Quadratic graphs

Quadratic graphs are the graphs of equations like y = x² − 2x − 3, where the highest power of x is 2. They are always a smooth U-shaped curve called a parabola. If the x² term is positive the curve is a U with a minimum point. If it is negative the curve is upside down, an ∩ shape with a maximum point. The key features are the roots, the y-intercept, the turning point and the line of symmetry.

What a quadratic graph looks like

Every quadratic can be written as y = ax² + bx + c, with a not zero. Drag the sliders to change a, b and c and watch the curve move. Make a negative and the parabola flips. Change b and the turning point slides sideways. Change c and the whole curve moves up or down.

Here x and y are the usual coordinates. It starts on y = x² − 2x − 3, the curve used in the examples below.

The features of a quadratic graph

The line of symmetry is always halfway between the two roots. Here, halfway between −1 and 3 is 1. AQA's GCSE Maths specification asks students to identify and interpret roots, intercepts and turning points graphically, including this symmetry.

How to plot a quadratic graph

To plot y = x² − 2x − 3 for x from −2 to 4, make a table of values first. Work out each y carefully: with negative x, squaring gives a positive number, so (−2)² = 4.

1. Plot each (x, y) pair as a small cross: (−2, 5), (−1, 0), (0, −3) and so on.
2. Join the points with a smooth curve, not straight lines. The bottom should be rounded, not pointed.
3. Check the symmetry: the y values should mirror each other either side of the turning point. Here 5, 0, −3 on one side match −3, 0, 5 on the other.

How to sketch a quadratic graph

A sketch doesn't need a table: it shows the shape and the key points, labelled. Work through four steps.

Step 1: which way does it open?

Look at the sign of a, the number in front of x². Positive gives a U with a minimum. Negative gives an ∩ with a maximum.

Step 2: find the y-intercept

Put x = 0. Everything vanishes except c, so the y-intercept is (0, c).

Step 3: find the roots

Set y = 0 and solve. Factorise if you can: x² − 2x − 3 factorises to (x + 1)(x − 3). One bracket must be zero, giving roots of −1 and 3. If it won't factorise, use the quadratic formula:

The part under the square root, b² − 4ac, is the discriminant. Positive means two roots, so the curve crosses the x-axis twice. Zero means one repeated root: the curve just touches the axis. Negative means no real roots: the curve never meets the x-axis. Try one with the formula, then press Mark to check in the course.

Step 4: find the turning point

The turning point is on the line of symmetry, x = −b ÷ 2a. For y = x² − 2x − 3, x = 2 ÷ 2 = 1, and putting x = 1 into the equation gives y = −4. At GCSE Higher, you can also find it by completing the square:

In the form (x − p)² + q, the turning point is (p, q), so here it is (1, −4). Watch the sign: (x − 1)² gives p = 1, positive. Now plot the four features, draw a smooth U through them, and label each one.

A negative quadratic graph

Sketch y = −x² + 4x + 5. Here a = −1, so it is an ∩ shape. The y-intercept is (0, 5). Factorise for the roots: −x² + 4x + 5 = −(x − 5)(x + 1), so x = 5 or x = −1. The line of symmetry is halfway between, x = 2, and at x = 2, y = −4 + 8 + 5 = 9. The maximum point is (2, 9).

Solving equations with a quadratic graph

A graph gives approximate solutions to equations, which AQA's specification includes alongside solving by factorising, completing the square and the formula.

• Equal to 0: the answers are the roots, the x-intercepts, here −1 and 3.
• x² − 2x − 3 = 2: draw the horizontal line y = 2 and read where it crosses the curve. The solutions are about x = −1.4 and x = 3.4. (The exact answers are 1 ± √6, which are −1.45 and 3.45 to 2 decimal places.)
• x² − 2x − 3 = −5: the line y = −5 is below the turning point at −4, so it never meets the curve. There are no real solutions.

The quadratic graph calculator plots any quadratic and marks its roots, turning point and intercepts, so you can check a sketch or a reading.

Common mistakes

• Squaring negatives wrongly. (−2)² = 4, not −4. On a calculator, use brackets.
• Joining points with a ruler. Quadratic graphs are smooth curves with a rounded bottom.
• Getting the turning point sign wrong. (x + 3)² − 1 has its turning point at (−3, −1).
• Unlabelled sketches. A sketch should show the intercepts and turning point with their coordinates.
• Misreading the scale. Check what one small square is worth on each axis before reading solutions.

To practise graphs among other topics in a short timed paper, try today's GCSE Maths Higher or GCSE Maths Foundation paper. The Graphing Functions course builds up from straight lines to parabolas, with an explorer for every part of the curve.

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Maths · AlgebraQuadratic formula calculator: roots, graph and workingSolve any quadratic equation with the quadratic formula: enter a, b and c to see the roots and the parabola, then follow the full working by hand.