A11, A12AlgebraFoundation & HigherGrades 4–7

Quadratic & Other Graphs

Q: How do I know whether the turning point is a maximum or a minimum?

Look at the sign of the x^2 term. Positive gives a U-shape with a minimum; negative gives a ∩-shape with a maximum.

Q: Do exponential graphs ever touch the x-axis?

No. A positive number raised to any power is positive, so y = 2^x gets closer and closer to zero as x decreases but never reaches it.

The graph of a quadratic such as $y = x^2 - 2x - 3$ is a smooth U-shaped curve called a parabola. Reading it tells you the solutions of the equation, where the curve crosses the axes, and its lowest or highest point. You also need to recognise the shapes of cubic, reciprocal and exponential graphs at a glance.

45 min AQA GCSE Maths

The big picture

A graph is a picture of every solution of its equation at once. The points where $y = x^2 - 2x - 3$ meets the $x$ -axis are where $y = 0$ — so reading those $x$ -values solves $x^2 - 2x - 3 = 0$ without any algebra.

Each family of graphs has a signature shape set by its highest power. Knowing those shapes lets you sketch quickly, check plotted work, and match equations to graphs — a regular AQA multiple-choice style question.

What you'll learn

Plot quadratic graphs from a table of values
Identify roots, the $y$ -intercept, the line of symmetry and the turning point
Solve quadratic equations approximately from a graph
Recognise and sketch cubic, reciprocal and exponential graphs

Plotting a quadratic

Make a table of values, substituting each $x$ carefully — square negative numbers in brackets. Plot the points and join them with a $\textbf{smooth curve}$ , not straight lines.

If the $x^2$ coefficient is positive the curve is U-shaped (a minimum point); if negative it is ∩-shaped (a maximum point).

Example 1Complete the table for

y = x^2 - 2x - 3

for

x = -2

4

x = -2

4 + 4 - 3 = 5

x = -1

1 + 2 - 3 = 0

x = 0

-3

x = 1

-4

x = 2

-3

x = 3

0

x = 4

5

Answer

y = 5, 0, -3, -4, -3, 0, 5

Your turnSame method, new numbers — try it before moving on.

For $y = x^2 + 3x - 1$ , find $y$ when $x = -4$ .

Your answer

Show hint

$(-4)^2 = 16$ and $3 \times (-4) = -12$ .

Show worked solution

Solution

1 $(-4)^2 + 3(-4) - 1 = 16 - 12 - 1 = 3$ .

Answer: $y = 3$

Tip — The $y$ -values of a quadratic table are symmetrical about the turning point — if they are not, recheck one.

Reading key features

$\textbf{Roots}$ are the $x$ -values where the curve crosses the $x$ -axis. The $\textbf{y-intercept}$ is where it crosses the $y$ -axis — the constant term.

The $\textbf{line of symmetry}$ is the vertical line halfway between the roots. The $\textbf{turning point}$ lies on it.

y = x^2 - 2x - 3

: roots

-1

and

3

y

-intercept

-3

, line of symmetry

x = 1

, turning point

(1, -4)

Example 1A quadratic graph has roots at

x = -5

and

x = 1

. Find the equation of its line of symmetry.

1Halfway between the roots:

\frac{-5 + 1}{2} = -2

Answer

x = -2

Your turnSame method, new numbers — try it before moving on.

A parabola crosses the $x$ -axis at $x = 2$ and $x = 8$ . What is the $x$ -coordinate of its turning point?

Your answer

Show worked solution

Solution

1The turning point is on the line of symmetry, halfway between the roots: $\frac{2 + 8}{2} = 5$ .

Answer: $x = 5$

Solving equations with graphs

To solve $f(x) = 0$ , read the $x$ -values where the curve meets the $x$ -axis. To solve $f(x) = k$ , draw the horizontal line $y = k$ and read the $x$ -values where it meets the curve.

Graphical solutions are estimates, so give them to a sensible accuracy, such as 1 decimal place.

Example 1Use the graph of

y = x^2 - 2x - 3

to solve

x^2 - 2x - 3 = 5

1Draw

y = 5

across the graph.

