AQA · GCSE · Mathematics
Course version: 8300 · Version from Sept 2015
Graphs: identify and interpret gradients and intercepts of linear functions graphically and algebraically
A linear function has the form y = mx + c.
- 4 key terms
Study notes
From the checked teaching notes for this specification point.
The idea
A linear function has the form y = mx + c. The gradient m measures steepness: it is the change in y divided by the change in x between two points, so m = (y₂ − y₁)/(x₂ − x₁). Intercepts are where the line crosses the axes. The y-intercept c is the y-value when x = 0. The x-intercept is the x-value when y = 0. Graphically, read intercepts where the line meets the axes and find the gradient by dividing vertical change by horizontal change between two points. Algebraically, rearrange into y = mx + c to identify m and c directly, and set y = 0 to find the x-intercept.
For example: 3y = 6x + 9 becomes y = 2x + 3; the gradient is 2, the y-intercept is 3, and setting y = 0 gives x = -1.5.
What earns the marks
- Gradient is the change in y divided by the change in x between two points on the line.
- The y-intercept is the value of y when x = 0, where the line crosses the y-axis.
- The x-intercept is the value of x when y = 0, where the line crosses the x-axis.
- In y = mx + c, m is the gradient and c is the y-intercept.
- An equation must be rearranged into the form y = mx + c before m and c can be read off.
Common mistakes and how to put them right
Reading the gradient as the change in x divided by the change in y.
Put it right by dividing the vertical change by the horizontal change.
Treating the coefficient of x as the intercept.
Put it right by rearranging to y = mx + c and identifying c as the constant term.
Confusing the x-intercept and y-intercept.
Put it right by remembering the x-intercept occurs when y = 0 and the y-intercept when x = 0.
Ignoring the sign of the gradient when the line falls from left to right.
Put it right by checking whether y decreases as x increases.
Exam tips
- Always rearrange an equation into y = mx + c before stating the gradient or y-intercept.
- To find the x-intercept algebraically, substitute y = 0 into the equation and solve for x.
- When reading from a graph, choose two points that lie exactly on grid intersections to reduce error.
Try the lesson
The start of the Graphs: identify and interpret gradients and intercepts of linear functions graphically and algebraically lesson, made from checked teaching notes for this specification point.
Have a go first
A straight line has equation y = 5x − 3. What is the y-intercept of this line?
Listen
A linear function has the form y = mx + c. The gradient m measures steepness: it is the change in y divided by the change in x between two points, so m = (y₂ − y₁)/(x₂ − x₁). Intercepts are where the line crosses the axes. The y-intercept c is the y-value when x = 0. The x-intercept is the x-value when y = 0. Graphically, read intercepts where the line meets the axes and find the gradient by dividing vertical change by horizontal change between two points. Algebraically, rearrange into y = mx + c to identify m and c directly, and set y = 0 to find the x-intercept. For example, 3y = 6x + 9 becomes y = 2x + 3; the gradient is 2, the y-intercept is 3, and setting y = 0 gives x = -1.5.
Key words
Say what each word means before you turn the card.
Gradient
Try answering aloud or in your head before turning the card.
y-intercept
Try answering aloud or in your head before turning the card.
x-intercept
Try answering aloud or in your head before turning the card.
y = mx + c
Try answering aloud or in your head before turning the card.
See every definition
- Gradient
- The change in y divided by the change in x between two points, so m = (y₂ − y₁)/(x₂ − x₁).
- y-intercept
- The y-value when x = 0, where the line crosses the y-axis.
- x-intercept
- The x-value when y = 0, where the line crosses the x-axis.
- y = mx + c
- The form of a linear function in which m is the gradient and c is the y-intercept.
There is more to this lesson
The full lesson goes on to check your understanding and help you earn the marks. Create a free account to carry on, with your progress saved.
7 days Premium · Then the free plan continues · No card, no charge
Test your understanding of Graphs: identify and interpret gradients and intercepts of linear functions graphically and algebraically
Active recall beats re-reading. Take an instant quiz now to test what you’ve learned and lock it into long-term memory.
7 days Premium · Then the free plan continues · No card, no charge