Positive gradient
The line rises from left to right.
Learning Objectives
- calculate the gradient of a straight line;
- interpret gradient as a rate of change;
- identify x- and y-intercepts;
- connect a line to y = mx + c;
- write the equation of a simple straight line.
Introduction
This lesson develops Gradient and Intercepts from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
🤔 Think About This...
Two roads rise by the same height, but one covers a much shorter horizontal distance.
Which road is steeper, and how could mathematics measure that steepness?
The gradient compares vertical change with horizontal change.
📐
See Gradient as Rise ÷ Run
Change the rise and run. The line and gradient update together.
What Is Gradient?
gradient = change in y ÷ change in x
Key idea: Gradient tells us how much y changes whenever x changes by one unit.
Negative gradient
The line falls from left to right.
Zero gradient
The line is horizontal.
Larger magnitude
The line is steeper.
Worked Example: Finding Gradient
Find the gradient through A(1, 3) and B(5, 11).
1
Change in y = 11 − 3 = 8.
2
Change in x = 5 − 1 = 4.
3
Gradient = 8 ÷ 4 = 2.
4
For every 1 unit increase in x, y increases by 2.
Intercepts
An intercept is where a graph crosses an axis.
- The y-intercept occurs where x = 0.
- The x-intercept occurs where y = 0.
y = mx + c
In this form, m is the gradient and c is the y-intercept.
Worked Example: Reading an Equation
For y = 3x − 4:
1
m = 3, so the gradient is 3.
2
c = −4, so the graph crosses the y-axis at (0, −4).
3
To find the x-intercept, set y = 0: 0 = 3x − 4, so x = 4/3.
Finding the Equation of a Line
Suppose a line has gradient 2 and y-intercept 5.
y = 2x + 5
Common mistake: Using a point as the gradient. Gradient
is a ratio of two changes, not a coordinate.
✅ Quick Check
State the gradient of y = 5x + 2.
State the y-intercept of y = −2x + 7.
Find the gradient through (2, 1) and (6, 9).
🧠 Let’s Think Together
Before calculating, identify the information given, the result required and the mathematical relationship that connects them. Predict whether the answer should be larger, smaller or unchanged, then use that prediction to check the result.
Practice Questions
Foundation- State m and c for y = 4x − 3.
- Find the gradient through (0, 2) and (3, 8).
- Write the equation of a line with gradient 5 and y-intercept −1.
- Find the x-intercept of y = 2x − 10.
- A line passes through (0, 4) and (5, 14). Find its equation.
- Explain the meaning of a negative gradient in a real-life graph.
- A line passes through (2, 7) and has gradient 3. Find its equation.
✅ Check Your Work
Attempt the questions before opening the solutions.
Show answers and working
Foundation
- m=4 and c=−3.
- m=(8−2)/(3−0)=2.
- y=5x−1.
Developing
- Set y=0: 0=2x−10, so x=5.
- Gradient=(14−4)/(5−0)=2 and intercept=4, so y=2x+4.
- The quantity represented by y decreases as x increases.
Challenge
- y−7=3(x−2), so y=3x+1.
Common Mistakes
- Choosing a rule before identifying what the question describes.
- Skipping working or changing notation part-way through a solution.
- Accepting an answer without checking its size, sign or units.
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.
Summary
- Gradient is change in y divided by change in x.
- The y-intercept occurs where x = 0.
- The x-intercept occurs where y = 0.
- In y = mx + c, m is gradient and c is y-intercept.