2It meets the curve where

x = -2

and

x = 4

(check:

16 - 8 - 3 = 5

✓).

Answer

x = -2

x = 4

Your turnSame method, new numbers — try it before moving on.

Using $y = x^2 - 2x - 3$ , the line $y = -3$ meets the curve at two points. What are their $x$ -values?

Your answer

Separate with a comma.

Show worked solution

Solution

1Solve $x^2 - 2x - 3 = -3$ : $x^2 - 2x = 0$ .
2 $x(x - 2) = 0$ , so $x = 0$ or $x = 2$ — matching the table.

Answer: $x = 0$ or $x = 2$

Two, one or none

A horizontal line can meet a U-shaped curve twice, once (touching the turning point) or not at all. That is the picture behind a quadratic equation having two, one or no solutions.

Cubic, reciprocal and exponential graphs

$\textbf{Cubic}$ graphs such as $y = x^3$ have an S-shape. $\textbf{Reciprocal}$ graphs $y = \frac{k}{x}$ have two separate branches and never touch either axis. $\textbf{Exponential}$ graphs $y = a^x$ (with $a > 1$ ) grow ever more steeply, pass through $(0, 1)$ and never touch the $x$ -axis.

Cubic (S-shape), reciprocal (two branches, never touching the axes) and exponential (through

(0, 1)

, steepening).

Example 1Which graph type is

y = \frac{6}{x}

, and what happens when

x = 0

1It has the form

\frac{k}{x}

, so it is a reciprocal graph.

\frac{6}{0}

is undefined, so there is no point with

x = 0

— the curve never meets the

y

-axis.

Answer

Reciprocal; there is no value at

x = 0

Think like an examiner

Draw curves in one smooth stroke through every point. Joining points with straight ruler lines loses the curve mark.
A turning point is a coordinate pair — give both values.
When asked to "use your graph", your answers must come from the graph, with any lines you drew left visible.

Graph facts

y = ax^2 + bx + c

a > 0

: U-shape;

a < 0

: ∩-shape;

y

-intercept

c

x = \tfrac{\text{root}_1 + \text{root}_2}{2}

Line of symmetry.

y = \tfrac{k}{x}, \quad y = a^x

Reciprocal and exponential shapes.

Watch out for these

Substituting

x = -3

into

x^2

-9

(-3)^2 = 9

Joining plotted points with straight lines, or a flat bottom between two lowest points.

The curve is smooth and dips below the lowest plotted points when the turning point is between them.

Reading the roots as the

y

-values where the curve crosses an axis.

Roots are

x

-values where

y = 0

Drawing

y = \frac{1}{x}

crossing the axes.

It gets closer and closer to both axes but never touches them.

Stretch yourself

A quadratic graph has a $y$ -intercept of $-8$ and roots at $x = -2$ and $x = 4$ . Find the coordinates of its turning point.

Hint — Roots $-2$ and $4$ mean the equation is $y = a(x + 2)(x - 4)$ . Use the intercept to find $a$ .

Reveal solution

SolutionAt

x = 0

a(2)(-4) = -8a = -8

, so

a = 1

and

y = (x + 2)(x - 4) = x^2 - 2x - 8

. The line of symmetry is

x = 1

, and

y = 1 - 2 - 8 = -9

. The turning point is

(1, -9)

\blacksquare

Questions students ask

How do I know whether the turning point is a maximum or a minimum?

Look at the sign of the

x^2

term. Positive gives a U-shape with a minimum; negative gives a ∩-shape with a maximum.

Why are graphical solutions only approximate?

You are reading values off a drawn curve, which is limited by the scale and the accuracy of plotting. Exact answers need algebra.

Do exponential graphs ever touch the

x

-axis?

No. A positive number raised to any power is positive, so

y = 2^x

gets closer and closer to zero as

x

decreases but never reaches it.

