1MA1 · A12AlgebraFoundation & HigherGrades 4–7

Cubic, Reciprocal & Exponential Graphs

Q: What is an asymptote?

A line that a curve gets closer and closer to without ever reaching.

Q: How many times can a cubic cross the x-axis?

At least once and at most three times.

Q: Why does an exponential graph never reach zero?

Multiplying a positive number by a positive factor always gives a positive number, however small.

Straight lines and parabolas are not the only graphs with recognisable shapes. Cubic graphs snake up and down, reciprocal graphs split into two pieces that never touch the axes, and exponential graphs start slowly and then shoot upwards. Recognising each shape from its equation — and the equation from its shape — is a regular Edexcel question.

45 min Graphs

The big picture

Each family has a signature. Cubics $y = ax^{3} + \dots$ have an S-shape and can cross the $x$ -axis up to three times. Reciprocals $y = \dfrac{k}{x}$ have $\textbf{asymptotes}$ — lines they approach but never meet. Exponentials $y = ab^{x}$ multiply by the same factor for each step in $x$ , which is why they model compound interest, populations and radioactive decay.

Knowing why each graph looks as it does — what happens for large $x$ , for negative $x$ , or near $x = 0$ — lets you sketch confidently and match graphs to equations without plotting.

What you'll learn

Recognise and sketch cubic graphs, including in factorised form
Recognise and sketch reciprocal graphs and identify their asymptotes
Recognise and sketch exponential graphs $y = ab^{x}$
Find the constants of an exponential model from two points (Higher)

Cubic graphs

$y = x^{3}$ passes through the origin, rising steeply for positive $x$ and falling for negative $x$ , with a flat point at the origin.

A cubic with a positive $x^{3}$ coefficient rises from bottom left to top right; a negative coefficient reverses this.

In factorised form $y = (x - p)(x - q)(x - r)$ , the graph crosses the $x$ -axis at $p$ , $q$ and $r$ . A squared factor such as $(x - p)^{2}$ means the curve touches the axis at $p$ without crossing.

Example 1Sketch

y = (x + 2)(x - 1)(x - 3)

1Roots at

x = -2

1

and

3

y

-intercept:

(2)(-1)(-3) = 6

3Positive

x^{3}

coefficient: rises from bottom left to top right, turning between the roots.

AnswerS-shaped curve through

(-2, 0)

(0, 6)

(1, 0)

and

(3, 0)

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

Where does $y = (x + 2)(x - 1)(x - 3)$ cross the $y$ -axis? Give the $y$ -value.

Your answer

Show worked solution

Solution

1Set $x = 0$ : $(2)(-1)(-3)$ .
2 $= 6$ .

Answer: $y = 6$

Reciprocal graphs

$y = \dfrac{1}{x}$ is undefined at $x = 0$ . As $x$ gets very large, $y$ gets very close to 0; as $x$ gets close to 0, $y$ becomes very large.

So the graph has two separate branches, in opposite quadrants, and approaches both axes without touching them. The axes are its asymptotes.

For $y = \dfrac{k}{x}$ with $k > 0$ , the branches are in the first and third quadrants; with $k < 0$ , in the second and fourth.

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

For $y = \dfrac{12}{x}$ , find $y$ when $x = -4$ .

Your answer

Show worked solution

Solution

1 $12 \div (-4) = -3$ .

Answer: $-3$

Why this works

Reciprocal graphs describe inverse proportion: doubling $x$ halves $y$ . The time to complete a journey against speed, or the number of workers against the time for a job, follows exactly this shape.

Exponential graphs

$y = b^{x}$ with $b > 1$ passes through $(0, 1)$ , increases ever more steeply, and approaches the $x$ -axis for negative $x$ without touching it.

With $0 < b < 1$ it is a decreasing curve — exponential decay — still through $(0, 1)$ .

$y = ab^{x}$ passes through $(0, a)$ and is multiplied by $b$ for every increase of 1 in $x$ .

y = ab^{x}

Starts at a when x = 0; multiplied by b each step.

Example 1The curve

y = ab^{x}

passes through

(0, 3)

and

(2, 48)

. Find

a

and

b

, given

b > 0

1At

x = 0

y = a

, so

a = 3

2At

x = 2

3b^{2} = 48

, so

b^{2} = 16

and

b = 4

Answer

a = 3

b = 4

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

The curve $y = ab^x$ passes through $(0, 5)$ and $(3, 40)$ . Find $b$ .

