Learn how to plot, recognise and interpret graphs confidently
GCSE Graphs Revision Guide
Graphs show mathematical relationships using coordinates, lines and curves. This GCSE Graphs guide explains how to plot points, draw linear and non-linear graphs, identify important features and interpret graphs in mathematical and real-life situations. It includes clear rules, lesson links, a worked example and practical exam advice for Foundation and Higher Tier students.
GCSE Maths
Foundation & Higher
Graphs & Coordinates
What You'll Learn
Plot and Read Coordinates
Plot ordered pairs accurately in all four quadrants and read coordinates from a correctly scaled grid.
Draw Linear Graphs
Complete tables of values, plot coordinates and join the points with an accurate straight line.
Recognise Non-Linear Graphs
Identify the typical shapes of quadratic, simple cubic and reciprocal functions from their equations or graphs.
Interpret Important Features
Locate roots, intercepts, turning points and gradients, and explain what these features represent.
Use Real-Life Graphs
Interpret distance-time, speed-time and other graphs that represent changing quantities in practical situations.
Transform Functions
Sketch translations and reflections of familiar functions using correct function notation and transformation rules.
Key Graph Rules
1
Coordinate Order
`(x, y)`
The first number gives the horizontal x-coordinate. The second number gives the vertical y-coordinate. For example, the point `(−3, 2)` is three units left and two units up from the origin.
Always move along the x-axis before moving vertically along the y-axis.
2
Straight-Line Equation
`y = mx + c`
In a straight-line equation, `m` represents the gradient and `c` represents the y-intercept. The y-intercept is the point where the line crosses the y-axis.
At the y-intercept, `x = 0`
3
Gradient Between Two Points
`m = (y₂ − y₁) ÷ (x₂ − x₁)`
Gradient measures the rate at which a line rises or falls. A positive gradient rises from left to right, while a negative gradient falls from left to right.
`x₂` must not equal `x₁`. A vertical line has an undefined gradient.
4
Standard Non-Linear Graphs
Quadratic:
`y = x²`
Simple cubic: `y = x³` Reciprocal:
`y = 1/x`
A quadratic graph is a parabola. The simple cubic graph `y = x³` passes through the origin and has a stationary point of inflection. The reciprocal graph has two separate branches.
For `y = 1/x`, `x ≠ 0`.
5
Roots, Intercepts &Turning Points
At an x-intercept or root: `y = 0`
At a y-intercept: `x = 0`
At a turning point: the graph changes from increasing to decreasing, or decreasing to increasing.
These features help students interpret graphs and find approximate solutions to equations.
A stationary point is not always a turning point. For example, the origin on `y = x³` is a stationary point of inflection.
GCSE Graphs Lessons
Distance-time Graphs
Learn how horizontal, rising and changing sections represent stopping, movement and different speeds during a journey.
Combined Transformations
Apply more than one transformation in sequence and describe each stage using clear mathematical language.
Reflections
Learn how function graphs change when reflected in the x-axis or y-axis.
Stretches
Explore how scale factors can change a graph’s width or height as an extension to standard GCSE transformations.
Translations
Understand how changes inside or outside a function move its graph horizontally or vertically.
Equation Of A Line
Find and use straight-line equations from gradients, intercepts, coordinates and graphical information.
Gradient And Intercept
Identify the gradient and y-intercept of a straight line from its graph or equation.
Parallel And Perpendicular Lines
Compare gradients to identify parallel lines and, at Higher Tier, perpendicular straight lines.
Plotting Coordinates
Plot and read positive and negative coordinates accurately across all four quadrants.
Real Life Graphs
Interpret graphs that represent journeys, costs, temperature, conversion rates and other practical situations.
Speed-time Graphs
Interpret speed, acceleration and stationary motion, and calculate distance using the area beneath a speed-time graph.
Graph Translations and Reflections
Rules:
- y = f(x) + a`: move upwards by `a`
- `y = f(x) − a`: move downwards by `a`
- `y = f(x − a)`: move right by `a`
- `y = f(x + a)`: move left by `a`
- `y = −f(x)`: reflect in the x-axis
- `y = f(−x)`: reflect in the y-axis
Core GCSE Graph Subtopics
Both
Quadratic Graphs
Recognise parabolas, identify roots and turning points, and understand how the coefficient of `x²` affects direction.
Both
Cubic & Reciprocal Graphs
Recognise the standard shapes of simple cubic and reciprocal functions and understand their important features.
Both
Graphical Solutions
Find approximate solutions to linear, quadratic or simultaneous equations by locating intersections and x-intercepts.
Higher
Exponential & Trigonometric Graphs
Recognise and interpret exponential, sine, cosine and tangent graphs over suitable intervals.
Higher
Gradients of Curves
Estimate the gradient of a curve at a point by drawing and calculating the gradient of a tangent.
Higher
Areas Under Graphs
Estimate or calculate areas beneath graphs and interpret the result in contexts such as speed-time graphs.
Worked Example: Drawing and Interpreting a Quadratic Graph
Example 1: Collect like terms
4x + 8 + 6x + 3 = 10x + 11
Step 1: Group like terms.
