What is the gradient of the line \(y = 3x + 2\)?
The gradient is the number in front of x.
In \(y = mx + c\), the gradient is m. Here, m = 3.
Graphs of Linear Functions
Linear graphs are straight lines that show relationships between variables. Understanding gradient and intercepts is essential for interpreting graphs in GCSE Maths.
Overview
A linear function produces a straight-line graph.
These graphs are usually written in the form y = mx + c.
\( y = mx + c \)
In this form, m represents the gradient and c represents the y-intercept.
What you should understand after this topic
- Understand what a linear graph is
- Understand what gradient means
- Understand what the y-intercept represents
- Draw a straight line from a table of values
- Read the equation of a line from a graph
Key Definitions
Linear Function
A function whose graph is a straight line.
Gradient
How steep the line is.
y-intercept
The point where the line crosses the y-axis.
x-intercept
The point where the line crosses the x-axis.
Coordinate
A point written as \( (x,y) \).
Table of Values
A table used to generate points for a graph.
Parallel Lines
Lines with the same gradient.
Equation of a Line
A rule that describes every point on the line.
Key Rules
\(m\) is the gradient
It tells you how much \(y\) changes when \(x\) increases by 1.
\(c\) is the y-intercept
It tells you where the line crosses the y-axis.
Positive gradient
The line rises from left to right.
Negative gradient
The line falls from left to right.
Quick Pattern Check
\( y = 2x + 3 \)
Gradient 2, y-intercept 3.
\( y = -x + 4 \)
Gradient -1, y-intercept 4.
\( y = 5 \)
Horizontal line.
Same gradient
Parallel lines never meet.
How to Solve
Step 1: Understand a linear graph
A linear graph is a straight line with a constant rate of change. This means \(y\) increases or decreases by the same amount each time \(x\) increases by 1. These graphs come from equations such as linear equations.
\( y = mx + c \)
Exam tip: Straight line = constant gradient.
Step 2: Identify gradient and intercept
The equation \( y = mx + c \) tells you everything about the line.
\( m = \text{gradient} \), \( c = \text{y-intercept} \)
Exam thinking: Always read \(m\) and \(c\) first before doing anything else.
Gradient (m)
Controls steepness and direction.
Positive gradient
Line goes up to the right.
Negative gradient
Line goes down to the right.
Intercept (c)
Where the line crosses the y-axis.
Step 3: Draw using gradient and intercept
The fastest method is to plot the intercept, then use the gradient.
\( y = 2x + 1 \)
Exam tip: This method is faster than making a full table.
- Plot the y-intercept: \( (0,1) \) using coordinates.
- Use the gradient: 2 means rise 2, run 1.
- From \( (0,1) \), go to \( (1,3) \), then \( (2,5) \).
- Join the points with a straight line.
Step 4: Drawing using a table (alternative method)
You can also create a table of values.
Draw \( y = 2x + 1 \)
- Choose values for \(x\): for example \(-1, 0, 1, 2\).
- Substitute into the equation to find \(y\).
- Plot the points.
- Join with a straight line.
Step 5: Parallel lines
Parallel lines have the same gradient.
\( y = 2x + 1 \), \( y = 2x - 4 \)
Both gradients are 2, so the lines are parallel.
Exam tip: Same \(m\) → parallel lines.
Step 6: Finding the equation of a line
You may need to find the equation from a graph or points. This builds directly on solving linear equations.
Exam tip: Always calculate gradient first, then find \(c\).
- Find the gradient (change in y ÷ change in x).
- Find the y-intercept (where the line crosses the y-axis).
- Write the equation in the form \( y = mx + c \).
Step 7: Special cases
Understanding gradient is essential for topics such as graph transformations.
Horizontal line
\( y = 4 \) means the graph is flat.
Through the origin
If \( c = 0 \), the line passes through \( (0,0) \).
Example Questions
Edexcel
Exam-style questions inspired by Edexcel GCSE Mathematics.
Edexcel
Complete the table of values for \( y = 2x + 1 \) when \( x = -1, 0, 1, 2 \).
Edexcel
Write down the gradient of the line \( y = 3x - 4 \).
Edexcel
Write down the y-intercept of the line \( y = -2x + 5 \).
Edexcel
Determine whether the point \( (2, 7) \) lies on the line \( y = 3x + 1 \).
Edexcel
Find the equation of the line with gradient 4 and y-intercept \( -3 \).
AQA
Exam-style questions based on the AQA GCSE Mathematics specification, focusing on interpretation and forming equations of straight lines.
AQA
Find the gradient of the line passing through the points \( (1, 2) \) and \( (5, 10) \).
AQA
Find the equation of the line passing through \( (0, 3) \) and \( (2, 7) \).
AQA
Find the equation of the line parallel to \( y = 2x - 1 \) that passes through the point \( (0, 4) \).
AQA
Find the equation of the line perpendicular to \( y = \frac{1}{2}x + 3 \).
AQA
Explain how the gradient and y-intercept can be identified from the equation of a straight line.
OCR
Exam-style questions aligned with OCR GCSE Mathematics, emphasising reasoning, interpretation, and real-life applications.
OCR
A taxi fare is given by the formula \( C = 3 + 2x \), where \( x \) is the distance in miles. Interpret the gradient and the intercept.