Your answer

Show worked solution

Solution

1At $x = 0$ : $a = 5$ .
2 $5b^3 = 40$ , so $b^3 = 8$ and $b = 2$ .

Answer: $b = 2$

Tip — In $y = ab^{x}$ , substitute the point with $x = 0$ first — it gives $a$ immediately, because $b^{0} = 1$ .

Matching graphs to equations

Check the shape family first (line, parabola, cubic, reciprocal, exponential), then use key points: intercepts, whether it passes through the origin, and behaviour for large or negative $x$ .

A curve through $(0, 1)$ that never crosses the $x$ -axis is exponential. A curve with two separate pieces is reciprocal. A curve crossing the $x$ -axis three times is cubic.

Tip — Substituting $x = 0$ into each candidate equation is often the fastest way to eliminate wrong options.

Think like an examiner

Label intercepts with coordinates on sketches.
Draw asymptotes as dashed lines and do not let the curve touch them.
Show the substitutions used to find constants in $y = ab^{x}$ .

Standard graphs

y = x^{3}

Cubic.

y = \tfrac{k}{x}

Reciprocal, asymptotes at the axes.

y = ab^{x}

Exponential, through

(0, a)

Watch out for these

Drawing

y = \tfrac1x

crossing the axes.

It approaches the axes but never meets them.

Sketching

y = 2^{x}

through the origin.

2^{0} = 1

, so it passes through

(0, 1)

Crossing the axis at a squared factor of a cubic.

A repeated root means the curve touches and turns.

Thinking

y = 2^{x}

becomes negative for negative

x

It stays positive, approaching 0.

Stretch yourself

A population of bacteria is modelled by $P = ab^{t}$ , where $t$ is time in hours. After 1 hour there are 600 bacteria and after 3 hours there are 5400. Find $a$ and $b$ , and interpret them.

Hint — Divide the two equations to eliminate $a$ .

Reveal solution

SolutionFrom the data:

ab = 600

and

ab^{3} = 5400

. Dividing:

b^{2} = 9

, so

b = 3

(it must be positive). Then

a = \dfrac{600}{3} = 200

. So

P = 200 \times 3^{t}

: the initial population was 200 bacteria, and it triples every hour. Check:

200 \times 27 = 5400

✓.

\blacksquare

Questions students ask

What is an asymptote?

A line that a curve gets closer and closer to without ever reaching.

How many times can a cubic cross the

x

-axis?

At least once and at most three times.

Why does an exponential graph never reach zero?

Multiplying a positive number by a positive factor always gives a positive number, however small.

Key takeaways

Cubics are S-shaped and can have up to three roots.
Reciprocal graphs $y = \tfrac{k}{x}$ have two branches and asymptotes along the axes.
Exponentials $y = ab^{x}$ pass through $(0, a)$ and never touch the $x$ -axis.
Use shape, intercepts and behaviour to match graphs to equations.
Find $a$ from $x = 0$ , then $b$ from another point.

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 42 marks

This question is about $y = x^3 - 3x$ .

(a)Work out $y$ when $x = -2$ .[1]
Show mark scheme (a)
- B1 $-2$
Answer: $-2$
(b)Work out $y$ when $x = 2$ .[1]
Show mark scheme (b)
- B1 $2$
Answer: $2$

Question 2Grade 63 marks

The value of an investment, in pounds, after $t$ years is $V = 800 \times 1.05^t$ .

(a)Write down the value of the investment at the start.[1]
Show mark scheme (a)
- B1£800
(b)Work out the value after 4 years.[2]
Show mark scheme (b)
- M1 $800 \times 1.05^4$
- A1£972.41
Answer: £972.41

Question 3Grade 73 marks

The curve $y = ab^x$ passes through $(1, 6)$ and $(3, 54)$ , where $b > 0$ . Find $a$ and $b$ .[3]
Show mark scheme
- M1 $ab = 6$ and $ab^3 = 54$
- M1Divides: $b^2 = 9$
- A1 $b = 3$ , $a = 2$
Answer: $a = 2$ , $b = 3$

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

Browse practice topics

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