GCSE MATHS • HIGHER TIER
Gradient and Intercept Made Simple
Understand straight-line graphs by identifying gradient, y-intercept and rates of change.
Clear method
Exam practice
Worked proofs
Gradient and Intercept in Four Moves
1
Identify
Write the line in the form y = mx + c whenever possible.
2
Read
Identify m as the gradient and c as the y-intercept.
3
Calculate
If coordinates are given, calculate gradient using change in y ÷ change in x.
4
Interpret
Explain what the gradient and intercept mean in the graph or real-life context.
What You’ll Learn
Understand what gradient and y-intercept mean.
Recognise gradient and intercept in y = mx + c.
Identify positive and negative gradients.
Find the y-intercept from an equation.
Calculate gradient from two coordinates.
Interpret gradient as a rate.
Interpret the intercept as a starting value.
Key Rules
Straight-line equation
y=mx+c
Gradient
m = gradient — it tells us how steep the line is.
Y-intercept
c = y-intercept — where the line crosses the y-axis, when \(x = 0\).
Gradient from coordinates
Gradient = Change in \(y\) ÷ Change in x
Worked Proof: Gradient and Intercept
Step-by-Step Method
- 1. Write or identify y = mx + c.
- 2. Find m if the gradient is required.
- 3. Find c if the y-intercept is required.
- 4. If two coordinates are given, calculate change in y.
- 5. Calculate change in x.
- 6. Divide change in y by change in x.
- 7. Check negative signs and interpret the result.
More Worked Examples
1
Finding Gradient and Intercept from an Equation
y = -3x + 7
Step 1: The coefficient of x is -3.
Gradient = -3
Step 2: The constant term is 7.
Intercept = 7
Gradient = -3
Step 2: The constant term is 7.
Intercept = 7
2
Finding Gradient and Intercept from an Equation
y = 6x + 8
Step 1: Compare with y = mx + c.
Step 2: Identify the number in front of x.
6 is the gradient.
Step 3: Identify the constant term.
8 is the intercept.
Step 2: Identify the number in front of x.
6 is the gradient.
Step 3: Identify the constant term.
8 is the intercept.
3
Finding Gradient from Coordinates
Find the gradient between (1, 5) and (7, 17).
Step 1: Change in y: 17 - 5 = 12
Step 2: Change in x: 7 - 1 = 6
Step 3: Gradient: 12 ÷ 6 = 2
Step 2: Change in x: 7 - 1 = 6
Step 3: Gradient: 12 ÷ 6 = 2
4
Find the gradient between (3, 4) and (7, 20).
Step 1: Change in y:
20 - 4 = 16
Step 2: Change in x: 7 - 3 = 4
Step 3: Gradient: 16 ÷ 4 = 4
Step 2: Change in x: 7 - 3 = 4
Step 3: Gradient: 16 ÷ 4 = 4
- Common GCSE Mistakes
- Ignoring negative signs.
- Confusing gradient and intercept.
- Using coordinates in the wrong order.
- Forgetting to divide change in y by change in x.
- Exam Tips
- Remember that m means gradient and c means y-intercept.
- Keep negative signs when reading an equation.
- Use the same coordinate order when calculating both changes.
- Always calculate change in y ÷ change in x, not the other way around.
- The y-intercept occurs when x = 0.
Skills Used in Gradient and Intercept
Linear Graphs
Gradient
Y-Intercept
Coordinates
Rates of Change
Graph Interpretation
Ready to Practise?
Printable Worksheet
Download and print practice questions with answers.
Interactive Quiz
Test your knowledge with instant feedback and hints.
Exam-Style Questions
Timed questions to build confidence for the real exam.
Watch the Gradient and Intercept Tutorial
Step-by-step walkthroughs of Gradient and Intercept with clear methods and exam tips.
Frequently Asked Questions
The gradient measures how steep a straight-line graph is and how much y changes for a change in x.
The y-intercept is where the graph crosses the y-axis, which happens when x = 0.
m is the gradient and c is the y-intercept.
Calculate change in y and divide it by change in x.
A negative gradient means the line falls as you move from left to right.
Gradient often represents a rate, while the intercept often represents a starting or fixed value.