Topics
3. Linear Graphs and Models
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VCE 11 General 2023

2.05 Graphs from tables

Lesson
Worksheet
Practice
Lesson

Introduction

Remember that a table of values can be constructed by substituting multiple values into a  linear equation  . Once a table of values has been constructed, the table can then be used to create a graph.

Plot points from a table of values

Each column in a table of values may be grouped together in the form (x, \, y). This pairing of numbers is known as an ordered pair.

Let's plot the ordered pairs from the following table of values:

x1234
y-2147
-2
-1
1
2
3
4
5
6
x
-3
-2
-1
1
2
3
4
5
6
7
8
y

The table of values has the following ordered pairs: \left(1, -2 \right) , \left(2, 1 \right) , \left(3, 4 \right) , \left(4, 7\right).

Each ordered pair becomes a point on the xy-plane.

The ordered pair, (a, b), is plotted on the number plane by first identifying where x=a is along the 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 along the x-axis and draw a vertical line through this point. Then we identify y=4 along the y-axis and draw a horizontal line through this point. Finally, we plot the point where the two lines meet, and this represents the ordered pair (3, 4).

By plotting points on the number plane that correspond to ordered pairs from the table of values, a straight line can then be drawn that passes through each of these points.

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

In the example, the line that passes through these points is given by this graph.

This straight line is the graph of y=3x-5 which was used to complete the table of values.

To draw a line from a table of values, it is useful to plot the significant points and draw the line that passes through them.

Consider the following linear equation:y = - 6 + 3x and the following table of values:

x0123
y-6-303

There are two significant ordered pairs, namely the x-intercept and the y-intercept.

  • The x-intercept has the form \left( a, 0 \right) which is a point that lies on the x-axis.

  • The y-intercept has the form \left( a, 0 \right) which is a point that lies on the y-axis.

The x-intercept in our example is \left( 2, 0 \right) and the y-intercept in \left( 0, -6 \right).

The line represented by equation y = - 6 + 3x can be graphed by drawing a line which passes through these two points.

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

Examples

Example 1

Consider the equation y=2x.

a

Complete the table of values.

x-1012
y⬚⬚⬚⬚
Worked Solution
Create a strategy

Substitute each x-value into 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

The results are:

x-1012
y-2024
b

Plot the points from the completed table of values.

Worked Solution
Create a strategy

Plot each set of coordinates.

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 give the coordinates (-1,-2),\,(0,0),\,(1,2),\,(2,4).

c

Is the relationship linear?

Worked Solution
Create a strategy

Join the points plotted from part (b) to determine if a straight line is formed.

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 is linear.

Idea summary

Each column in a table of values may be grouped together in the form (x, \, y). This pairing of numbers is known as an ordered pair.

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.

There are two significant ordered pairs, namely the x-intercept and the y-intercept.

  • The x-intercept has the form \left( a, 0 \right) which is a point that lies on the x-axis.

  • The y-intercept has the form \left( a, 0 \right) which is a point that lies on the y-axis.

Outcomes

U1.AoS4.1

the properties of linear functions and their graphs

U1.AoS4.3

the forms, rules, graphical images and tables for linear relations and equations

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