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Middle Years

11.03 Graphs of linear relationships

Worksheet
Linear relationships
1

Do the points on each of the following number planes form a linear relationship? Explain your answer.

a
-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
b
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
y
c
-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
x
-4
-3
-2
-1
1
2
3
4
y
d
-4
-3
-2
-1
1
2
3
4
x
-3
-2
-1
1
2
3
4
5
6
7
8
y
2

Consider the three points A \left(3, 6\right), B \left(1, 0\right), and C \left(2, 3\right).

a

Plot the points on a number plane.

b

Do the points on each of the following number planes form a linear relationship? Explain your answer.

3

Consider the three points X \left(3, - 7 \right), Y \left(4, - 9 \right), and Z \left(1, - 2 \right).

a

Plot the points on the same number plane.

b

Do the points on each of the following number planes form a linear relationship? Explain your answer.

Equations, tables and graphs
4

For each of the following equations and their table of values:

i

Plot the points from the table on a number plane.

ii

Determine whether the points form a linear relationship.

a

y = x + 5

x1234
y6789
b

y = x

x1234
y1234
c

y = 3

x-2-101
y3333
d

y = x^{2}

x-2-101
y4101
e

y = x + 4

x-4-3-2-1
y0123
f

x = - 6

x-6-6-6-6
y-6-303
g

y = - x

x-3-2-101
y3210-1
h

y = 4x

x-2-1012
y-8-4048
5

For each of the following equations:

i

Complete the table of values provided.

ii

Plot the points from the table on a number plane.

a

y = 3 x

x-1012
y
b

y = x - 5

x4567
y
c

y = x + 3

x13710
y
d

y = 2 x + 7

x-2-101
y
e

y = 3

x-6-3-22
y
f

y = - 2x + 1

x-2-101
y
6

Consider the given table of values:

a

Plot the points in the table on a number plane.

b

Is the relationship between x and y linear?

x- 3- 2-10
y151050
c

If we extend the table of values to include x = 1, find the corresponding value of y.

7

Consider the given table of values:

a

Plot the points in the table on a number plane.

b

Is the relationship between x and y linear?

x- 8- 404
y- 2-101
c

If we extend the table of values to include x = 8, find the corresponding value of y.

8

Consider the given table of values:

a

Plot the points in the table on a number plane.

b

Is the relationship between x and y linear?

x- 1012
y- 3159
c

If we extend the table of values to include x = 3, find the corresponding value of y.

9

Consider the given table of values:

Does the ordered pair \left( - 4 , 8\right) satisfy the linear relationship between x and y?

x- 2-101
y2-1- 4- 7
Applications
10

After each week, the number of apples on an apple tree increases according to the diagram:

a

Complete the following table of values:

Week1234
No. Apples
b

Plot the number of apples against each week on a number plane.

c

Is the relationship between the number of weeks and the number of apples linear?

11

The following pattern was created using matchsticks. Each figure is labelled with a number:

a

Complete the following table of values:

Figure number1234
No. Matchsticks
b

Plot the number of matchsticks used at each step on a number plane.

c

Is the relationship between the step number and the number of matchsticks linear?

12

A staircase with 5 steps is shown below. The height of each step is 15\text{ cm} and the length of each step is 10\text{ cm}.

a

Complete the following table of values:

No. Steps2345
Height of staircase (cm)30
b

Plot the height of a staircase against the number of steps on a number plane.

c

Is the relationship between the height of a staircase and the number of steps linear?

d

Is it possible to have a staircase with height 105 \text{ cm} with steps that have the same height and length as in the image?

e

Complete the following table of values:

No. Steps2345
Length of staircase (cm)20
f

Plot the length of a staircase against the number of steps on a number plane.

g

Is the relationship between the length of a staircase and the number of steps in a staircase linear?

h

Is it possible to have a staircase with length 70 cm with steps that have the same height and length as in the image?

13

A water tank initially starts with 65 L of water. Each day, 10 L of water is taken from the water tank.

a

Plot the volume (in L) of water in the tank against the number days, from 0 to 4 days.

b

Is the relationship between the volume of water in the water tank and the number of daysd linear?

c

How many litres of water will there be after 5 days?

14

The points on the number plane show the relationship between water temperature, x, and surface air temperature, y:

-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
a

Complete the table of values:

\text{Water Temperature in } \degree \text{C},\, x- 3- 2- 10123
\text{Surface Air Temperature in } \degree \text{C},\, y
b

Find the surface air temperature when the water temperature is 4 \degree \text{C}.

c

Find the surface air temperature when the water temperature is 14 \degree \text{C}.

15

Let the height of a candle be h \text{ cm}. If the candle is lit, the height of the candle decreases according to the equation \\ h = - 2 t + 9, where t is the time in minutes. A table of values is shown:

Plot the points in the table on a number plane.

\text{Time} \\ \text{(}t \text{ mins)}0123
\text{Height of candle} \\ \text{(}h\text{ cm)}9753
16

A dead tree is 7 \text{ m} tall and its height does not change over time. Let the height of the tree be y metres and the time passed be x years.

a

State whether the following coordinates satisfy the growth of the tree, where the x-coordinate is the time passed, and the y-coordinate is the height of the tree:

i
\left(3, 6\right)
ii
\left(0, 7\right)
iii
\left(7, 21\right)
iv
\left(10, 1\right)
v
\left(4, 7\right)
vi
\left(2, 7\right)
vii
\left(7, 0\right)
viii
\left(7, 7\right)
b

Plot the points from part (a) that satisfy the growth of the tree on a number plane.

17

Let the volume of water in a tank be V \text{ L}, and the time taken to fill up the tank be t minutes. Suppose the tank is initially empty and the number of litres in the tank is always three times the number of minutes that have passed.

a

Complete the table of values:

\text{Time (}t \text{ mins)}0123
\text{Volume of water (}V \text{ L)}
b

Which of the following equations represents the relationship between the time taken, t, and the volume of the tank, V?

A

V = 3

B

V = 3 t

C

V = 3 t + 2

D

V = t + 3

c

Plot the points in the table on a number plane.

18

Let the height of a pot plant be y \text{ cm}. As the pot plant grows, the height increases according to the equation y = 2 t + 9, where t is the elapsed time in weeks.

a

Complete the table of values:

\text{Time (}t \text{ weeks)}0123
\text{Height of pot plant (}y \text{ cm)}
b

Plot the points in the table on a number plane.

19

Max is 21 years old and Sebastian is 14 years old. Let M represent Max's age and S represent Sebastian's age.

a

Write an equation for M in terms of S.

b

State whether following ordered pairs, in the form \left(S, M\right), satisfy the relationship between Max and Sebastian's ages:

i
\left(14, 21\right)
ii
\left(36, 29\right)
iii
\left(20, 27\right)
iv
\left(25, 25\right)
v
\left(24, 31\right)
vi
\left(16, 23\right)
vii
\left(38, 46\right)
c

Plot the points from part (b) that satisfy the relationship between their ages on a number plane.

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