Year 9 · Mathematics · Algebra
Gradient and linear graphs
Calculate the gradient of a line from two points, interpret y = mx + c, and sketch straight-line graphs from their equations.
Awaiting educator review. This material was drafted by our content team with AI assistance and is waiting for review by a qualified educator. Please check it suits your child before using it. Created 10 October 2026.
Learning goals
Learners will
- Calculate the gradient of a straight line using rise over run.
- Identify the gradient and y-intercept from an equation of the form y = mx + c.
- Write the equation of a line given its gradient and y-intercept.
Success looks like
- I can calculate the gradient between two points.
- I can describe a line as having positive, negative or zero gradient.
- I can read m and c from y = mx + c and use them to sketch the line.
- I can test whether a point lies on a line by substitution.
The big idea
The gradient of a line tells us how steep it is. It is found by dividing the vertical change (the rise) by the horizontal change (the run) between two points on the line: gradient = rise ÷ run.
Given two points (x1, y1) and (x2, y2), the gradient is m = (y2 − y1) ÷ (x2 − x1). It does not matter which point you call the first, as long as you are consistent on the top and bottom of the fraction.
- A line going up from left to right has a positive gradient.
- A line going down from left to right has a negative gradient.
- A horizontal line has a gradient of zero, because the rise is 0. A vertical line has an undefined gradient, because the run is 0.
Every non-vertical straight line can be written as y = mx + c, where m is the gradient and c is the y-intercept. For example, y = −3x + 5 has a gradient of −3 and crosses the y-axis at 5.
To sketch y = mx + c, plot the y-intercept (0, c), then use the gradient to find another point. For a gradient of 2, move 1 unit right and 2 units up. Join the points with a ruled line.
A point lies on a line if its coordinates make the equation true. Substitute the x-value and check whether you get the y-value.
Worked examples
Gradient from two points
Find the gradient of the line through (1, 3) and (4, 9).
- Rise: 9 − 3 = 6.
- Run: 4 − 1 = 3.
- Gradient = 6 ÷ 3 = 2.
Answer: The gradient is 2.
Reading an equation
State the gradient and y-intercept of y = −3x + 5, and describe the line.
- Compare with y = mx + c: m = −3 and c = 5.
- The gradient is negative, so the line goes down from left to right.
- It crosses the y-axis at the point (0, 5).
Answer: Gradient −3, y-intercept 5; the line slopes downwards.
Writing an equation
A line has gradient 4 and passes through (0, −2). Write its equation.
- The point (0, −2) is on the y-axis, so the y-intercept is c = −2.
- The gradient is m = 4.
- Substitute into y = mx + c.
Answer: y = 4x − 2
Practice check
Have a go, then check your answers. Each answer comes with an explanation.
Answer guide for parents
What is the gradient of the line y = 5x − 7?
5 — In y = mx + c the gradient is m, the coefficient of x, which is 5.
What is the y-intercept of the line y = 2x + 9?
9 — In y = mx + c the y-intercept is c, which is 9. Check: when x = 0, y = 9.
Find the gradient of the line through (2, 1) and (6, 9).
2 — Rise = 9 − 1 = 8 and run = 6 − 2 = 4, so the gradient is 8 ÷ 4 = 2.
A line slopes downwards from left to right. Which statement is true?
It has a negative gradient — Lines that fall from left to right have a negative gradient.
Which point lies on the line y = 3x + 1?
(2, 7) — Substitute x = 2: y = 3 × 2 + 1 = 7, so (2, 7) is on the line. The other points do not satisfy the equation.
Find the gradient of the line through (−1, 4) and (3, −4).
−2 — Rise = −4 − 4 = −8 and run = 3 − (−1) = 4, so the gradient is −8 ÷ 4 = −2.
For the line y = 4x − 3, find the value of y when x = 5.
17 — y = 4 × 5 − 3 = 20 − 3 = 17.
What is the gradient of the horizontal line y = 6?
0 — A horizontal line has no rise, so its gradient is 0 ÷ run = 0.
Watch out for
- Calculating run ÷ rise instead of rise ÷ run. Link gradient to steepness: a bigger rise for the same run means a steeper line.
- Mixing up the order of subtraction, such as (y2 − y1) ÷ (x1 − x2), which flips the sign. Label the points before substituting.
- Thinking the y-intercept is the number in front of x. Substitute x = 0 to confirm where the line crosses the y-axis.
Tips for parents
- Look for gradients in everyday life: ramps, driveways and roof pitches. Ask which is steeper and how you could measure it.
- Ask your learner to sketch a quick graph for each equation. A picture often reveals sign errors.
- When checking work, substitute the given points into their final equation together.
Go further
- Lean a plank on a stack of books and measure the rise and run. Calculate the gradient, then add books and predict the new gradient before measuring.
- A taxi charges a $4 flagfall plus $2 per kilometre. Write the cost as an equation, graph it and explain what the gradient and y-intercept mean.
- Investigate what is special about two lines with the same gradient, and about two lines whose gradients multiply to −1.