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11 Transformations and the number plane
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Stage 4

11.03 Graphs of linear relationships

Lesson
Worksheet
Practice
Lesson

Introduction

When an  equation  has exactly two variables in it, we can express the relationship between those variables on the number plane.

Equations, tables, and graphs

We can make a table of values for any equation by substituting some test values for one of the variables and recording the values that the other variable must be for the equation to be true. Doing this can help us see the relationship between the two variables.

x-2-1012
y12345

Let's start with the equation y=x+3. By substituting a range of values for x into the equation, we can find the corresponding y-values to complete this table.

We can now plot these as ordered pairs (x,\,y) on the number plane:

\left(-2,\,1\right),\, \left(-1,\,2\right),\, \left(0,\,3\right),\, \left(1,\,4\right),\, \left(2,\,5\right)

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

We can start by plotting the point (-2,\,1). Since this point has a negative x-value and a positive y-value, we know that it will be in the second quadrant of the number plane. Knowing this, we can plot the point (-2,\,1) by starting at the origin and moving 2 units to the left, then 1 unit up.

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

After plotting all the ordered pairs onto the number plane, we form a line.

Examples

Example 1

Consider the equation y=3x.

a

Complete the table of values.

x-5-3-11
y
Worked Solution
Create a strategy

Substitute each x-value into the equation.

Apply the idea
\displaystyle y\displaystyle =\displaystyle 3\times(-5)Substitute x=-5
\displaystyle =\displaystyle -15Evaluate
\displaystyle y\displaystyle =\displaystyle 3\times(-3)Substitute x=-3
\displaystyle =\displaystyle -9Evaluate
\displaystyle y\displaystyle =\displaystyle 3\times(-1)Substitute x=-1
\displaystyle =\displaystyle -3Evaluate
\displaystyle y\displaystyle =\displaystyle 3\times(1)Substitute x=1
\displaystyle =\displaystyle 3Evaluate
x-5-3-11
y-15-9-33
b

Plot the points in the table of values.

Worked Solution
Create a strategy

Plot each point by using the coordinates to know how to move from the origin.

Apply the idea
-6
-5
-4
-3
-2
-1
1
2
x
-15
-14
-13
-12
-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
y

The points from the table have the coordinates (-5,-15),\,(-3,-9),\,(-1,-3),\,(1,3).

Idea summary

To complete a table of values for an equation, substitute each given x-value into the equation to find the corresponding y-value.

To plot the points from a table, move horizontally along the x-axis according to the x-value, and move vertically along the y-axis according to the corresponding y-value for each pair of values in the table.

Linear relationships

When plotting a set of points onto a number plane, the relationship between the points will be linear if they all lie on a straight line, and non-linear otherwise.

A relationship is linear if we can draw a straight line on the number plane that passes through all the plotted points, otherwise the relationship is non-linear.

Let's have a look at some equations and their graphs.

The image shows the tables and graphs of 3 equations. Ask your teacher for more information.

For each of the equations, we have filled in a table of values and plotted the ordered pairs onto the number plane. On each graph, we have also drawn a straight line through two of the points to check if the relationship is linear or not.

As we can see from the graphs above, (a) and (c) represent linear relationships while (b) represents a non-linear relationship.

Is there a connection between an equation and whether or not the relationship between its variables is linear?

Any linear relationship can be represented by an equation of the form y=mx+c, where m and c are numeric values.

For example: y=-3x+4 is linear with m=-3 and c=4, but y=x^{2}-1 is not linear because it includes the term x^{2}.

Examples

Example 2

Consider the equation y=x+5. A table of values is given below.

x1234
y6789
a

Plot the points in the table of values.

Worked Solution
Create a strategy

Plot each point by using the coordinates to know how to move from the origin.

Apply the idea
1
2
3
4
5
6
7
8
9
x
1
2
3
4
5
6
7
8
9
y

The points from the table have the coordinates (1,6),\,(2,7),\,(3,8),\,(4,9).

b

Do the points on the plane form a linear relationship?