(4x + 6x) + (8 + 3)
Step 2: Add the coefficients.
10x + 11
(4x + 6x) + (8 + 3)
Step 2: Add the coefficients.
10x + 11
Example 2: Expand brackets
2(x + 5) = 2x + 10
Step 1: Multiply 2 by each term.
2 × x = 2x and 2 × 5 = 10
Step 2: Write without brackets.
2x + 10
2 × x = 2x and 2 × 5 = 10
Step 2: Write without brackets.
2x + 10
Example 3: Substitution
If a = 6, then 2a + 4 = 16
Step 1: Replace a with 6.
2 × 6 + 4
Step 2: Calculate.
12 + 4 = 16
2 × 6 + 4
Step 2: Calculate.
12 + 4 = 16
The Question:
Complete a table of values for:`y = x² − 4`
Use values of `x` from `−3` to `3`. Plot the graph and state:
1. The coordinates of the roots
2. The y-intercept
3. The turning point
1. The coordinates of the roots
2. The y-intercept
3. The turning point
Given Information
- Equation: `y = x² − 4`
- x-values: `−3, −2, −1, 0, 1, 2, 3`
- The graph should be drawn as a smooth curve.
Method
Substitute each x-value into the equation, plot the resulting coordinates and interpret the completed parabola.
Step 1: Calculate the y-values
- | x | −3 | −2 | −1 | 0 | 1 | 2 | 3 |
- | - | -: | -: | -: | -: | -: | -: | -: |
- | y | 5 | 0 | −3 | −4 | −3 | 0 | 5 |
For example:
- `y = (−3)² − 4`
- `y = 9 − 4`
- `y = 5`
Step 2: Write the coordinates
The coordinates are:
- `(−3, 5), (−2, 0), (−1, −3), (0, −4), (1, −3), (2, 0), (3, 5)`
Step 3: Plot the points
Plot each coordinate carefully using a consistent scale on both axes.
Step 4: Draw the graph
Join the points with a smooth U-shaped curve. Do not connect them using separate straight-line segments.
Step 5: Identify the roots
The graph crosses the x-axis at:
- `(−2, 0)` and `(2, 0)`
- Therefore, the roots are:
- `x = −2` and `x = 2`
Step 6: Identify the y-intercept and turning point
- The graph crosses the y-axis at:`(0, −4)`
- The lowest point is also:`(0, −4)`
Therefore:
- Y-intercept: (0, −4)
- Turning point: (0, −4)
Final Answer:
- Roots: `x = −2` and `x = 2`
- Y-intercept: `(0, −4)`
- Turning point: `(0, −4)`
Independent Answer Check
To check the roots, substitute `x = 2` and `x = −2` into the original equation:
- 2² − 4 = 4 − 4 = 0
- (−2)² − 4 = 4 − 4 = 0
- Common GCSE Mistakes
- Reversing the Coordinates
- Using an Inconsistent Scale
- Joining Curves With Straight Segments
- Confusing Roots and Y-Intercepts
- Plotting a Reciprocal Graph at x = 0
- Moving Horizontal Translations the Wrong Way
- Exam Tips
- Check the coordinate order.
- Choose a sensible scale.
- Show the table of values.
- Use the correct drawing method.
- Check important features.
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Watch the Graph Tutorial
Clear, step-by-step explanation of expanding brackets, simplifying complex expressions, and solving for x. Perfect for visual learners who need a breakdown of the rules in action.
- 12 minute comprehensive guide
- High-definition whiteboard walk-through
Continue Your GCSE Maths Revision
Linear Equations
Quadratics
Sequences
Graphs
Fractions
Number
Linear Graphs
Percentages
Frequently Asked Questions
A graph is a visual representation of the relationship between two variables. Values of the independent variable are normally shown on the horizontal x-axis, while corresponding values are plotted on the vertical y-axis. GCSE questions may ask students to draw, recognise, compare or interpret graphs.
Choose suitable x-values and substitute each one into the equation to calculate the corresponding y-values. Write the results as coordinates, plot them using a consistent scale and join them appropriately. Straight-line graphs require a ruler, while quadratic and other non-linear graphs normally require smooth curves.
A linear graph is a straight line. A quadratic graph is a parabola that opens upwards or downwards. The simple cubic graph `y = x³` has a characteristic S-shaped curve. The reciprocal graph `y = 1/x` has two separate branches and is undefined when `x = 0`.
Both tiers include coordinates, linear graphs, quadratic graphs, simple cubic graphs, reciprocal graphs and the interpretation of graphs in context. Higher Tier may also include exponential and trigonometric graphs, function transformations, gradients of curves and areas beneath non-linear graphs.
Substitute selected x-values back into the equation and check that the plotted points match. Look for expected features such as intercepts, symmetry and turning points. Make sure the scale is consistent and confirm that straight lines are straight and curves are smooth.
Under the current arrangements for GCSE Mathematics examinations in England, students receive a formulae sheet. However, the sheet does not replace understanding. Students should still know what graph notation means, how to identify the correct method and how to apply relevant rules accurately.
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