Measure how steep a line is: gradient is the change in y divided by the change in x, with a sign for direction. It is on Foundation and Higher, aimed at grade 5. Free questions with answers are on this page.
Gradient is change in xchange in y=x2−x1y2−y1. Positive slopes up left to right; negative slopes down; zero is horizontal; undefined is vertical. Simplify the fraction; leave it as a fraction rather than a rounded decimal if it is not integer.
From a graph, pick two clear grid points and count rise over run. A gradient of 32 means 'along 3, up 2'. Parallel lines have equal gradients; that fact is the next topic's whole job.
Gradient =change in xchange in y. Horizontal is 0; vertical is undefined. A negative gradient goes down as you read left to right. Keep the order consistent — both subtractions the same way round.
Objectives
Find the gradient through two points.
Say whether a gradient is positive or negative.
Find the change in y, and use a gradient from (0, c).
Key points
Find the gradient from two coordinates using change in xchange in y.
Find the gradient from a drawn line by making a right-angled triangle on the grid.
Decide the sign: uphill left-to-right is positive, downhill is negative.
Recognise a horizontal line has gradient 0 and parallel lines share the same gradient.
Worked examples
Example 1
Work out the gradient of the line through (1, 2) and (5, 10).
2 marks
Find vertical change dy: 10 - 2 = 8
Find horizontal change dx: 5 - 1 = 4
Divide dy by dx: 8 / 4 = 2
Answer: 2
Common slips
Slip: Amber calculated dx only Gradient requires both vertical and horizontal changes.
Slip: Amber inverted the fraction Gradient is dy divided by dx, not dx / dy.
Slip: Austin gave 8 as the final answer Do not forget to divide by the change in x.
Example 2
Is the gradient through (1, 1) and (3, 5) positive or negative?
2 marks
Compare y-values: y increases from 1 to 5 as x increases from 1 to 3
Check direction: rising from left to right means a positive gradient
State the conclusion: positive
Answer: positive
Common slips
Slip: Boris saw positive numbers and guessed negative Track whether the function rises or falls.
Slip: Bonnie said negative Both dy and dx are positive, so their quotient is positive.
Slip: Bonnie gave a numerical value The question asks whether it is positive or negative.
Example 3
A line has gradient 3 and passes through (0, 1). Work out y when x=2.
3 marks
Identify equation of the line: y = 3x + 1
Substitute x = 2: y = 3(2) + 1 = 6 + 1
Evaluate to find y: y is 7
Answer: 7
Common slips
Slip: Chloe omitted the y-intercept A line passing through (0, 1) has intercept c = 1.
Slip: Chloe calculated 3 × 2 = 6 Remember to add the initial intercept value.
Slip: Chloe wrote 3 + 2 = 5. The gradient multiplies x: 3 × 2 = 6. Then add the intercept 1 to get 7.
Questions and answers
GCSE Maths gradient of a line questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
Work out the gradient of the line through (0, 1) and (3, 7).
Show answer
Answer: 2
Change in y = 7 - 1 = 6, change in x = 3 - 0 = 3.
Gradient = 6 / 3 = 2.
M1 — Change in y = 7 - 1 = 6, change in x = 3 - 0 = 3.
A1 — Gradient = 6 / 3 = 2.
Question 2 Easy2 marks
Work out the gradient of the line through (2, 3) and (5, 12).
Show answer
Answer: 3
Change in y = 12 - 3 = 9, change in x = 5 - 2 = 3.
Gradient = 9 / 3 = 3.
M1 — Change in y = 12 - 3 = 9, change in x = 5 - 2 = 3.
A1 — Gradient = 9 / 3 = 3.
Question 3 Medium2 marks
Is the gradient through (1, 5) and (3, 1) positive or negative?
Show answer
Answer: negative
y falls from 5 to 1 as x increases.
A downward slope means negative gradient.
M1 — y falls from 5 to 1 as x increases.
A1 — A downward slope means negative gradient.
Question 4 Medium2 marks
Is the gradient through (0, 2) and (2, 8) positive or negative?
Show answer
Answer: positive
y rises from 2 to 8 as x increases.
An upward slope means positive gradient.
M1 — y rises from 2 to 8 as x increases.
A1 — An upward slope means positive gradient.
Question 5 Hard2 marks
A line has gradient 2 and passes through (0, 4). Work out y when x=3.
Show answer
Answer: 10
Equation: y = 2x + 4.
When x = 3: y = 2(3) + 4 = 10.
M1 — Equation: y = 2x + 4.
A1 — When x = 3: y = 2(3) + 4 = 10.
Question 6 Hard2 marks
A line has gradient 4 and passes through (0, 1). Work out y when x=2.
Show answer
Answer: 9
Equation: y = 4x + 1.
When x = 2: y = 4(2) + 1 = 9.
M1 — Equation: y = 4x + 1.
A1 — When x = 2: y = 4(2) + 1 = 9.
Question 7 Exam-style2 marks
Work out the gradient of the line through (1, 3) and (4, 15).