3 4 5 6 x
Worked Example 1 y
Plot the points with coordinates 9
(3, 8), (6, 1) and (2, 5)
8 (3, 8)
Solution 7
5 (2, 5)
8
For (6, 1) move 6 across and 1 up.
4
3
For (2, 5) move 2 across and 5 up. 5
2
1 (6, 1)
2 3 1
6
0
0 1 2 3 4 5 6 x
Shading (Section 27.6) denotes that the topic is not on the current CXC/CSEC Mathematics
syllabus and therefore not examined, but is of relevance to the content of the Unit.
Worked Example 2
Write down the coordinates of each point in the diagram below.
y
6
B A
5
3 C
0 D
0 1 2 3 4 5 6 7 8 9 x
Solution
A is 6 across and 5 up, so the coordinates are (6, 5).
B has no movement across and is straight up 5, so the coordinates are (0, 5).
C is 6 across and 3 up, so the coordinates are (6, 3).
D is 8 across and no movement up, so the coordinates are (8, 0).
Exercises
1. Write down the coordinates of each point on the diagram below.
A
4
3 B E
2 F
C
1
0 D
0 1 2 3 4 5 x
Rocky Point
8
3 Old Tower
Sandy Beach
Landing Stage
2
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 x
3. On a grid, join the points with the following coordinates and write down the name
of the shape you draw.
(a) (4, 2) (8, 2) (8, 5) (4, 5)
(b) (2, 1) (6, 1) (4, 6)
(c) (1, 4) (3, 7) (5, 4) (3, 1)
(d) (4, 0) (3, 2) (5, 4) (7, 2) (6, 0)
(e) (1, 1) (0, 3) (1, 5) (3, 5) (4, 3) (3, 1)
y
6
5
4
3
2
1
0
0 1 2 3 4 5 6 7 x
5. The pattern below is made up of 5 circles. Write down the coordinates of the centre
of each circle.
0
0 1 2 3 4 5 6 7 x
6. y
5 B
4
3
2
A
1
0
1 2 3 4 5 6 x
27.2 Coordinates
The coordinates of a point are written as a pair of numbers, y
(x, y), which describe where the point is on a set of axes. 3
The x-axis is always horizontal (i.e. across the page) and 2
the y-axis always vertical (i.e. up the page). 1
Worked Example 1
On a grid, plot the point A which has coordinates (–2, 4), the point B with coordinates
(3, –2) and the point C with coordinates (– 4, –3).
Solution
y
For A, begin at (0, 0), where the two axes
5
cross. A (–2, 4)
4
Move –2 in the x direction.
3
Move 4 in the y direction. 4 2
1
Points B and C are plotted in a similar way. –4 –2 3
–5 –4 –3 –2 –1 0 1 2 3 4 5 x
For B, move 3 in the x direction and –2 in –1 –2
the y direction. –3 –2
B
For C, move – 4 in the x direction and –3 –3
C
in the y direction. –4
–5
Worked Example 2
Write down the coordinates of each place on the map of the island.
y
5 Light-
house
4
3 Jetty
Shop 2
Telephone 1
box
–8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 x
–1
Café –2 Church
Lifeboat
station –3 Campsite
–4
–5
–6
Solution
Lighthouse (7, 5) ⎫
⎬ All coordinates positive
Jetty (1, 3) ⎭
Church (3, − 2) ⎫
⎬ Negative y - coordinates
Camp Site (1, − 3) ⎭
Shop ( − 4, 2) ⎫
⎬ Negative x - coordinates
Telephone Box ( − 4, 1) ⎭
© CIMT and e-Learning Jamaica 5
27.2 MEP Jamaica: STRAND H UNIT 27 Coordinates: Student Text
Café ( − 5, − 2) ⎫
⎬ All coordinates negative
Lifeboat Station ( − 2, − 2) ⎭
Exercises
1. Write down the coordinates of each point marked on the grid below.
y A
5
G
4
H B
3
I
2
C
1
–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 x
–1
J D
–2
K E
–3
F
–4
L
–5
y
6
5
4
3 Cairns
Broome
2
1
–7 –6 –5 – 4 –3 –2 –1 0 1 2 3 4 5 6 7 8 x
–1
Alice –2
Springs
–3 Brisbane
–4 Canberra
Perth –5
Albany –6
3. The positions of some of the tors (rocky outcrops) on Dartmoor in the south west
of England are marked on the grid below.
