Plotting and Interpreting Quadratic, Cubic and Reciprocal Graphs — AQA GCSE Maths Revision Notes
What you'll learn
This topic covers the curved graphs on the specification: quadratics, cubics, reciprocals and exponentials. By the end of this guide you should be able to complete a table of values, plot the points accurately and draw a smooth curve through them.
You should also be able to recognise each type of graph from its equation or its shape, identify key features such as roots and turning points, use a graph to estimate solutions, and avoid the plotting errors that cost the most marks.
The organising idea is that the highest power tells you the shape before you plot a single point. An x² term gives one turning point and a symmetrical parabola. An x³ term gives up to two turning points and a curve that runs from bottom-left to top-right. A term with x on the bottom gives two separate branches that never touch the axes. Knowing the expected shape first means the table of values is checking a picture you already have in mind, rather than producing one blindly — and a plotted point that breaks the shape is then obviously an arithmetic slip rather than a feature.
Key terms and definitions
Quadratic — an equation with x² as its highest power, giving a parabola.
Cubic — an equation with x³ as its highest power.
Reciprocal — an equation with x in the denominator, such as y = 1/x.
Exponential — an equation with x in the index, such as y = 2ˣ.
Root — a value of x where the curve crosses the x-axis, so where y = 0.
Turning point — a maximum or minimum where the curve changes direction.
Asymptote — a line the curve approaches but never reaches.
Core concepts
Completing a table of values
Substitute each x-value into the equation and record the y-value.
The care is entirely in the negatives. For y = x² − 3 at x = −4: square first, giving 16, then subtract 3 to get 13. Writing −4² and reading it as −16 is the classic error, because the index attaches only to the 4 unless brackets say otherwise.
Cubes keep their sign, so (−3)³ = −27 while (−3)² = +9. That difference is what makes cubic tables harder than quadratic ones.
A useful check on a quadratic table is symmetry: the y-values either side of the turning point must match. If they do not, one of them is wrong.
Plotting and drawing
Plot each point carefully and join them with a smooth curve, drawn freehand in a single confident stroke rather than as a series of straight segments.
Three faults lose accuracy marks: joining with straight lines, drawing a sharp point at a turning point instead of a smooth curve, and forcing the curve through a plotted point that is clearly out of line with the rest. That last one is worth stating plainly — if one point breaks an otherwise smooth pattern, it is almost certainly a table error, and the curve should be drawn through the others.
Turn the paper so your hand moves comfortably along the curve, and use a sharp pencil.
Quadratic graphs
A quadratic gives a parabola, symmetrical about a vertical line through its turning point.
A positive x² term opens upwards with a minimum; a negative x² term opens downwards with a maximum.
The roots are where it crosses the x-axis, the y-intercept is the constant term, and the line of symmetry sits halfway between the roots.
A parabola may cross the x-axis twice, touch it once, or miss it entirely, depending on where its turning point sits.
Cubic graphs
A cubic has x³ as its highest power and a characteristic double bend.
With a positive x³ term the curve comes up from the bottom left and continues to the top right. With a negative x³ term it runs the other way, from top left down to bottom right.
A cubic may have two turning points, or none at all if the curve rises throughout — y = x³ is the simplest example, rising steadily with only a flattening at the origin.
A cubic crosses the x-axis at least once, and may cross up to three times, which is why cubic equations can have three solutions.
Reciprocal graphs
A reciprocal graph such as y = 1/x has x in the denominator, and its defining feature is what happens near zero.
The curve has two separate branches and never touches either axis. Dividing by zero is undefined, so there is no point at x = 0, and as x grows large the value shrinks towards zero without reaching it.
Both axes are therefore asymptotes — lines the curve approaches indefinitely.
For y = 1/x the branches sit in the top-right and bottom-left. For y = −1/x they sit in the other two quadrants.
The commonest plotting error is joining the two branches through the origin. They are separate, and the curve must break at x = 0.
Exponential graphs
An exponential has x in the index, as in y = 2ˣ.
The curve rises increasingly steeply as x increases, and approaches the x-axis without meeting it as x becomes negative — so the x-axis is an asymptote.
Every exponential of the form y = aˣ passes through (0, 1), since anything to the power zero is 1.
These model growth and decay, so context questions about populations, investments or radioactive decay usually involve one.
Recognising the type
From the equation, look at the highest power of x, or whether x sits on the bottom or in the index.