OCR
Find the coordinates of the point where the line \( y = 4x - 2 \) crosses the y-axis.
OCR
Find the coordinates of the point where the line \( y = 5 - x \) crosses the x-axis.
OCR
Determine whether the lines \( y = 2x + 3 \) and \( y = 2x - 5 \) are parallel. Give a reason for your answer.
OCR
The cost of producing items is given by \( C = 50 + 4x \). Explain what the gradient and intercept represent in this context.
Exam Checklist
Step 1
Check whether the equation is in the form \( y = mx + c \).
Step 2
Identify the gradient and y-intercept carefully.
Step 3
If needed, make a table of values and plot accurate points.
Step 4
Draw a straight line through the points.
Most common exam mistakes
Gradient mistake
Reading the coefficient of \( x \) incorrectly.
Intercept mistake
Using the wrong sign for the constant term.
Plotting mistake
Plotting inaccurate points from the table.
Parallel lines
Forgetting that parallel lines must have the same gradient.
Common Mistakes
These are common mistakes students make when working with graphs of linear functions in GCSE Maths.
Mixing up gradient and intercept
Incorrect
A student confuses the gradient with the y-intercept in \(y = mx + c\).
Correct
In \(y = mx + c\), m is the gradient (slope of the line) and c is the y-intercept (where the line crosses the y-axis).
Plotting points incorrectly
Incorrect
A student makes errors when plotting coordinates from a table.
Correct
Plot each point carefully by moving along the x-axis first, then up or down the y-axis. Check each coordinate before drawing the line.
Not drawing a straight line
Incorrect
A student joins points with uneven or curved lines.
Correct
Linear functions produce straight lines. Use a ruler to draw a single straight line through the points.
Incorrect handling of negative gradients
Incorrect
A student draws a line increasing when the gradient is negative.
Correct
A negative gradient means the line goes down from left to right. Always check the direction of the slope.
Misunderstanding parallel lines
Incorrect
A student thinks parallel lines can have different gradients.
Correct
Parallel lines have the same gradient but different y-intercepts. If the gradients differ, the lines will intersect.
Try It Yourself
Practise plotting and interpreting straight-line graphs.
Foundation
Foundation Practice
Understand gradients and intercepts of straight lines.
What is the y-intercept of \(y = 4x + 1\)?
The y-intercept is the value of y when x = 0.
In \(y = mx + c\), the intercept is c. Here, c = 1.
Which equation has gradient 5?
Look for the number multiplying x.
The gradient is the coefficient of x. Only \(y = 5x + 2\) has gradient 5.
Write the equation of a line with gradient 2 and intercept 3.
Use the form \(y = mx + c\).
Gradient m = 2 and intercept c = 3, so \(y = 2x + 3\).
Which graph is horizontal?
A horizontal line has no x term.
The equation \(y = 4\) has no x term, so it is a horizontal line.
What is the gradient of \(y = -2x + 5\)?
Look at the coefficient of x.
The gradient is -2.
Which equation passes through (0, 4)?
At x = 0, y equals the intercept.
The intercept is 4, so the equation must be \(y = x + 4\).
Write the equation of a line with gradient -3 and intercept 2.
Use \(y = mx + c\).
m = -3 and c = 2, so \(y = -3x + 2\).
A student says the gradient of \(y = 2x + 5\) is 5. What is the mistake?
Check m and c.
The gradient is the coefficient of x (2), not the constant (5).
What is the y-intercept of \(y = -x + 6\)?
Look at the constant term.
The intercept is 6.
Higher
Higher Practice
Find equations, gradients between points and interpret graphs.
Find the gradient between (1, 2) and (3, 6).
Use (change in y) / (change in x).
Gradient = (6 − 2) / (3 − 1) = 4/2 = 2.
Find the gradient between (2, 5) and (6, 13).
Use (y₂ − y₁)/(x₂ − x₁).
(13 − 5) / (6 − 2) = 8 / 4 = 2.
Find the equation of a line with gradient 3 passing through (0, 2).
Use y = mx + c.
m = 3 and intercept is 2, so \(y = 3x + 2\).
Find the equation of a line with gradient 4 and passing through (0, -1).
Use the y-intercept.
Intercept = -1, so \(y = 4x - 1\).
Which line is parallel to \(y = 2x + 3\)?
Parallel lines have the same gradient.
Both have gradient 2.
Find the gradient of a line passing through (3, 7) and (5, 11).
Use change in y over change in x.
(11 − 7) / (5 − 3) = 4 / 2 = 2.
Find the equation of a line through (0, 5) with gradient -2.
Use intercept and gradient.
m = -2, c = 5 → \(y = -2x + 5\).
Find the equation of a line with gradient 1 passing through (0, 4).
Gradient is 1, intercept is 4.
\(y = x + 4\).
A student says two lines are parallel if they have the same intercept. What is wrong?
Think about slope.
Parallel lines have equal gradients, not equal intercepts.
Find the equation of a horizontal line passing through y = 7.
Horizontal lines have no x term.
The equation is \(y = 7\).
Games
Practise this topic with interactive games.
Games coming soon.
Frequently Asked Questions
What does the gradient represent?
It shows how steep the line is.
What is the y-intercept?
The point where the line crosses the y-axis.
How do I plot a straight line?
Use a table of values or identify gradient and intercept.