Quadratic Graphs — AQA GCSE Maths Revision Notes

What you'll learn

This topic covers the graphs of quadratic functions: the symmetrical curves given by y = ax² + bx + c. By the end of this guide you should be able to plot one from a table of values, describe its shape, and find its roots, its y-intercept, its turning point and its line of symmetry.

You should also be able to read solutions of an equation off a graph, say how many roots a curve has without drawing it, and interpret a quadratic graph set in a real context such as the flight of a projectile.

The organising idea is symmetry. A parabola is a mirror image of itself about a vertical line through its turning point, and almost every feature can be found from that one fact. The line of symmetry sits exactly halfway between the roots. The turning point sits on that line. Two points at the same height are always the same distance from it. Once you find the line of symmetry, the rest of the curve's features follow with very little extra work.

Key terms and definitions

Quadratic graph (parabola) — the curve of y = ax² + bx + c, always symmetrical.

Roots (or zeros) — the x-values where the curve crosses the x-axis, so where y = 0.

y-intercept — where the curve crosses the y-axis, found by putting x = 0.

Turning point (vertex) — the lowest point of a U-shaped curve or the highest point of a ∩-shaped one.

Minimum / maximum — the two kinds of turning point.

Line of symmetry — the vertical line through the turning point, written as x = a number.

Core concepts

The shape, and what decides it

The sign of the x² coefficient decides which way up the curve sits, and nothing else does.

A positive x² term gives a U-shaped curve with a minimum turning point. A negative x² term gives a ∩-shaped curve with a maximum.

So y = x² − 4x + 1 opens upwards, while y = −x² + 3 opens downwards. The other terms shift and stretch the curve but never flip it.

Plotting from a table of values

Substitute each x-value into the equation and record the y-value, then plot the points and join them with a smooth curve — never with straight line segments, and never as a V at the bottom.

The care is all in the negatives. For y = x² − 3 at x = −2: square first, so (−2)² = 4, then 4 − 3 = 1. Writing −2² and getting −4 is the single commonest table-of-values error, because the index attaches only to the 2 unless brackets say otherwise.

Choose enough x-values to show the turning point and both crossings. A table that stops before the curve turns hides the most important feature on the graph.

The y-intercept

Put x = 0. Every term with an x in it disappears, so the y-intercept is simply the constant term.

For y = x² + 5 the curve crosses the y-axis at 5; for y = x² − 4 it crosses at −4. This takes no working at all and is often worth a mark.

Roots

The roots are where y = 0, which means solving the quadratic equation.

When the expression factorises, the roots can be read straight off. For y = (x − 2)(x − 5), the curve is zero when either bracket is zero, so the roots are 2 and 5. Note the sign change: a bracket of (x − 2) gives a root at positive 2.

For y = x² − 9, the roots are 3 and −3. For y = x² − 6x + 8, factorising to (x − 2)(x − 4) gives roots at 2 and 4.

How many roots?

A parabola can meet the x-axis twice, once, or not at all.

Two roots: the curve crosses the axis at two separate points.

One repeated root: the turning point sits exactly on the axis, so the curve touches without crossing.

No real roots: the curve stays entirely above or entirely below the axis. This is what happens with y = x² + 1, whose lowest point is at y = 1, so it never reaches zero.

You can often answer this by thinking about the turning point rather than by calculating. A U-shaped curve whose minimum is above the axis has no roots.

The turning point and the line of symmetry

The line of symmetry is the vertical line through the turning point, and the quickest way to find it is to average the roots.

For roots at 2 and 5, the line of symmetry is x = 3.5. For roots at 3 and −3, it is x = 0 — which is why y = x² is symmetrical about the y-axis.

Once you have that x-value, substitute it back into the equation to find the turning point's height. For y = x² − 6x + 8 with roots 2 and 4, the line of symmetry is x = 3, and substituting gives y = 9 − 18 + 8 = −1, so the minimum is at (3, −1).

Where the equation is written in completed-square form, the turning point can be read directly: y = (x − 3)² − 1 has its turning point at (3, −1), with the signs reversed inside the bracket and kept outside.

Reading solutions from a graph

A drawn curve answers equations as well as showing shape. The solutions of ax² + bx + c = 0 are where the curve meets the x-axis.

