1.06aPureCore

Exponential Functions

Q: What is the difference between x^n and a^x?

In x^n the base varies and the index is fixed — a polynomial. In a^x the base is fixed and the index varies — an exponential. They behave completely differently, and exponentials eventually grow faster than any polynomial.

Q: Why can the base not be negative?

Because (-2 )^x has no real value for most x — try x = 12. Restricting to a>0 keeps the function defined for every real x, which is what makes it a usable curve.

Q: Where does e actually come from?

It is the base for which the gradient of y = a^x equals the value of y at every point. That condition pins down a single number, 2.71828, which is irrational. It also arises as the limit of (1+ 1n )^n — compound interest compounded ever more often.

Q: How do I solve 2^x = 7?

Not with indices alone, because 7 is not a power of 2. This is what logarithms are for, and it is the next thing the specification covers after this.

Every index you have met so far has been a fixed number. Move the variable into the index instead — $y = 2^{x}$ rather than $y = x^{2}$ — and you get an $exponential function$ , whose behaviour is unlike anything polynomial. OCR requires you to know the shape of $y = a^{x}$ , recognise growth and decay, and understand what makes $e$ the special base.

30 min Indices and Surds

The big picture

Exponential functions describe anything whose rate of change is proportional to how much of it there currently is: compound interest, radioactive decay, cooling coffee, spreading infection, charging capacitors. That single property — the bigger it is, the faster it changes — is what separates exponential growth from the steady growth of a straight line, and it is why exponential curves eventually outrun every polynomial no matter how high the power. When you meet differential equations later, $e^{x}$ turns out to be the function that solves the simplest one of all, and the reason will be the gradient property in the last section of this lesson.

What you'll be able to do

Sketch $y = a^{x}$ and describe its key features for $a > 1$ and $0 < a < 1$
Distinguish exponential growth from exponential decay
Interpret the parameters of $y = A b^{x}$ in a modelling context
State what makes $e$ special and why $e^{x}$ is the natural exponential

Swapping the variable into the index

Compare $y = x^{2}$ and $y = 2^{x}$ at $x = 1, 2, 3, 4$ . The first gives $1, 4, 9, 16$ — differences of $3, 5, 7$ , growing steadily. The second gives $2, 4, 8, 16$ — each value is $double$ the last. Multiplying rather than adding is the whole difference, and it is why exponentials eventually win: by $x = 10$ the polynomial is at 100 and the exponential is at 1024.

For $y = a^{x}$ with $a > 0$ , the graph passes through $(0, 1)$ for every base, because $a^{0} = 1$ . It never touches the $x$ -axis: $a^{x}$ is positive for all $x$ , so the $x$ -axis is a horizontal $asymptote$ .

Why this works

The constant ratio is the identifying feature. If a table of values multiplies by the same factor for each equal step in $x$ , the relationship is exponential — however the numbers are dressed up. A constant $difference$ would mean linear.

Growth, decay and the shape of the curve

When $a > 1$ the function grows: rising slowly at first, then steeply, with the $x$ -axis as an asymptote to the left. When $0 < a < 1$ each step multiplies by a number less than one, so the function decays — falling steeply at first, then flattening towards the axis on the right.

The two cases are reflections of each other. Since $(\frac{1}{2})^{x} = 2^{- x}$ , the decay curve for base $\frac{1}{2}$ is exactly the growth curve for base 2 reflected in the $y$ -axis. Any decay can be written as growth with a negative index, which is often the more useful form.

y = a^{x}, a > 1

Growth: through

(0, 1)

, asymptotic to the negative

x

-axis.

y = a^{x}, 0 < a < 1

Decay: through

(0, 1)

, asymptotic to the positive

x

-axis.

(\frac{1}{a})^{x} = a^{- x}

Decay is growth reflected in the

y

-axis.

1The base is between 0 and 1, so this is decay.

2As

x \to \infty

, repeated multiplication by 0.8 drives

y

towards 0 without reaching it.

3As

x \to - \infty

0. 8^{x} = 1.2 5^{- x}

grows without bound.

Answer

y \to 0^{+}

x \to \infty

;

y \to \infty

x \to - \infty

Tip — Whatever the base, the curve goes through $(0, 1)$ and stays strictly positive. A sketch that crosses or touches the $x$ -axis is wrong.

The modelling form $y = Ab^{x}$

Real models rarely start at 1, so they are written $y = A b^{x}$ . Setting $x = 0$ gives $y = A$ , so $A is the initial value$ . The base $b$ is the $multiplier per unit of x$ : $b = 1.05$ means a 5% increase each step, $b = 0.9$ means a 10% decrease.

Reading the numbers back into English is where most modelling marks are. A model $P = 2000 (1.03)^{t}$ with $t$ in years says: 2000 to begin with, growing 3% a year.

1At

t = 0

P = 500 \times 1 = 500

2The multiplier 1.12 is a 12% annual increase.

P = 500 \times 1.1 2^{3} = 500 \times 1.404928

AnswerInitial 500; growth 12% per year;

P \approx 702

after 3 years.