y
7 Higher Tor
6 Row Tor
5
4 West Mill Tor
3
2
Yes Tor 1
Oke Tor
–17 –16 –15 –14 –13 –12 – 11–10 –9 –8 –7 –6 –5 – 4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 x
Shelstone –1
Sourton Tor –2
Tor –3
Black Tor
–4
–5
Steeperton
Tor
–6
Dinger –7
Tor
–8
–9
–10
Kitty Tor –11
(b) The highest tor marked on the grid is Yes Tor. Write down the
coordinates of this tor.
(c) A boy and his dog walk from Oke Tor to Kitty Tor. Write down the
coordinates of the point where they start and the point where they
finish.
(d) Sourton Tor is the tor that is the farthest west on this grid. What are the
coordinates of this tor?
(e) Higher Tor is the tor that is farthest north. What are the coordinates of this
tor?
(c) (–2, 0), (–1, 1), (–1, 2), (–2, 3), (–3, 3), ( −4 , 2), ( −4 , 1), (–3, 0).
(d) (3, –1), (2, –3), (0, −4 ), (–2, –3), (–3, –1), (–1, –2), (1, –2), (3, –1).
6. (a) Draw a set of axes with x-values from −4 to 4 and y-values from –5 to 4.
(b) Plot the following points and join them in the order listed.
(3, –5), (2, –5), ( −4 , –2), (–2, –3), (0, –2), (0, 0), (3, 2), (3, 3), (4, 3),
(4, 2), (3, 2), (3, 0), (–2, 2).
7. Three corners of a square have coordinates (4, 2), (–2, 2) and (4, −4 ).
(a) Draw a set of axes with x-values from –2 to 4 and y-values from −4 to 2.
Plot the three points and draw the square.
(b) Write down the coordinates of the centre of the square.
8. Two corners of a rectangle have coordinates (–3, –1) and (–3, 3). The centre of
the rectangle has coordinates (1, 1).
(a)
Plot the three points given and draw the rectangle.
(b) Write down the coordinates of the other two corners of the rectangle.
9. A set of axes are arranged so that the x-axis runs from west to east and the y-axis
from south to north.
A ship is at the point A which has coordinates (4, 2).
N
y
W x
E
(a) How far south can the ship travel before its y-coordinate becomes negative?
(b) How far west can the ship travel before its x-coordinate becomes negative?
(c) If the ship travels SW, how far does it travel before both coordinates become
negative?
Worked Example 1
(a) Copy and complete the following pairs of coordinates using the relationship
y = x − 2.
(4, ? ), (1, ? ), (–1, ? )
(b) Plot the points on a set of axes.
(c) Draw a straight line through these points.
Solution
(a) For the first point x = 4 , so
y =4−2
=2
So the first point is (4, 2).
–3 –2 –1 0 1 2 3 4 x
–1
(1, –1)
–2
(–1, –3) –3
–3 –2 –1 0 1 2 3 4 x
–1
–2
–3
Worked Example 2
Draw the graph of y = 2 x − 1.
Solution
The first step is to find the coordinates of three points on the line. Choose any three
x-values for the coordinates. Three possible values are given below.
(4, ? ), (2, ? ), (–2, ? )
These points are now plotted on the axes below. Then a straight line can be drawn
through them.
y (4, 7)
7
6
5 y = 2x −1
4
(2, 3)
3
2
1
–3 –2 –1 0 1 2 3 4 x
–1
–2
–3
(–2, –5) –4
–5
Note In fact, just two points uniquely define a straight line but it is safer to use
three as a check.
Exercises
1. (a) Copy and complete the coordinates below using the relationship y = x + 2 .
(4, ? ), (1, ? ), (–3, ? )
(b) Draw a set of axes with x-values from –3 to 4 and y-values from –1 to 6.
(c) Plot the points with the coordinates found in (a).
(d) Check that the points lie on a straight line and draw a straight line through
the points.
(e) Write down the coordinates of the point where the line crosses the y-axis.
4. Use a different set of axes with x-values from – 5 to 5 to draw the graph of each
relationship. You will need to decide what y-values to use for each graph.
