Algebra

Quadratic graphs

Q: What are the roots of a quadratic graph?

They are the x values where the curve crosses the x-axis. They are the solutions of the equation y = 0.

12 past-paper questions, easiest first. Try each one, then reveal the answer.

Foundation & Higher11 of 12 from Foundation papers

Revision guide by The Full Marks

Quadratic graphs: what you need to know

A quadratic graph, such as $y = x^2 - 2x - 3$ , is a smooth U-shaped curve called a parabola. Exam questions ask you to complete a table, draw the curve and read off its roots, turning point and $y$ -intercept.

Both tiers need this topic, including reading where the curve meets a horizontal line $y = k$ ; drawing a sloping straight line to solve a different equation from the same curve normally appears only on Higher papers.

Key facts

If the $x^2$ term is positive the curve is U-shaped; if negative it is upside down (∩).
Roots are where the curve crosses the $x$ -axis ( $y = 0$ ). The $y$ -intercept is where $x = 0$ .
The turning point is the lowest (or highest) point. It lies on the line of symmetry, halfway between the roots when there are two.
You can find the roots algebraically by factorising and setting each bracket to 0.

Drawing a quadratic graph

Complete the table, putting negative $x$ values in brackets, e.g. $(-2)^2 = 4$ .
Plot each point accurately.
Join them with one smooth curve, not straight lines, and keep the bottom rounded.

Finding key points algebraically

Roots: factorise and solve $= 0$ .
$y$ -intercept: the constant term.
Turning point: $x$ is halfway between the roots; substitute to find $y$ .

Worked examples

Example 1 Foundation · grade 4

Here is part of a table of values for $y = x^2 - 2x - 3$ : when $x = 0, 1, 2, 3, 4$ , $y = -3, -4, -3, 0, 5$ . (a) Find $y$ when $x = -2$ and when $x = -1$ . (b) Use the table to write down the roots and the turning point of the graph.

Show the solution

(a) $x = -2$ : $(-2)^2 - 2(-2) - 3 = 4 + 4 - 3 = 5$
$x = -1$ : $(-1)^2 - 2(-1) - 3 = 1 + 2 - 3 = 0$
(b) $y = 0$ when $x = -1$ and $x = 3$ , so the roots are $x = -1$ and $x = 3$ .
$y = -3$ at both $x = 0$ and $x = 2$ , so the line of symmetry is $x = 1$ .
The turning point is $(1, -4)$ .

Example 2 Foundation · grade 5

Without drawing, find the roots, $y$ -intercept and turning point of $y = x^2 + 2x - 8$ .

Show the solution

Factorise: $(x + 4)(x - 2) = 0$ , so the roots are $x = -4$ and $x = 2$ .
$y$ -intercept: $-8$ , so $(0, -8)$
Turning point $x = \frac{-4 + 2}{2} = -1$
$y = (-1)^2 + 2(-1) - 8 = -9$
Turning point $(-1, -9)$

Common mistakes

Working out $(-1)^2$ as $-1$ . A negative number squared is positive; on a calculator, use brackets.
Getting the sign of $-2x$ wrong for negative $x$ : when $x = -2$ , $-2x = -2 \times (-2) = +4$ .
Giving the $y$ -intercept when asked for roots. Roots are on the $x$ -axis.
Joining the points with straight lines or a flat bottom. Draw one smooth curve; if a point is out of line with the curve, recheck it in the table.

Exam tips

Roots read from a graph are estimates, so answers within a small range of the true value get the marks.
To solve an equation like $x^2 - 2x - 3 = 2$ from the graph, read the $x$ values where the curve meets the line $y = 2$ .

Questions people ask

What are the roots of a quadratic graph?

They are the $x$ values where the curve crosses the $x$ -axis. They are the solutions of the equation $y = 0$ .

How do you find the turning point of a quadratic?

Read the lowest or highest point from the graph. Algebraically, its $x$ value is halfway between the roots; Higher students can also complete the square.

Exam questions: Quadratic graphs (12, easiest first)

Quadratic graphs4 marksGrade 4CalculatorOn both tiers

November 2024 · Paper 2 Foundation · Q24

Complete a table of values for a quadratic and draw its graph.

Show answer

(a)	$5, -3, -4, 0, 5$ (for $x = -2, 0, 1, 3, 4$ )	2 marks
(b)	Smooth U-shaped curve through the points	2 marks

Worked solution Checked against the mark scheme

(a) $x = -2$ : $4 + 4 - 3 = 5$ ; $x = 0$ : $-3$ ; $x = 1$ : $1 - 2 - 3 = -4$ ; $x = 3$ : $9 - 6 - 3 = 0$ ; $x = 4$ : $16 - 8 - 3 = 5$ .

(b) Plot $(-2, 5), (-1, 0), (0, -3), (1, -4), (2, -3), (3, 0), (4, 5)$ and join them with a smooth curve.

