What you'll learn
Straight-line graphs are described by the equation y = mx + c, with a gradient and a y-intercept. In this guide you will learn how to plot straight lines, find the gradient and intercept, write the equation of a line, find the equation from two points, and understand parallel and perpendicular lines. These are core GCSE algebra and graph skills.
Key terms and definitions
Gradient (m) — the steepness of a line; the change in y divided by the change in x.
y-intercept (c) — where the line crosses the y-axis.
y = mx + c — the equation of a straight line, with gradient m and y-intercept c.
Parallel lines — lines with the same gradient.
Perpendicular lines — lines that meet at 90°; their gradients multiply to −1.
Core concepts
The equation y = mx + c
Every straight line can be written as y = mx + c, where m is the gradient (steepness) and c is the y-intercept (where it crosses the y-axis). For example, y = 2x + 3 has gradient 2 and crosses the y-axis at 3.
Finding the gradient
The gradient measures steepness: gradient = change in y ÷ change in x (rise over run) between two points on the line. A positive gradient slopes upwards left to right; a negative gradient slopes downwards. A steeper line has a larger gradient magnitude.
Plotting a line
To plot y = mx + c, you can use the y-intercept (start at (0, c)) and then use the gradient to find more points (for gradient 2, go up 2 for every 1 across). Alternatively, make a table of values and plot the points, joining them with a straight line using a ruler.
Finding the equation from two points
Given two points, find the gradient first (change in y ÷ change in x), then substitute one point into y = mx + c to find c. For example, through (1, 5) and (3, 11): gradient = (11 − 5)/(3 − 1) = 3; substitute (1, 5): 5 = 3(1) + c → c = 2, so y = 3x + 2.
Parallel and perpendicular lines
Parallel lines have the same gradient. Perpendicular lines have gradients whose product is −1 — each is the negative reciprocal of the other. For example, a line perpendicular to y = 2x + 1 has gradient −½.
Worked examples
Example 1: Reading m and c
State the gradient and y-intercept of y = −4x + 7.
Gradient = −4; y-intercept = 7 (crosses at (0, 7)).
Example 2: Gradient from two points
Find the gradient through (2, 3) and (6, 11).
(11 − 3)/(6 − 2) = 8/4 = 2.
Example 3: Perpendicular gradient
What is the gradient of a line perpendicular to y = 3x − 5?
The negative reciprocal of 3 is −1/3.
Common mistakes and how to avoid them
Mixing up m and c. m is the gradient (coefficient of x); c is the y-intercept (constant).
Gradient the wrong way up. Gradient = change in y ÷ change in x, not the reverse.
Sign errors with negative gradients. A downward slope has a negative gradient.
Forgetting to find c. After the gradient, substitute a point to find the intercept.
Perpendicular gradient errors. Use the negative reciprocal (flip and change sign).
Exam technique for Straight-line Graphs
Identify m and c directly from y = mx + c.
Find the gradient as change in y over change in x.
Plot using the intercept and gradient, or a table of values.
Derive the equation from two points (gradient then intercept).
Use equal gradients for parallel and negative reciprocals for perpendicular lines.
Quick revision summary
A straight-line graph has equation y = mx + c, where m is the gradient (steepness) and c is the y-intercept (where it crosses the y-axis). The gradient = change in y ÷ change in x (rise over run): positive for an upward slope, negative for a downward slope, larger magnitude for steeper lines. Plot a line using the intercept (0, c) and the gradient, or from a table of values joined with a ruler. To find the equation from two points, work out the gradient first, then substitute one point into y = mx + c to find c (through (1, 5) and (3, 11) gives y = 3x + 2). Parallel lines share the same gradient; perpendicular lines have gradients that multiply to −1 (negative reciprocals, so 2 ↔ −½). The common errors are confusing m and c, inverting the gradient, sign slips, forgetting to find c, and perpendicular gradient mistakes. Read m and c, compute gradient as y over x, plot or derive the equation, and use the parallel/perpendicular gradient rules.