(a) y=x+3
(b) y=x−5
(c) y = 3x
(d) y = 2x + 2
(e) y = 3x − 1
m 0 10 25
p
(b) Complete the coordinates below using the contents of your table.
(0, ? ), (10, ? ), (25, ? )
(c) Using m on the horizontal axis and p on the vertical axis, plot the points with
the coordinates you obtained in (b).
(d) Copy the following table and then use the graph to fill in the missing entries.
Mark Percentage
Tyson 15
Carlyle 21
Shanice 18
Carl 80
Winston 60
Temperature in °Fahrenheit
7. y
7
0 1 2 3 4 5 6 7 x
(a) Plot the points (0, 6), (1, 4), (2, 2) and (3, 0) on a copy of the grid above.
Draw a straight line through the four points.
x y
(b) Plot the values from the table onto a copy of the following coordinate grid.
Join your points with a straight line.
0 1 2 3 4 5 6 7 x
(d) Use your rule from part (c) to calculate the value of x when y = 9 .
A 2
–5 –4 –3 –2 –1 0 1 2 3 4 x
–1
–2
–3
–4 B
x –2 0 2 3 4
y –2 2
Worked Example 1
(a) Complete the table below using the relationship y = x 2 − 2.
x –3 –2 –1 0 1 2 3
y
Solution
(a) For each value of x the y value can be calculated using y = x 2 − 2.
If x =1 then y = 12 − 2
=1− 2
= −1
2
If x = – 2 then y = ( −2) − 2
=4−2
=2
x –3 –2 –1 0 1 2 3
y 7 2 –1 –2 –1 2 7
(b) The coordinates of the points to plot are (–3, 7), (–2, 2), (–1, –1), (0, –2), (1, –1),
(2, 2) and (3, 7).
(c) These points are plotted on the following graph and have been joined by a
smooth curve.
(–3, 7) 7 (3, 7)
6
5
4
3
(–2, 2) 2 (2, 2)
1
–3 –2 –1 0 1 2 3 x
–1
(–1, –1) (1, –1)
–2
(0, –2)
Worked Example 2
Draw the graph of y = x 3 − 4 x for values of x from –3 to 3.
Solution
The first step is to draw up and complete a table of values using the relationship
y = x 3 − 4 x , as below.
x –3 –2 –1 0 1 2 3
y –15 0 3 0 –3 0 15
3
For example: If x = −3 then y = ( −3) − 4 × ( −3)
= − 27 + 12
= − 15
If x =2 then y = 23 − 4 × 2
=8−8
=0
(–3, –15), (–2, 0), (–1, 3), (0, 0), (1, –3), (2, 0), (3, 15)
These can then be plotted and a smooth curve drawn through the points as shown in the
following graph.
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
–3 –2 –1 0 1 2 3 x
–1
–2
–3
–4
–5
–6
–7
–8
–9
–10
–11
–12
–13
–14
–15
Exercises
1. (a) Complete a copy of the table below for y = x 2 − 5.
x –3 –2 –1 0 1 2 3
y –4 4
(b) Write a list of coordinates. Plot these points and draw a smooth curve
through the points.
x –2 –1 0 1 2 3 4
y 3 8
x –2 –1 0 1 2 3
y 0 –2
x –2 –1 0 1 2
y 0 6
x 5 10 15 20 25 30
y 12 4
120
(c) Draw a graph of y = .
x
(d) Use your graph to find the width of a postcard that has a height of 7 cm.
(e) If the height of a postcard must be no greater than 14 cm, what is the least
width it can have?
(c) If the base of a tank is a square sheet of metal 2.2 m by 2.2 m, find from the
graph the volume of the tank.
(d) What should be the size of the base in order to give a volume of
7. The height, h, in metres, of the distance travelled by a ball hit straight up into the
air is given by
h = 18t − 5t 2
where t is the time in seconds.
(a) Complete a copy of this table and draw a graph of h against t.
t 0 1 2 3 4
h 16
(b) Use your graph to estimate when the ball hits the ground.
