Year 10 · 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.
These lessons were written around Australian Curriculum strands and matched to New Zealand year levels by age (NZ Year 1 is the first year of school). They are not yet mapped to The New Zealand Curriculum, so check them against your own learning programme.
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.