From the shape: one bend means quadratic; two bends means cubic; two separate branches avoiding both axes means reciprocal; a single curve rising steeply from a flat approach means exponential.
Questions frequently show several sketches and ask which matches a given equation, and the shape alone settles it.
Using a graph to solve equations
A drawn curve answers equations without algebra.
The roots of the equation are where the curve crosses the x-axis, so y = x² − 4 crossing at −2 and 2 tells you the solutions of x² − 4 = 0.
To solve something else, draw the appropriate line and read off the crossings. For x² − 4 = 3, draw y = 3 and read where it meets the curve.
Answers read from a graph are estimates, and questions acknowledge this by asking for a value to one decimal place rather than an exact one.
Interpreting in context
Curved graphs model real situations: the height of a projectile against time, the volume of a container against depth, or the cooling of a drink.
The maximum gives the greatest value reached, the roots give when the quantity is zero, and the y-intercept gives the starting value.
Only part of the curve usually makes sense — negative times or negative heights should be discounted — and saying so is often worth a mark.
Worked examples
Example 1: A table of values with negatives
Complete the table for y = x³ − 2x at x = −2, −1, 0, 1, 2.
At x = −2: (−2)³ = −8, and 2x = −4, so y = −8 − (−4) = −4.
At x = −1: (−1)³ = −1, and 2x = −2, so y = −1 + 2 = 1.
At x = 0: y = 0.
At x = 1: 1 − 2 = −1.
At x = 2: 8 − 4 = 4.
The values are −4, 1, 0, −1, 4. Note that this cubic has two turning points, visible in the rise to 1 and the dip to −1, which is the shape the highest power predicted.
Example 2: Recognising a graph
A curve has two separate branches, one in the top-right and one in the bottom-left, and never touches either axis. Which equation does it match: y = x², y = 1/x, y = 2ˣ or y = x³?
Two separate branches that avoid both axes is the signature of a reciprocal graph, so the answer is y = 1/x.
The others are ruled out by shape: y = x² is a single parabola, y = x³ is one continuous curve through the origin, and y = 2ˣ is a single rising curve passing through (0, 1).
Example 3: Using a graph to solve an equation
The graph of y = x² − x − 3 has been drawn. Explain how to use it to solve x² − x − 3 = 2, and say why the answer is approximate.
Draw the horizontal line y = 2 across the grid.
Read the x-coordinates where that line crosses the curve. Those two values are the solutions.
They are approximate because the readings depend on the accuracy of the drawn curve and on reading a scale by eye — which is why such questions ask for an answer to one decimal place rather than an exact value.
Common mistakes and how to avoid them
Mishandling negatives in the table. Square or cube first: (−3)² = 9 but (−3)³ = −27.
Joining points with straight lines. These are smooth curves.
Drawing a sharp point at a turning point. The curve turns smoothly.
Forcing the curve through an off-line point. Recheck that value; it is almost certainly a table error.
Joining the two branches of a reciprocal graph. They are separate and the curve breaks at x = 0.
Expecting a reciprocal or exponential curve to touch an axis. Those lines are asymptotes.
Treating a graphical solution as exact. Readings from a graph are estimates.
Exam technique for "Plotting and Interpreting Curved Graphs"
Decide the expected shape from the highest power before plotting anything, so an error in the table shows up immediately as a point out of place.
Check a quadratic table for symmetry either side of the turning point. It is a free check on half the values.
Draw curves freehand in one stroke with a sharp pencil, turning the page if it helps.
When solving an equation graphically, draw the required line clearly and mark the crossing points. Marks are awarded for the line as well as for the readings.
Give graphical answers to the accuracy the question asks for, and describe them as estimates.
In context questions, state the units and reject any part of the curve that makes no physical sense.
Quick revision summary
The highest power tells you the shape. x² gives a parabola with one turning point; x³ gives a cubic with up to two; x on the bottom gives a reciprocal with two separate branches; x in the index gives an exponential.
In a table of values, square or cube before anything else: (−3)² = 9 but (−3)³ = −27. Check a quadratic table for symmetry.
Join points with a smooth freehand curve, never straight segments, and never force the curve through a point that breaks the pattern.
A reciprocal graph never touches either axis — both are asymptotes — and its two branches must not be joined.
An exponential y = aˣ always passes through (0, 1) and approaches the x-axis without meeting it.
To solve an equation from a graph, draw the line the right-hand side names and read the crossings. The answers are estimates.
In context, the maximum is the greatest value, the roots are when the quantity is zero, and the y-intercept is the starting value.