(c) What is the maximum height reached by the ball?
x –3 –2 –1 0 1 2 3
y 0 3 5
x –3 –2 –1 0 1 2 3
y 4
y
9
–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 x
–1
–2
–3
x –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7
y 49 36 16 9 4 1 0 1 4 9 16 49
50
40
30
20
10
–7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 x
10. The annual cost of the heat lost through a wall depends on the length of the wall.
When the wall is a square of length x m the annual cost, $y, is given by the equation
y = 5x 2
(a) Calculate the cost, $y, when x is 8 m.
(b) The table shows the cost, $y, for different values of x m.
Length, x (m) 3 4 5 6 7
Cost, y ($) 45 80 125 180 245
Use the table of values to draw the graph of y = 5 x 2 on a copy of the grid
on the following page.
y
400
350
300
250
Cost ($)
200
(£)
Cost ($)
Cost
150
100
50
0 x
1 2 3 4 5 6 7 8
Length (m)
(c) The annual cost of the heat lost through a square wall is $150.
Use your graph to estimate the length of the wall
1 cm
2 cm
3 cm
30
25
20
Area (cm2)
15
10
5
x
0
1 2 3 4 5 6
Length of side (cm)
(b) The side of a square is x cm. The area of a square is A cm 2 . Write down the
formula which may be used to calculate the area from the length of the side.
(c) A square has an area of 12 cm 2 . Angela wants to use the graph to find the
length of the side of this square.
She draws a line on the graph to help her do this.
(i) Draw this line.
(ii) Write down the length of the side of the square whose area is 12 cm 2 .
9
(6, 8)
8
7
6
Mid-point (4, 5)
5
4
3
2
(2, 2)
1
0
0 1 2 3 4 5 6 7 8 x
The value of the x-coordinate of the mid-point of the line segment AB is the mean
value of the two x-coordinates of the end points A and B.
Similarly for the y-coordinate of the mid-point, it is the mean of the y-coordinates
of the end points A and B.
The coordinates of the mid-point could have been calculated directly as shown
below.
⎛ 2 + 6 , 2 + 8 ⎞ = ⎛ 8 , 10 ⎞
⎝ 2 2 ⎠ ⎝2 2 ⎠
= ( 4, 5)
Generally, for any two points, the coordinates of the mid-point of the line segment
a + c b + d⎞
joining the points (a, b) and (c, d) is given by ⎛ , .
⎝ 2 2 ⎠
Worked Example 1
The diagram shows the points A, B, C and D.
A
7
5
4
C B
3
2
1
6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 x
1
2
3
4
D
5
Solution
(a) The coordinates of A are (2, 7).
The coordinates of B are (5, 3).
2 + 5 7 + 3⎞
The coordinates of the mid-point of AB = ⎛ ,
⎝ 2 2 ⎠
7 10 ⎞
= ⎛ ,
⎝2 2 ⎠
= (3.5, 5)
(b) The coordinates of C are (–5, 3).
⎛ 2 + ( − 5) 7 + 3 ⎞
The coordinates of the mid-point of AC = ⎜ , ⎟
⎝ 2 2 ⎠
− 3 10 ⎞
= ⎛ ,
⎝ 2 2⎠
= ( −1.5, 5)
= (0.5, −1)
Exercises
1. (a) Draw a set of axes and mark on them the points A and B which have
coordinates (1, 4) and (7, 6).
(b) Draw the line segment AB and mark its mid-point.
(c) Write down the coordinates of the mid-point of AB.
10
A
9
8
7
D
6
5
4
B
3
C
2
E
1
0
0 1 2 3 4 5 6 7 8 9 10 11 x
3. Determine the coordinates of the mid-point of the line segment joining the two
points given in each case.
(a) (4, 7) (8, 11) (b) (6, 2) (18, 8)
(c) (3, 2) (9, 4) (d) (6, 3) (10, 11)
(e) (4, 1) (3, 4) (f) (6, 6) (1, 7)
(g) (2, 15) (13, 2) (h) (24, 2) (13, 3)
7 A
6
5
E
4
3 B
2
1
–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 x
–1
–2
–3
D
–4
C
–5
–6
–7
5. Determine the coordinates of the mid-point of the line segment joining each pair of
points listed below.
