Australian Curriculum v9 / ACiQ Year 9 Mathematics - Unit 2 - Linear relations, gradient and intercepts

Linear Relations, Gradient and Intercepts | Year 9 Mathematics

Connect tables, graphs, equations, gradient and intercepts so linear relationships are more than a drawn line.

Updated 2026-06-15 - 4 min read

A linear relation is a relationship that makes a straight-line graph. Year 9 students need to connect four views of the same relationship: a context, a table, an equation and a graph.

The point is not just drawing a line. The point is explaining what the line says.

What makes a relation linear?

A relation is linear when equal changes in $x$ create equal changes in $y$.

Example:

| $x$ | 0 | 1 | 2 | 3 | | --- | --- | --- | --- | --- | | $y$ | 4 | 7 | 10 | 13 |

Each time $x$ increases by 1, $y$ increases by 3. The rate of change is constant, so the relationship is linear.

The equation is:

$y = 3x + 4$

The $3$ tells us the output goes up by 3 for every increase of 1 in $x$. The $4$ tells us the value of $y$ when $x = 0$.

Gradient is a rate, not just steepness

Gradient describes how much $y$ changes compared with $x$.

$\text{gradient} = \frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x}$

If a graph represents cost over distance, gradient might mean dollars per kilometre. If a graph represents water level over time, gradient might mean centimetres per minute.

Intercepts have meaning

The y-intercept is where the graph crosses the y-axis. It is the value of $y$ when $x = 0$.

For $y = 3x + 4$, the y-intercept is 4.

In a context, this might mean a starting fee, initial height, starting distance, initial savings or fixed cost.

The x-intercept is where the graph crosses the x-axis. It is the value of $x$ when $y = 0$. In some contexts this makes sense, and in others it may not.

If $y$ is the number of litres left in a tank, the x-intercept might mean the time when the tank is empty. If $x$ is number of students, negative or fractional intercepts might not make practical sense.

Domain matters in contexts

In pure graphing, a line can continue forever. In a real context, only some values make sense.

If $x$ is the number of tickets sold, $x$ cannot be negative or a decimal like $2.5$. If $x$ is time after a tank starts draining, the model may stop being useful once the tank is empty.

This is an important bridge to Year 10 modelling. A line can be mathematically correct but still unreasonable outside the sensible domain.

Reading equations

The common form is:

$y = mx + c$

Here $m$ is the gradient and $c$ is the y-intercept.

If $y = -2x + 9$, then the gradient is $-2$ and the y-intercept is 9. The negative gradient means $y$ decreases as $x$ increases.

Moving between representations

From a table, find the constant change.

From a graph, read the rise and run between two clear points.

From an equation, identify the coefficient of $x$ and the constant term.

From a context, ask what changes each time and what starts before the changing part begins.

Quick check

  1. In $y = 5x + 2$, what is the gradient?
  2. What is the y-intercept?
  3. A taxi costs \$4 plus \$1.80 per kilometre. Write an equation.
  4. What does the gradient mean in that equation?

Answers:

  1. $5$
  2. $2$
  3. $C = 1.80d + 4$
  4. Each extra kilometre adds \$1.80 to the cost.

Transfer task

Find a real situation with a fixed starting amount and a constant change: a subscription, savings plan, walking distance, tank filling or phone plan. Write a table, equation and sentence explaining the gradient and intercept.

Sources