What you'll learn
Plotting and interpreting non-linear graphs covers quadratic, cubic, reciprocal and other curved graphs. In this guide you will learn how to complete a table of values, plot the points and draw a smooth curve, recognise the characteristic shapes of common non-linear graphs, and read information from them. These skills appear in graph-drawing and interpretation questions across GCSE Maths.
Key terms and definitions
Non-linear graph — a graph that is a curve, not a straight line.
Quadratic graph — the graph of y = ax² + bx + c, a parabola.
Cubic graph — the graph of y = ax³ + …, with a characteristic S-shape.
Reciprocal graph — the graph of y = a/x, with two separate curves.
Table of values — a set of x-values with the matching y-values, used to plot points.
Core concepts
Completing a table of values
To plot a non-linear graph, substitute each x-value into the equation to find y, recording the pairs in a table. Take care with negative values and squaring/cubing — for example, (−3)² = 9, but (−3)³ = −27. Accurate values give an accurate curve.
Plotting and drawing
Plot each (x, y) point carefully, then join them with a smooth curve, not straight line segments. The curve should pass through all the plotted points. Use a sharp pencil and draw freehand smoothly; do not "join the dots" with rulers.
Quadratic graphs (parabolas)
A quadratic y = ax² + bx + c gives a parabola. If a is positive, it is U-shaped (lowest point at the bottom); if a is negative, it is ∩-shaped (highest point at the top). The parabola is symmetrical about a vertical line through its turning point. The turning point is the minimum or maximum.
Cubic and reciprocal graphs
A cubic graph (y = x³ and similar) has a characteristic S-shape, increasing steeply at both ends with possible bends in the middle. A reciprocal graph (y = a/x) has two separate curves in opposite quadrants; it never touches the axes (the axes are asymptotes), because x cannot be zero.
Reading information from graphs
You can read off values (find y for a given x, or solve y = 0 by reading where the curve crosses the x-axis), find turning points (maximum or minimum), and estimate solutions to equations by reading intersections. Always use the scales carefully.
Worked examples
Example 1: Table of values
Complete the table for y = x² − 2 at x = −2, 0, 2.
x = −2: (−2)² − 2 = 2; x = 0: 0 − 2 = −2; x = 2: 4 − 2 = 2. So y-values are 2, −2, 2.
Example 2: Identifying shape
What shape is the graph of y = −x² + 3?
Because the x² term is negative, it is an ∩-shaped (upside-down) parabola with a maximum point.
Example 3: Reading a root
A parabola crosses the x-axis at x = 1 and x = 4. What does this tell you?
These are the solutions of the quadratic (where y = 0): x = 1 and x = 4.
Common mistakes and how to avoid them
Errors squaring/cubing negatives. (−3)² = 9 (positive); (−3)³ = −27 (negative).
Joining points with straight lines. Non-linear graphs need a smooth curve.
Wrong parabola orientation. Positive x² → U-shape; negative x² → ∩-shape.
Misreading scales. Check how much each square represents before reading values.
Forcing the curve through every dot rigidly. Draw a smooth curve; a stray point may signal an arithmetic error to recheck.
Exam technique for Non-linear Graphs
Complete the table accurately, watching negative values.
Plot points precisely and draw a smooth curve.
Recognise standard shapes — parabola, cubic S-shape, reciprocal pair of curves.
Read values and roots using the scales carefully.
Recheck any point that doesn't fit the smooth curve.
Quick revision summary
Non-linear graphs are curves. To plot one, complete a table of values by substituting x-values into the equation, taking care with negatives ((−3)² = 9, (−3)³ = −27), then plot the points and join them with a smooth curve — never straight segments. A quadratic gives a parabola: U-shaped if the x² coefficient is positive, ∩-shaped if negative, symmetrical about its turning point (a minimum or maximum). A cubic has an S-shape, and a reciprocal graph (y = a/x) has two separate curves that approach but never touch the axes. From a graph you can read off values, find roots where it crosses the x-axis (the solutions of the equation), and identify turning points. Watch for sign errors when squaring or cubing, draw smooth curves, recognise the standard shapes, and read the scales carefully. Recheck any point that breaks the smooth curve, since it usually signals an arithmetic slip in the table.