F1PureCore

Exponential Functions

Q: What is the difference between an exponential and a power function?

In an exponential like 2^x the variable is the exponent and the base is fixed. In a power function like x^2 the variable is the base and the exponent is fixed. Exponentials grow far faster in the long run.

Q: Why does every exponential graph pass through (0, 1)?

Because anything to the power 0 is 1: a^0 = 1 for any positive base a. So the y-intercept is always 1.

Q: Can an exponential function be negative?

No. For a positive base, a^x is always positive. The graph approaches the x-axis (y = 0) as an asymptote but never touches or crosses it.

Q: What decides whether the graph grows or decays?

The base. If a > 1 the function grows; if 0 < a < 1 it decays. A decay curve is a growth curve reflected in the y-axis, since ( 1a )^x = a^-x.

An exponential function has the variable in the power, like $y = 2^{x}$ . That one structural change produces explosive growth (or rapid decay) — the mathematics behind populations, interest, viruses and radioactive decay.

24 min Video by TLMaths Exponentials and Logarithms

The big picture

You have met powers where the base varies and the exponent is fixed, like $x^{2}$ . An $exponential$ turns this around: the exponent varies and the base is fixed, like $2^{x}$ . That swap changes everything. A polynomial grows steadily; an exponential $doubles$ (or halves) at a constant rate, so it eventually outgrows any polynomial — the reason “exponential growth” is a byword for “fast”. This chapter builds the language for all such change, and this first lesson establishes the shape and behaviour that logarithms (the inverse) will later let you control.

A-Level Maths F1-01: Introducing the function a^xWatch the walkthrough, then work through the notes below.

Open on YouTube

What you'll be able to do

Recognise an exponential function $y = a^{x}$ and distinguish it from a power function
Describe the graph of $y = a^{x}$ for $a > 1$ and $0 < a < 1$
Identify the key features: the point $(0, 1)$ and the asymptote $y = 0$
Interpret exponential growth and decay in context

Variable in the power

An $exponential function$ has the form $y = a^{x}$ , where the base $a$ is a positive constant and the variable $x$ is the $exponent$ . Compare $2^{x}$ (exponential — the power changes) with $x^{2}$ (a power function — the base changes). They behave completely differently.

Because $a > 0$ , an exponential output is always positive, and $a^{0} = 1$ means every exponential graph passes through $(0, 1)$ .

Why this works

The defining feature is that the function multiplies by the same factor for each unit increase in $x$ : $2^{x}$ doubles every time $x$ rises by 1. Constant multiplicative growth is what makes exponentials outrun any polynomial in the long run.

Growth and decay

The base decides the direction. If $a > 1$ , the function $grows$ — rising ever more steeply as $x$ increases (exponential growth). If $0 < a < 1$ , it $decays$ — falling towards zero (exponential decay). In both cases the graph hugs the $x$ -axis on one side: $y = 0$ is a $horizontal asymptote$ the curve approaches but never reaches.

y = a^{x} : a > 1 grows, 0 < a < 1 decays

Always positive; passes through

(0, 1)

; asymptote

y = 0

y = 3^{x}

: base

> 1

, so it grows — through

(0, 1)

, rising steeply, asymptote

y = 0

to the left.

y = (\frac{1}{3})^{x}

: base

< 1

, so it decays — through

(0, 1)

, falling towards

y = 0

to the right.

3They are reflections of each other in the

y

-axis, since

(\frac{1}{3})^{x} = 3^{- x}

AnswerOne grows, one decays; both pass through

(0, 1)

with asymptote

y = 0

Tip — $(\frac{1}{a})^{x} = a^{- x}$ , so a decay curve is just the growth curve reflected in the $y$ -axis. One shape, two directions.

Modelling with exponentials

Exponentials model any quantity that changes by a fixed percentage per step: savings with compound interest, a population reproducing, a drug clearing from the blood, radioactive decay. A model like $P = P_{0} a^{t}$ has $P_{0}$ as the starting amount and $a$ as the per-unit growth (or decay) factor.

1Start: 200. Doubling each hour means multiplying by 2 each hour.

2After

t

hours:

N = 200 \times 2^{t}

Answer

N = 200 \times 2^{t}

Think like an examiner

Distinguish $a^{x}$ (exponential) from $x^{a}$ (power) — they are asked about differently and behave oppositely.
State the key features when sketching: passes through $(0, 1)$ , positive everywhere, asymptote $y = 0$ .
For a decay curve, note the base is between 0 and 1 (or write it as $a^{- x}$ ).
In modelling, identify the starting value and the growth/decay factor separately.

Common misconceptions

Treating

2^{x}

and

x^{2}

as similar.

2^{x}

has the variable in the power (exponential);

x^{2}

has it in the base (power function).

Thinking an exponential graph can be negative or cross the

x

-axis.

For

a > 0

a^{x} > 0

always;

y = 0

is an asymptote it never reaches.

Forgetting every

y = a^{x}

passes through

(0, 1)

a^{0} = 1

, so the

y

-intercept is always 1.

Confusing which base gives growth vs decay.

a > 1

grows;

0 < a < 1

decays.

Exponential facts

y = a^{x}, a > 0

Exponential function.

a^{0} = 1

Always through

(0, 1)

a > 1 \Rightarrow growth

Base above 1.

0 < a < 1 \Rightarrow decay

Base below 1.

Stretch yourself

Explain why $y = 2^{x}$ eventually exceeds $y = x^{10}$ , even though $x^{10}$ is much larger for small $x$ .

Hint — Compare how each grows when $x$ increases by 1. What does each get multiplied by?

Questions students ask

What is the difference between an exponential and a power function?

In an exponential like

2^{x}

the variable is the exponent and the base is fixed. In a power function like

x^{2}

the variable is the base and the exponent is fixed. Exponentials grow far faster in the long run.

Why does every exponential graph pass through

(0, 1)

Because anything to the power 0 is 1:

a^{0} = 1

for any positive base

a

. So the

y

-intercept is always 1.

Can an exponential function be negative?

No. For a positive base,

a^{x}

is always positive. The graph approaches the

x

-axis (

y = 0

) as an asymptote but never touches or crosses it.

What decides whether the graph grows or decays?

The base. If

a > 1

the function grows; if

0 < a < 1

it decays. A decay curve is a growth curve reflected in the

y

-axis, since

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

Key takeaways

An exponential $y = a^{x}$ has the variable in the power.
It is always positive, passes through $(0, 1)$ , with asymptote $y = 0$ .
$a > 1$ gives growth; $0 < a < 1$ gives decay.
Exponentials model constant-percentage change and outgrow any polynomial.
Logarithms (next lessons) are the tool for solving exponential equations.

How this fits the course

Build on

Index Laws

Leads to

The Number e and eˣ

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