Algebra · GCSE Maths
Graphs
Plot and interpret y = mx + c, read gradient and intercept, and recognise linear, quadratic and other standard GCSE graphs.
Gradient is rise over run. The intercept is where the line meets the y-axis, when x is zero.
The important bits
What you need to know
- 1
The equation y = mx + c describes a straight line. m is the gradient; c is the y-intercept.
- 2
Gradient between two points is (y₂ − y₁) / (x₂ − x₁). Include the sign: down from left to right is negative.
- 3
Parallel lines have equal gradients. Perpendicular lines have gradients whose product is −1, so m₂ = −1/m₁.
- 4
To plot y = 2x − 1, make a small table of x values, calculate y, plot the points, and draw a single straight line through them.
- 5
Quadratic graphs are parabolas. Read roots from the x-intercepts and the turning point from the vertex, not from a guessed sketch.
- 6
y = k/x is a reciprocal graph with two branches and the axes as asymptotes. y = x³ has rotational symmetry about the origin.
- 7
The solution of two simultaneous equations is the intersection point of their graphs. Read it from a grid only as accurately as the scale allows.
- 8
Distance–time graphs: gradient is speed. Speed–time graphs: gradient is acceleration and area under the line is distance.
Go deeper
Gradient is a rate, not a decoration
Rise over run is a ratio of change: how much y increases when x increases by 1. For the points (1, 2) and (4, 8), the rise is 6 and the run is 3, so m = 2. The line climbs two units for every one unit right. If you reverse the points, 2 − 8 over 1 − 4 is still 2; the order must be consistent, not random. A horizontal line has gradient 0 because y never changes. A vertical line has undefined gradient because the run is zero; it cannot be written as y = mx + c and is instead x = a constant. When a question gives a line through (0, 5) with gradient −3, write y = −3x + 5 immediately. The intercept is a point, (0, c), not a number floating beside the graph.
Go deeper
Parallel and perpendicular from the gradient
Two lines never meet if they have the same gradient and different intercepts. y = 2x + 1 and y = 2x − 4 are parallel. A line perpendicular to gradient 2 has gradient −1/2, because 2 × (−1/2) = −1. The same rule turns 3/4 into −4/3: invert and change sign. Students often change the sign without inverting, or invert without changing the sign. Both are needed. To write the perpendicular through a given point, keep the new gradient and substitute the point to find c. For gradient −1/2 through (4, 1): 1 = (−1/2)(4) + c, so 1 = −2 + c, hence c = 3, and y = −½x + 3. The algebra is ordinary; the geometry sits entirely in the choice of m.
Go deeper
Read a graph with the scale, not with hope
Quadratic sketches in exams are often drawn on a grid. Roots are where the curve cuts the x-axis; estimate to the nearest half square unless the question says otherwise. The turning point is the highest or lowest plotted point, not a label you invent from the equation unless you have completed the square. For y = x² − 4x + 3 the curve crosses at 1 and 3 and turns at (2, −1). On kinematics graphs, the story matters. A distance–time graph that is horizontal means stopped, not “no graph”. A sloping straight line means constant speed. The area of a trapezium under a speed–time graph is a distance; forgetting to use the actual units on the axes is a regular dropped mark. Always write the scale in words: “each square is 2 m/s.”
See the idea in action
A line passes through (0, 4) and (3, 10). Gradient m = (10 − 4)/(3 − 0) = 2. The y-intercept is 4, so y = 2x + 4. A perpendicular line would have gradient −1/2. Through (3, 10) that would be 10 = (−1/2)(3) + c, so c = 11.5 and y = −½x + 11.5.
Exam technique
Turn knowledge into marks
When reading an intersection from a printed graph, write the coordinates in the order (x, y) and check they sit on both lines. Reversing them is a cheap way to lose the mark.
Common mistakes
Do not give these marks away
- 01
Calculating run over rise, or subtracting coordinates in inconsistent order so the sign of the gradient flips.
- 02
Stating that perpendicular to 2 is −2, forgetting to invert.
- 03
Reading a quadratic turning point from the y-intercept because that is the only labelled number on the sketch.
What is the gradient of the line through (1, 2) and (4, 8)?
A2
B3
C1/2
D6
Show the answer
2. (8 − 2)/(4 − 1) = 6/3 = 2. 1/2 would be run over rise, the usual inversion error.
Quick questions
If this is the bit you searched
How do I find the equation of a line from a graph?
Read c where the line meets the y-axis. Then pick a second clear point and calculate m as rise over run. Write y = mx + c.
What does a negative gradient look like?
The line falls as you read from left to right. Each step right corresponds to a step down.