Watch out: Working out $(-2)^2$ as $-4$ , or joining the points with straight lines.

Quadratic graphs6 marksGrade 4CalculatorOn both tiers

June 2024 · Paper 2 Foundation · Q24

Complete a table for a quadratic, draw its graph and use it to estimate solutions of an equation.

Show answer

(a)	$y$ values: $6,\ 2,\ 0,\ 0,\ 2,\ 6$ (missing: $2$ at $x=-1$ , $0$ at $x=1$ , $6$ at $x=3$ )	2 marks
(b)	Smooth U-shaped curve through the points	2 marks
(c)	$x \approx -1.6$ and $x \approx 2.6$ (accept $-1.7$ to $-1.5$ and $2.5$ to $2.7$ )	2 marks

Worked solution Checked against the mark scheme

(a) $x = -1$ : $1 + 1 = 2$ . $x = 1$ : $1 - 1 = 0$ . $x = 3$ : $9 - 3 = 6$ .

(b) Plot $(-2,6), (-1,2), (0,0), (1,0), (2,2), (3,6)$ and join with a smooth curve.

Watch out: Working out $(-1)^2 - (-1)$ as 0 instead of 2.

Quadratic graphs4 marksGrade 4CalculatorFoundation

November 2023 · Paper 2 Foundation · Q20

Show answer

(a)	$0, -2, 0, 4$ (for $x = -1, 0, 2, 3$ )	2 marks
(b)	A smooth U-shaped curve through $(-2,4), (-1,0), (0,-2), (1,-2), (2,0), (3,4)$	2 marks

Worked solution Checked against the mark scheme

(a) Substitute each $x$ into $x^2 - x - 2$ :

$x = -1$ : $1 + 1 - 2 = 0$ ; $x = 0$ : $-2$ ; $x = 2$ : $4 - 2 - 2 = 0$ ; $x = 3$ : $9 - 3 - 2 = 4$

(b) Plot the six points and join them with a smooth curve. The lowest point is between $x = 0$ and $x = 1$ , just below $-2$ .

Watch out: Working out $(-1)^2$ as $-1$ , giving the wrong value at $x = -1$ .

Quadratic graphs2 marksGrade 4CalculatorOn both tiers

June 2023 · Paper 3 Foundation · Q26

Read the turning point and estimate a root from the graph of a quadratic.

Show answer

(a)	$(1, -3)$	1 mark
(b)	$-0.7$ or $2.7$	1 mark

Worked solution Checked against the mark scheme

(a) The lowest point of the curve is at $(1, -3)$ .

(b) The roots are where the curve crosses the $x$ -axis: about $x = -0.7$ or $x = 2.7$ .

Watch out: In (b), reading where the curve crosses the $y$ -axis ( $-2$ ).

Quadratic graphs4 marksGrade 4CalculatorOn both tiers

November 2022 · Paper 2 Foundation · Q24

Read the y-intercept, turning point and approximate roots from the graph of a quadratic.

Show answer

(a)	4	1 mark
(b)	$(3, -5)$	1 mark
(c)	$x \approx 0.8$ and $x \approx 5.2$	2 marks

Worked solution Checked against the mark scheme

(a) The curve crosses the y-axis where $x = 0$ : $y = 4$ .

(b) The lowest point is at $(3, -5)$ .

Watch out: Giving the roots as coordinates on the y-axis.

Quadratic graphs6 marksGrade 4CalculatorOn both tiers

November 2021 · Paper 2 Foundation · Q24

Complete a table for a quadratic, draw its graph and use the graph to solve a quadratic equation.

Show answer

(a)	5, 1, 2, 10	2 marks
(b)	A smooth curve through the plotted points	2 marks
(c)	$x \approx -0.7$ and $x \approx 2.7$	2 marks

Worked solution Checked against the mark scheme

(a) $x=-1$ : $1 + 2 + 2 = 5$ . $x=1$ : $1 - 2 + 2 = 1$ . $x=2$ : $4 - 4 + 2 = 2$ . $x=4$ : $16 - 8 + 2 = 10$ .

(b) Plot $(-2,10), (-1,5), (0,2), (1,1), (2,2), (3,5), (4,10)$ and join with a smooth curve.

Watch out: Working out $(-1)^2$ as $-1$ .

Quadratic graphs6 marksGrade 4CalculatorOn both tiers

June 2018 · Paper 2 Foundation · Q24

Complete a table for a quadratic, plot its graph and use it to estimate solutions of an equation.

Show answer

(a)	$0, -4, -6, -4, 0$ (for $x = -2, -1, 1, 2, 3$ )	2 marks
(b)	Smooth U-shaped curve through the points	2 marks
(c)	$x \approx 2.6$ and $x \approx -1.6$	2 marks

Worked solution Checked against the mark scheme

(a) e.g. $x = -2$ : $4 + 2 - 6 = 0$ ; $x = 1$ : $1 - 1 - 6 = -6$ ; $x = 2$ : $4 - 2 - 6 = -4$

(b) Plot $(-3, 6), (-2, 0), (-1, -4), (0, -6), (1, -6), (2, -4), (3, 0)$ and join with a smooth curve.

