Quadratic graphs

Foundation and Higher

The idea

The graph of $y = x^2 + bx + c$ is a smooth U-shaped curve. Its roots are where it meets the $x$ -axis. When it crosses twice, the turning point sits halfway between the roots.

Make a table of values, including negative $x$ , and plot the points.
Join them with one smooth curve.
Read the roots where $y = 0$ ; with two roots, the turning point's $x$ -value is midway between them.

Watch out Draw a smooth curve through the points, never straight lines between them.

See it worked

Worked exampleNon-calculator2 marks

Here is the equation of a curve: $y = x^2 + 1$ .

(a) Work out the value of $y$ when $x = -2$ .

(b) Which point is the turning point of the curve?

(a) $(-2)^2 + 1 = 4 + 1 = 5$ .
(b) $x^2$ is smallest when $x = 0$ , so the lowest point is $(0, 1)$ .

Answer (a) $5$ ; (b) $(0, 1)$

Your turn

Question 1Non-calculator1 mark

The graph of y = (x + 1)(x $-$ 5) crosses the x-axis at two points. What are the x-coordinates of these points?

Question 2Non-calculator4 marks

A ball is thrown from ground level. Its height, h metres, after t seconds is $h = 12t - 3t^2$ for $0 \le t \le 4$ .

(a) Find the two times when the ball is at ground level.

(b) Find the greatest height of the ball.

Question 3Non-calculator5 marks

The curve $y = (x - 3)(x + 5)$ is a parabola.

(a) Write down the coordinates of the points where the curve crosses the $x$ -axis.

(b) Find the coordinates of the turning point of the curve.

In past papers

92 official questions in 88 papers since 2017. The latest:

All GCSE Maths past papers

GCSE Maths Pro

£119 once

At least 12 months of access

By paying you agree to our Terms and get access straight away, which ends your 14-day cancellation right.

Start free

No VAT is charged.