(a) (2, 3) (4, –6) (b) (–2, 1) (3, –6)
(c) (–2, –3) (–8, –10) (d) (–2, 4) (5, –6)
(e) (–3, –2) (4, 7) (f) (6, –2) (8, –10)
(g) (2, –6) (7, –5) (h) (8, –3) (–10, –11)
E (0, 0, 5) F (7, 0, 5) C
A
G (7, 6, 5) H (0, 6, 5)
B
x
(a) Explain why the coordinates of the mid-point of DG are (3.5, 6, 2.5).
(b) Determine the coordinates of the mid-point of each of the following line
segments.
(i) AB
(ii) CD
(iii) DH
(iv) FG
7. The points A, B and C have coordinates (4, 2, 1), (6, 10, 7) and (4, 8, 11).
(a) Determine the coordinates of the mid-points of AB and AC.
(b) The mid-point of AB is joined to the mid-point of AC by a line segment.
Determine the coordinates of the mid-point of this line segment.
Worked Example 1
The following diagram shows a cuboid.
z
y
H
5
4 8
E
7
3 6
D 5
2 4
3
1
2
G
A 1
0
1 F
2
3
C
4
5
6
B
7
8
Solution
A (0, 0, 0)
B (8, 0, 0)
C (8, 4, 0)
D (0, 4, 0)
E (0, 0, 3)
F (8, 0, 3)
G (8, 4, 3)
H (0, 4, 3)
Worked Example 2
The diagram shows a triangular-based pyramid, ABCD, in which D is vertically above A.
The base ABC is horizontal. Write down the coordinates of each of the vertices.
z
y
D
4 8
7
3 6
5
2 C
4
3
1
2
1
A
0
1
2
3
4
5 B
6
7
Solution
A (1, 1, 0)
B (6, 1, 0)
C (2, 6, 0)
D (1, 1, 4)
Worked Example 3
The diagram shows a cuboid and the x, y and z axes.
z
H
y
D
E G
A F C
B
x
If AB = 6, AD = 5 and AE = 2 , write down the coordinates of each vertex of the
cuboid.
Solution
A (0, 0, 0) B (6, 0, 0) C (6, 5, 0)
D (0, 5, 0) E (0, 0, 2) F (6, 0, 2)
G (6, 5, 2) H (0, 5, 2)
Exercises
1. Write down the coordinates of each of the vertices of the following cuboids.
(a)
z
H
4
y
3 G
D 6
E
5
2 4
3
1 C
2
A 1 F
0
1
2
B
3
4
5
6
x
(b) P
z
M
4
y
3 6
5
2 4 O
L N
3
1
2
I 1
0
1
2
3
4 K
5 J
6
x
(c)
X
z
4 U y
W
3 V 6
5
2 4
3 T
1
2
1
0 Q
S
1
2
3 R
4
5
6
(d) x
H
z
E
5
4 G
F
y
3 6
5
2 4
3 D
1
2
1
A
0
1
2 C
3 B
4
5
6
x
© CIMT and e-Learning Jamaica 31
27.6 MEP Jamaica: STRAND H UNIT 27 Coordinates: Student Text
2. Write down the coordinates of each of the vertices of the following solids.
(a)
z
D
4
y
3 6
5
2 4
3
1
2
1
0 A
1 C
2
3
4 B
5
x
(b)
z G
4
y
3 6
5
2 4
3 F
1
2
1
0
1 D
2
3
4
5 E
6
x
(c)
z
L
5
4
y
3 6
5
2 4 K
3
1
2
1
H
0
1
2 J
3 I
4
5
x
(d)
z Q
4
y
3 R
6
5
2 4
P
3
1
M 2
1
0
1
O
2
3
4 N
5
6
x
y
C
B
x
H
y
D
E G
A F C
B
x
5. In the diagram below, the x- and y-axes are horizontal and the z-axis is vertical.
The diagram shows a triangular prism with a horizontal base, ABCD. The faces of
the prism, AED and BFC, are vertical.
z
E
y
F
D
A
C
B
x
6. In the diagram below, the x- and y-axes are horizontal and the z-axis is vertical.
The diagram shows a triangular prism with horizontal faces, ABC and DEF. The
edges BE and CF are vertical.
z
F
D
E
y
C
B x
If AB = 4, AC = 5 and AD = 8 , write down the coordinates of each vertex of
the prism.