To solve something like x² − 4 = 2, draw the horizontal line y = 2 and read off where it crosses the curve. The question tells you which line to draw by what sits on the right-hand side.

Quadratics in context

A quadratic often models the height of a thrown or launched object against time. The y-intercept gives the starting height, the maximum point gives the greatest height reached and when it happened, and the roots give the times at which the height is zero — typically launch and landing.

Only part of the curve usually makes sense in context. Negative time, or height below ground, should be discounted when answering.

Worked examples

Example 1: Finding every feature from a factorised form

For y = (x − 1)(x − 7), find the roots, the line of symmetry, the turning point and the y-intercept.

The roots are where each bracket is zero: x = 1 and x = 7.

The line of symmetry is halfway between them: (1 + 7) ÷ 2, so x = 4.

Substituting x = 4 gives y = (3)(−3) = −9, so the turning point is (4, −9). The x² term is positive, so this is a minimum.

The y-intercept comes from x = 0: y = (−1)(−7) = 7.

Example 2: Counting roots without drawing

How many times does the graph of y = x² + 2 cross the x-axis?

The x² term is positive, so the curve is U-shaped with a minimum.

The smallest possible value of x² is 0, at x = 0, so the smallest value of y is 2.

The whole curve therefore lies at or above y = 2, never reaching zero. It crosses the x-axis no times, so the equation has no real roots.

Example 3: A table of values

Complete the table for y = x² − 3 at x = −2, −1, 0, 1, 2.

Square each value first, then subtract 3.

At x = −2: 4 − 3 = 1. At x = −1: 1 − 3 = −2. At x = 0: 0 − 3 = −3. At x = 1: −2. At x = 2: 1.

The values read 1, −2, −3, −2, 1 — symmetrical about x = 0, exactly as the shape of the curve requires. That symmetry is a free check on the arithmetic: if the two ends do not match, a negative has been squared wrongly.

Common mistakes and how to avoid them

Squaring a negative incorrectly. (−3)² is 9, not −9. Square before applying any other operation in the table.

Joining the points with straight lines. A parabola is a smooth curve, and a sharp point at the bottom loses the drawing mark.

Getting the sign of a root wrong. The bracket (x − 2) gives a root at +2, because that is the value making the bracket zero.

Confusing the turning point with the y-intercept. These coincide only when the line of symmetry is the y-axis.

Assuming every parabola has roots. A curve whose turning point lies above the axis has none at all.

Stopping the table before the turning point. Extend the range of x until the curve has clearly turned.

Giving the line of symmetry as a number. It is a line, so the answer is x = 3, not 3.

Exam technique for "Quadratic Graphs"

Find the roots first when the quadratic factorises. They give the line of symmetry by averaging, and the line of symmetry gives the turning point by substitution, so one piece of work produces three answers.

Write the y-intercept straight down from the constant term. It takes no working and is frequently worth a mark.

Use the symmetry of the table as a check. Matching values either side of the turning point mean the arithmetic is almost certainly right.

When asked to use a graph to solve an equation, draw the horizontal line the right-hand side names and mark the crossing points clearly. The marks are for the line and the readings, not for algebra.

In a context question, state the units and reject any answer that makes no physical sense, such as a negative time.

Answer "how many roots" by thinking about the turning point rather than by solving, when the shape makes it obvious.

Quick revision summary

A quadratic graph is a symmetrical parabola. A positive x² term gives a U shape with a minimum; a negative one gives a ∩ shape with a maximum.

The y-intercept is the constant term, found by putting x = 0.

The roots are where y = 0. From a factorised form they are read off with the signs reversed, so (x − 2)(x − 5) has roots at 2 and 5.

The line of symmetry is halfway between the roots — average them — and the turning point sits on it, found by substituting that x-value back in. In completed-square form, y = (x − 3)² − 1 gives the turning point directly as (3, −1).

A parabola has two roots, one repeated root, or none, according to whether the turning point is below, on, or above the x-axis.

When plotting, square negatives carefully, extend the table past the turning point, and join the points with a smooth curve.

In context, the y-intercept is the starting value, the maximum is the greatest value reached, and the roots are when the quantity is zero.