AC9M7A05 · Year 7 · Algebra

Tables of values and the Cartesian plane

ACARA v9 CONTENT DESCRIPTION “generate tables of values from visually growing patterns or the rule of a function; describe and plot these relationships on the Cartesian plane”

A rule, a table and a graph are three different ways of showing the same relationship. Earlier you read relationships off finished graphs; now you learn to build a graph yourself, starting from a rule, generating a table of values, and plotting those values on the Cartesian plane. Moving smoothly between these three forms is one of the most useful skills in algebra.

This year you learn to generate tables from a rule or a growing pattern and to plot the values as points. The plotted points reveal the shape of the relationship, and for the linear rules you meet this year, that shape is always a straight line.

From a rule to a table

A rule such as y equals 2x plus 1 is a recipe for turning any x into a y. To build a table, you choose some x values, often 0, 1, 2 and 3, and work out the y for each. When x is 3, the rule gives 2 times 3 plus 1, which is 7. Doing this for every x fills the table, and the table is simply the rule applied several times over. Looking down the y column, you can often spot a pattern: here each y is 2 more than the last, matching the 2 in the rule.

A rule fills a table

Substitute each x value into the rule to find the matching y.

A rule like y equals 2x plus 1 generates a table of values. Put each x into the rule and calculate the matching y: x of 3 gives 2 times 3 plus 1, which is 7. The table is just the rule worked out for several inputs.

The relationship between adjacent values is worth noticing. Because the rule multiplies x by 2, every step of 1 in x produces a step of 2 in y. This steady step is what makes the pattern linear, and it is the same constant change that will show up as a straight line when you plot the points. Reading a table for these step-by-step patterns deepens your sense of how the rule behaves.

Plotting on the Cartesian plane

The Cartesian plane has two axes, a horizontal x-axis and a vertical y-axis, meeting at the origin. Every row of your table is a coordinate pair, written (x, y), that marks a single point. The pair (3, 7) means go 3 along the x-axis and 7 up the y-axis. Plotting each pair from the table places a series of points on the plane.

Plotting the pairs

Each row of the table is a point; plotting them all reveals the line.

Each row of the table is a coordinate pair. Plotting (0, 1), (1, 3), (2, 5) and (3, 7) on the Cartesian plane and joining them reveals a straight line. The rule, the table and the graph are three views of the same relationship.

When the points come from a linear rule, they line up perfectly, and joining them produces a straight line, the graph of the rule. This completes the journey from rule to table to graph, and you can travel it in any direction: read a rule from a pattern of points, build a table to find missing values, or plot a table to picture a relationship. Holding all three views together, and moving between them with ease, is exactly the fluency this part of the curriculum is building toward.

Builds on: Relationships between variables in graphs (AC9M7A04). That unit read relationships off graphs; this unit generates the points and plots them.

Quick self-check

1. Using the rule y = 2x + 1, what is y when x = 4?

2. On the Cartesian plane, what does the coordinate pair (3, 5) mean?

3. A table from a rule gives x values 0, 1, 2 and y values 3, 5, 7. What is the rule?

4. When the points from a linear rule are plotted, they always

5. Why is making a table a useful first step before drawing a graph?