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6 Linear relationships
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Stage 4

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

MA4-11NA

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

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