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6 Linear relationships
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AustraliaVIC
Level 8

6.02 Visualising a table of values

Lesson
Worksheet
Practice
Lesson

Introduction

We know how to identify if a table of values represents  a linear equation  , and now we will look at how to display the same information on a number plane.

Create a table and plot points

A table of values, created using an equation, forms a set of points that can be plotted on a number plane. A line, drawn through the points, becomes the graph of the equation.

We'll begin by creating a table of values for the following equation:

y=3x-5

The first row of the table will contain values for the independent variable (in this case, x). The choice of x-values is often determined by the context, but in many cases they will be given. To find the corresponding y-value, we substitute each x-value into the equation y=3x-5.

x1234
y

Substituting x = 1:

\begin{aligned} y & = 3 \times 1 - 5\\ & = -2 \end{aligned}

Substituting the remaining values of x, allows us to complete the table:

x1234
y-2147
Table of x and y values forming ordered pairs. Ask your teacher for more information.

The x and y value in each column of the table can be grouped together to form the coordinates of a single point, (x,y).

To plot a point, (a, b), on a number plane, we first identify where x=a lies along the x-axis, and where y=b lies along the y-axis.

-2
-1
1
2
3
4
5
6
x
-3
-2
-1
1
2
3
4
5
6
7
8
y

For example, to plot the point (3, 4), we identify x=3 on the x-axis and construct a vertical line through this point. Then we identify y=4 on the y-axis and construct a horizontal line through this point. The point where the two lines meet has the coordinates (3, 4).

-2
-1
1
2
3
4
5
6
x
-3
-2
-1
1
2
3
4
5
6
7
8
y

If we sketch a straight line through the points, we get the graph of y=3x-5.

Notice that when sketching a straight line through a set of points, the line should not start and end at the points, but continue beyond them, across the entire coordinate plane.

To sketch a straight line graph we actually only need to identify two points.

  • When checking if a set of points forms a linear relationship, we can choose any two of the points and draw a straight line through them. If the points form a linear relationship then any two points will result in a straight line passing through all the points.

Examples

Example 1

Consider the equation y=2x.

a

Fill in the blanks to complete the table of values.

x-1012
y
Worked Solution
Create a strategy

Substitute each x-value to the given equation.

Apply the idea
\displaystyle y\displaystyle =\displaystyle 2\times(-1)Substitute x=-1
\displaystyle =\displaystyle -2Evaluate
\displaystyle y\displaystyle =\displaystyle 2\times(0)Substitute x=0
\displaystyle =\displaystyle 0Evaluate
\displaystyle y\displaystyle =\displaystyle 2\times(1)Substitute x=1
\displaystyle =\displaystyle 2Evaluate
\displaystyle y\displaystyle =\displaystyle 2\times(2)Substitute x=2
\displaystyle =\displaystyle 4Evaluate
x-1012
y-2024
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
-4
-3
-2
-1
1
2
3
4
x
-3
-2
-1
1
2
3
4
5
y

The points from the table have the coordinates (-1,-2),\,(0,0),\,(1,2),\,(2,4).

c

Is the relationship linear?

Worked Solution
Create a strategy

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

Apply the idea
-4
-3
-2
-1
1
2
3
4
x
-3
-2
-1
1
2
3
4
5
y

Since the line formed is a straight line, the relationship linear.

Idea summary

We can complete a table of values by substituting each x-value into the given equation.

To plot a point, (a, b), on a number plane, we first identify where x=a lies along the x-axis, and where y=b lies along the y-axis.

When checking if a set of points forms a linear relationship, we can choose any two of the points and draw a straight line through them. If the points form a linear relationship then any two points will result in a straight line passing through all the points.

Outcomes

VCMNA283

Plot linear relationships on the Cartesian plane with and without the use of digital technologies

VCMNA284

Solve linear equations using algebraic and graphical techniques. Verify solutions by substitution

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