Worked Solution
Create a strategy

Connect the points plotted from part (a) to check if it makes a straight line.

Apply the idea
1
2
3
4
5
6
7
8
9
x
1
2
3
4
5
6
7
8
9
y

Since the line formed is a straight line, the points on the plane form a linear relationship.

Idea summary

A relationship is linear if we can draw a straight line on the number plane that passes through all the plotted points, otherwise the relationship is non-linear.

Any linear relationship can be represented by an equation of the form y=mx+c, where m and c are numeric values.

Constant relationships

So far in this lesson we have looked at equations that have exactly two variables, x and y. However, we can also plot points for equations that have only one variable.

Consider the equation y=4.

Notice that the value of x does not affect the value for y, since x is not even in the equation. No matter what value x takes, the equation is only true if the value for y is 4.

x-2-1012
y44444

So our table of values will look like this.

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

If we plot these points onto the number plane we can see that this relationship is linear.

As we can see, the line y=4 is a horizontal line through the marker for 4 on the y-axis.

What about the equation x=-2?

x-2-2-2-2-2
y01234

Using the same logic, we can get this table of values.

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

Which gives this graph, of a line through the marker for 2 on the x-axis.

As we can see, this relationship is also linear, except in this case the line x=-2 is vertical.

When we talk about the number plane, we usually think of the xy-plane. However, the axes of the number plane are not limited to representing just x and y-values.

1
2
3
4
5
\text{time }(t)
1
2
3
4
5
\text{distance }(d)

Suppose we wanted to plot the relationship between time and distance, with t representing time and d representing distance. To do this, we would need to change the axis names from x and y to be time and distance.

Notice that we also include d next to distance and t next to time. This is because those are the variables that are used to represent them.

For these axes, any ordered pair should be in the form \left(t,\,d\right).

Unless specified otherwise, the left number in an ordered pair will correspond to the variable on the horizontal axis.

Examples

Example 3

A dead tree is 7 metres tall but since the tree is dead, its height does not change over time.

a

Let the height of the tree be h metres and the time passed be y years.

Which three of the following ordered pairs, in the form \left(y,\,h\right), match the growth of the dead tree?

A
(3,\,6)
B
(0,\,7)
C
(7,\,21)
D
(10,\,1)
E
(4,\,7)
F
(2,\,7)
Worked Solution
Create a strategy

Since the height is not changing we should choose points where h=7.

Apply the idea

The ordered pairs (y,\,h) should be in the form (y,\,7) for the height to be 7.

The answers are options B (0,\,7), E (4,\,7), and F (2,\,7).

b

Which of the following graphs shows the growth of the dead tree?

A
1
2
3
4
5
\text{years } (y)
1
2
3
4
5
6
7
8
9
\text{height } (h)
B
1
2
3
4
5
\text{years } (y)
1
2
3
4
5
6
7
8
9
\text{height } (h)
C
1
2
3
4
5
\text{years } (y)
1
2
3
4
5
6
7
8
9
\text{height } (h)
D
1
2
3
4
5
\text{years } (y)
1
2
3
4
5
6
7
8
9
\text{height } (h)
Worked Solution
Create a strategy

Plot the points from part (a) and draw a line through them.

Apply the idea
1
2
3
4
5
\text{years } (y)
1
2
3
4
5
6
7
8
9
\text{height } (h)

Plot the points (0,7), (2,\,7), and (4,\,7) on the number plane and draw a line through them.

This line matches the horizontal line from option B, so the answer is option B.

Idea summary

A constant relationship is a kind of linear relationship where the value of the variable does not change. When we express this relationship using an equation, we only use one variable with no powers. When we draw this relationship on the number plane, it will be a horizontal or vertical line.

Unless specified otherwise, the left number in an ordered pair will correspond to the variable on the horizontal axis.

Outcomes

MA4-11NA

creates and displays number patterns; graphs and analyses linear relationships; and performs transformations on the Cartesian plane

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