Watch out: In (a), working out $-x$ wrongly for negative $x$ , e.g. $(-2)^2 - (-2) = 2$ .

Quadratic graphs5 marksGrade 4CalculatorOn both tiers

November 2018 · Paper 3 Foundation · Q22

Complete a table of values for a quadratic, draw its graph and use it to estimate a root.

Show answer

(a)	$2, -4, 2, 8$ (for $x = -3, -1, 2, 3$ )	2 marks
(b)	Smooth U-shaped curve through the points	2 marks
(c)	$x \approx 1.6$ or $x \approx -2.6$	1 mark

Worked solution Checked against the mark scheme

(a) $x = -3$ : $9 - 3 - 4 = 2$ ; $x = -1$ : $1 - 1 - 4 = -4$ ; $x = 2$ : $4 + 2 - 4 = 2$ ; $x = 3$ : $9 + 3 - 4 = 8$

(b) Plot all seven points and join with a smooth curve, dipping just below $-4$ between $x = -1$ and $x = 0$ .

Watch out: In (a), working out $-3^2$ as $-9$ .

Quadratic graphs6 marksGrade 5No calculatorOn both tiers

June 2022 · Paper 1 Foundation · Q28

Complete a table of values for a quadratic, plot its graph, and read off where it crosses the x-axis.

Show answer

(a)	5, −1, 1, 5 (for x = −1, 2, 3, 4)	2 marks
(b)	Smooth U-shaped curve through the plotted points	2 marks
(c)	$x \approx 0.4$ and $x \approx 2.6$	2 marks

Worked solution Checked against the mark scheme

(a) $x = -1$ : $1 + 3 + 1 = 5$ ; $x = 2$ : $4 - 6 + 1 = -1$ ; $x = 3$ : $9 - 9 + 1 = 1$ ; $x = 4$ : $16 - 12 + 1 = 5$

(b) Plot $(-1,5), (0,1), (1,-1), (2,-1), (3,1), (4,5)$ and join with a smooth curve.

Watch out: Working out $(-1)^2$ as $-1$ , giving the wrong value at x = −1.

Quadratic graphs3 marksGrade 5CalculatorOn both tiers

November 2020 · Paper 3 Foundation · Q28

Read the solution of simultaneous equations from the intersection of two lines, and estimate the roots of a quadratic from its graph.

Show answer

(a)	$x = -2$ , $y = 4$	1 mark
(b)	$x \approx 0.6$ and $x \approx 3.4$ (accept 0.55 to 0.65 and 3.35 to 3.45)	2 marks

Worked solution Checked against the mark scheme

(a) The lines cross at $(-2, 4)$ , so $x = -2$ , $y = 4$ .

Check: $5(-2) - 9(4) = -46$ ✓

(b) Read where the curve meets the $x$ -axis: $x \approx 0.6$ and $x \approx 3.4$

Watch out: In (b), reading where the curve crosses the $y$ -axis (2).

Quadratic graphs3 marksGrade 5No calculatorFoundation

June 2019 · Paper 1 Foundation · Q29

Read the turning point and the roots of a quadratic from its graph.

Show answer

(a)	$(1, -4)$	1 mark
(b)	$x = -1$ and $x = 3$	2 marks

Worked solution Checked against the mark scheme

(a) The lowest point of the curve is at $(1, -4)$ .

(b) The roots are where the curve crosses the $x$ -axis ( $y = 0$ ): $x = -1$ and $x = 3$ .

Check: $(-1)^2 - 2(-1) - 3 = 0$ and $3^2 - 6 - 3 = 0$

Watch out: Giving the $y$ -intercept, $-3$ , as a root.

Quadratic graphs4 marksGrade 8CalculatorHigher

November 2019 · Paper 3 Higher · Q17

Draw a suitable straight line on a given quadratic graph to estimate the roots of a different quadratic equation.

Show answer

Answer	$x \approx 2.7$ and $x \approx -0.7$	4 marks

Worked solution Checked against the mark scheme

$x^2 - 2x - 2 = 0$ is the same as $x^2 - 3 = 2x - 1$ .

Draw $y = 2x - 1$ on the graph of $y = x^2 - 3$ .

The line and curve cross at about $x = 2.7$ and $x = -0.7$ .

Watch out: Drawing $y = 2x + 1$ instead of $y = 2x - 1$ .

Also tested in these questions

Their main topic is different, but part of each one tests this topic.

November 2024 · Paper 3 Higher · Q23: Complete the square to find the turning point of a quadratic involving a constant p, then sketch it with its y-intercept.