Key takeaways

Plot quadratics from a table and join with a smooth curve.
Roots: where $y = 0$ . $y$ -intercept: the constant term.
The line of symmetry is halfway between the roots and passes through the turning point.
Solve $f(x) = k$ by drawing $y = k$ and reading the $x$ -values.
Recognise cubic (S), reciprocal (two branches) and exponential (through $(0,1)$ ) shapes.

Exam-style questions

Practise it the way it is examined

3 original questions · 8 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 43 marks

This question is about the graph of $y = x^2 - 4x + 1$ .

(a)Work out the value of $y$ when $x = -1$ and when $x = 3$ .[2]
Show mark scheme (a)
- B1 $x = -1$ : $y = 6$
- B1 $x = 3$ : $y = -2$
(b)Write down the coordinates of the turning point of $y = x^2 - 4x + 1$ .[1]
Show mark scheme (b)
- B1 $(2, -3)$
Answer: $(2, -3)$

Question 2Grade 53 marks

The graph of $y = x^2 + x - 6$ crosses the $x$ -axis at $A$ and $B$ , and the $y$ -axis at $C$ . Find the coordinates of $A$ , $B$ and $C$ .[3]
Show mark scheme
- M1Solves $x^2 + x - 6 = 0$ , e.g. $(x + 3)(x - 2) = 0$
- A1 $(-3, 0)$ and $(2, 0)$
- B1 $C = (0, -6)$
Answer: $A(-3, 0)$ , $B(2, 0)$ , $C(0, -6)$

Question 3Grade 62 marks

Match each equation to its graph shape: (i) $y = x^3 - 4x$ (ii) $y = -\frac{2}{x}$ (iii) $y = 3^x$ (iv) $y = 4 - x^2$ . Shapes: A — ∩-shaped parabola; B — S-shaped curve; C — curve through $(0, 1)$ that grows steeply; D — two branches in the second and fourth quadrants.[2]
Show mark scheme
- B1Two correct
- B1All correct: (i) B, (ii) D, (iii) C, (iv) A

Unlimited practice

Skill drill

A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

0 / 0 first-time correct Streak 0Best at this level: 0

Generating a question…

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

What shape is the graph of $y = x^2 + 1$ ?
- U-shaped parabola
- ∩-shaped parabola
- Straight line
- S-shaped curve
Show answer and explanation
Answer: U-shaped parabola
Positive $x^2$ coefficient.
For $y = x^2 - 5$ , what is $y$ when $x = -3$ ?
- $4$
- $-14$
- $-8$
- $14$
Show answer and explanation
Answer: $4$
$9 - 5 = 4$ .
Where does $y = x^2 + 4x - 7$ cross the $y$ -axis?
- $(0, -7)$
- $(0, 4)$
- $(-7, 0)$
- $(0, 7)$
Show answer and explanation
Answer: $(0, -7)$
Set $x = 0$ .
A parabola has roots $x = -4$ and $x = 2$ . Its line of symmetry is…
- $x = -1$
- $x = 1$
- $x = -2$
- $y = -1$
Show answer and explanation
Answer: $x = -1$
Halfway between $-4$ and $2$ .
The roots of $y = (x - 3)(x + 5)$ are…
- $3$ and $-5$
- $-3$ and $5$
- $3$ and $5$
- $-15$
Show answer and explanation
Answer: $3$ and $-5$
Each bracket equals zero.
Which graph passes through $(0, 1)$ and never touches the $x$ -axis?
- $y = 2^x$
- $y = x^2$
- $y = \frac{1}{x}$
- $y = x^3$
Show answer and explanation
Answer: $y = 2^x$
$2^0 = 1$ and $2^x > 0$ .
The turning point of $y = x^2 - 6x + 5$ is…
- $(3, -4)$
- $(-3, 32)$
- $(3, 5)$
- $(6, 5)$
Show answer and explanation
Answer: $(3, -4)$
Roots 1 and 5, symmetry $x = 3$ , $y = 9 - 18 + 5$ .
How many times does $y = 4$ meet the curve $y = x^2 + 5$ ?
- $0$
- $1$
- $2$
- $4$
Show answer and explanation
Answer: $0$
The minimum of $x^2 + 5$ is 5, so it never reaches 4.

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