Tip — The base is the multiplier, not the percentage. A 12% rise gives $b = 1.12$ ; a 12% fall gives $b = 0.88$ . Writing $b = 0.12$ for a 12% fall is the standard slip.

Why $\mathrm{e}$ is the natural base

Every exponential curve $y = a^{x}$ has a gradient proportional to its own height — that is the defining property. What changes with the base is the constant of proportionality. For $y = 2^{x}$ the gradient is about $0.69$ times the height; for $y = 3^{x}$ it is about $1.10$ times.

Somewhere between 2 and 3 there is a base where that constant is exactly 1 — where the gradient at every point equals the value at that point. That base is $e \approx 2.71828$ , an irrational number, and $y = e^{x}$ is called the natural exponential function.

This is the reason $e$ appears everywhere in the rest of the course. Differentiating $e^{x}$ leaves it unchanged, which makes it the natural language for anything whose rate of change is proportional to its size.

e \approx 2.718281828 \dots

The base whose gradient equals its own value.

\frac{d}{d x} (e^{x}) = e^{x}

Proved later — but this is why

e

was chosen.

Why this works

$e$ is not an arbitrary constant someone picked. It is forced, in the same way $a^{0} = 1$ was forced: it is the unique base that makes the gradient property as simple as it can possibly be.

Think like an examiner

A sketch of $y = a^{x}$ must show the intercept at $(0, 1)$ and the $x$ -axis asymptote. Both are commonly credited marks.
In modelling questions, interpret every constant in context and in words — "500 is the initial population" scores where "500" alone does not.
State clearly whether a model shows growth or decay by referring to the base being greater or less than 1.
Keep $e$ exact through the working. Answers involving $e$ are usually wanted in exact form unless a decimal is explicitly requested.

Common misconceptions

Confusing

y = x^{2}

with

y = 2^{x}

In an exponential the

variable is the index

x^{2}

is a parabola;

2^{x}

is an exponential curve through

(0, 1)

Sketching

y = a^{x}

crossing the

x

-axis.

a^{x} > 0

for every real

x

. The axis is an asymptote the curve approaches but never reaches.

Writing a 7% annual decrease as

b = 0.07

It is

b = 0.93

— the multiplier is what remains, not what is lost.

Treating

e

as approximately 2.7 and rounding early.

Carry

e

symbolically. Rounding it at the start compounds through every later line.

Exponential functions

y = a^{x} passes through (0, 1)

Every base, same intercept.

a > 1 \Rightarrow growth

Multiplier above one.

0 < a < 1 \Rightarrow decay

Multiplier below one.

y = A b^{x}

A

initial value,

b

multiplier per step.

(\frac{1}{a})^{x} = a^{- x}

Decay as reflected growth.

Stretch yourself

Two colonies start together at 300 cells. Colony A follows $N = 300 (1.4)^{t}$ and colony B follows $N = 300 + 240 t$ , with $t$ in hours. Which is larger at $t = 2$ , and which is larger at $t = 8$ ? What does the comparison show?

Hint — Evaluate both at each time. Do not assume the one that leads early keeps leading.

Questions students ask

What is the difference between

x^{n}

and

a^{x}

x^{n}

the base varies and the index is fixed — a polynomial. In

a^{x}

the base is fixed and the index varies — an exponential. They behave completely differently, and exponentials eventually grow faster than any polynomial.

Why can the base not be negative?

Because

(- 2)^{x}

has no real value for most

x

— try

x = \frac{1}{2}

. Restricting to

a > 0

keeps the function defined for every real

x

, which is what makes it a usable curve.

Where does

e

actually come from?

It is the base for which the gradient of

y = a^{x}

equals the value of

y

at every point. That condition pins down a single number,

2.71828 \dots

, which is irrational. It also arises as the limit of

(1 + \frac{1}{n})^{n}

— compound interest compounded ever more often.

How do I solve

2^{x} = 7

Not with indices alone, because 7 is not a power of 2. This is what logarithms are for, and it is the next thing the specification covers after this.

Key takeaways

An exponential function has the variable in the index: $y = a^{x}$ , not $y = x^{a}$ .
Every $y = a^{x}$ passes through $(0, 1)$ and has the $x$ -axis as an asymptote.
$a > 1$ gives growth, $0 < a < 1$ gives decay, and the two are reflections in the $y$ -axis.
In $y = A b^{x}$ , $A$ is the initial value and $b$ the multiplier per unit step.
$e \approx 2.718$ is the base whose gradient equals its own value, making $e^{x}$ the natural exponential.

How this fits the course

Build on

Logarithms (later in 1.06)soon

Leads to

Expanding Brackets

Test yourself

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Next step

Practise this topic

Real past-paper questions on Exponential Functions, marked mark-by-mark. How you do feeds straight into your weak-topic list, so your revision keeps targeting what actually needs work.

Practise this topic

Exponential Functions

Swapping the variable into the index

Growth, decay and the shape of the curve

The modelling form $y = Ab^{x}$

Why $\mathrm{e}$ is the natural base

Practise this topic

Related topics

Exponential Functions

Swapping the variable into the index

Growth, decay and the shape of the curve

The modelling form $y = Ab^{x}$

Why $\mathrm{e}$ is the natural base

Practise this topic

Related topics