Curriculum guide
Graphing linear relationships: a Year 8 guide
Learn to graph linear relationships from a rule or table, understand gradient and y-intercept, and interpret straight-line models with practice.
What you'll be able to do
- Build a table of values from a linear rule and plot the ordered pairs
- Read the gradient and y-intercept from a rule in the form y = mx + b
- Interpret a straight-line model, including where the model stops making sense
CurriculumAustralian Curriculum v9 - Year 8 Mathematics, Algebra strand (AC9M8A02, AC9M8A03) · view it on the curriculum site
A linear relationship has a constant rate of change, so its graph is a straight line. To graph one, choose values for the independent variable, calculate the matching dependent values, plot the ordered pairs on a Cartesian plane and draw the line through them. Then interpret the graph: its gradient shows the rate of change, while its intercept can show an initial value.
Four ways to represent the same relationship
Consider the rule
y = 2x − 3
It can be represented as:
- an equation: y = 2x − 3
- a table of values
- a set of ordered pairs, such as (0, −3)
- a straight-line graph containing every point that satisfies the rule.
Each representation reveals something different. The rule makes calculation efficient, the table lists selected values and the graph makes patterns and comparisons visible.
How to graph a linear relationship from a rule
Step 1: make a table of values
Choose several useful x-values and substitute each into the rule.
| x | Calculation | y |
|---|---|---|
| −1 | 2(−1) − 3 | −5 |
| 0 | 2(0) − 3 | −3 |
| 1 | 2(1) − 3 | −1 |
| 2 | 2(2) − 3 | 1 |
Step 2: write the ordered pairs
The table gives (−1, −5), (0, −3), (1, −1) and (2, 1). The x-coordinate always comes first.
Step 3: prepare the Cartesian plane
- Put x on the horizontal axis and y on the vertical axis.
- Choose a consistent scale that includes all values.
- Label both axes.
- If the graph represents a practical situation, include variable names and units.
Step 4: plot and check the points
Plot each ordered pair. The points should line up. If one does not, substitute its x-value into the rule again before drawing the line.
Step 5: draw the line
Use a ruler to draw the straight line through the points. Add the rule as a label if more than one relationship appears on the axes.
| x | y |
|---|---|
| -1 | -5 |
| 0 | -3 |
| 1 | -1 |
| 2 | 1 |
Understanding gradient and y-intercept
Straight-line rules are often written in the form
y = mx + b
- m is the gradient: how much y changes when x increases by 1.
- b is the y-intercept: the y-value where x = 0, so the line crosses the y-axis at (0, b).
For y = 2x − 3, the gradient is 2 and the y-intercept is −3. Every time x increases by 1, y increases by 2.
A negative gradient means y decreases as x increases. For example, y = −3x + 10 has gradient −3: move 1 unit to the right and the line falls 3 units.
A practical linear model
A delivery service charges a $7 booking fee plus $4 per kilometre. Let C be the cost in dollars and d the distance in kilometres:
C = 4d + 7
| Distance d (km) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| Cost C ($) | 7 | 11 | 15 | 19 |
Here the gradient 4 represents $4 per kilometre, and the intercept 7 represents the booking fee at 0 kilometres.
The graph can estimate costs between or beyond the table values. However, a model has limits. Negative distance is not meaningful here, and an estimate far beyond the service's normal range may not reflect real pricing.
If the total cost is $23, solve 23 = 4d + 7. Then 16 = 4d, so d = 4. On the graph, this is where the horizontal line C = 23 meets the cost line. This shows how graphs and algebra can solve the same question - the same connection used in solving linear equations.
ACARA's Year 8 linear-relationships work sample similarly shows students working between rates, tables, equations and graphs in a practical context.
Common mistakes
- Reversing the coordinatesThe point (2, −1) means move 2 along the x-axis and then to −1 on the y-axis. It is not the same as (−1, 2).
- Using an inconsistent scaleEqual physical gaps on an axis must represent equal numerical changes. A broken or changing scale distorts the relationship.
- Joining points in table order rather than x-orderPlot each point in its correct coordinate position. The straight line follows the relationship, not the order in which values were calculated.
- Reading the intercept as “where the graph starts”The y-intercept is specifically where x = 0. A displayed graph may begin elsewhere because of the chosen viewing window.
- Confusing steepness with heightGradient describes change, not how high the line sits. Two parallel lines can have the same gradient but different y-intercepts.
- Extending a model without checking the contextA straight line can be extended mathematically, but its real-world meaning may only apply within a sensible domain. Always interpret axes, units and restrictions.
Try it yourself
Have a go on paper before revealing anything - the working is what makes it stick. The hint is there if you stall.
Complete a table for y = −x + 4 when x = 0, 1, 2
Hint
Show the answer
State the gradient and y-intercept of y = 3x − 5
Hint
Show the answer
Find the gradient through (2, 3) and (6, 11)
Hint
Show the answer
A membership costs $12 plus $5 per visit. Write a rule for total cost C after n visits, then find the cost after 6 visits.
Hint
Show the answer
Sources
- Australian Curriculum v9: Year 8 Mathematics (ACARA)australiancurriculum.edu.au
- Year 8 work sample: linear relationships in the real world (ACARA)australiancurriculum.edu.au