14.1PureCore

Exponential Functions

An exponential function has the variable in the power, like $y = 2^x$ . These functions model rapid growth and decay, and their graphs all share a characteristic shape passing through (0, 1).

25 min Exponentials and Logarithms

Edexcel AS Level Maths: 14.1 Exponential FunctionsCurated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Recognise exponential functions of the form aˣ
Sketch the graph of y = aˣ
Identify the key point (0, 1) and the asymptote
Compare growth and decay

The form aˣ

An exponential function is $y = a^x$ where the base $a$ is a positive constant and the variable is in the $\textbf{exponent}$ . This is different from a power function like $x^2$ where the variable is the base.

The graph

Every graph $y = a^x$ (with $a > 1$ ) passes through $(0, 1)$ , rises steeply to the right, and approaches the $x$ -axis (a horizontal asymptote at $y = 0$ ) to the left — it never touches it.

y = a^{x}: \quad \text{through } (0, 1), \ \text{asymptote } y = 0

Anything to the power 0 is 1, so all such curves share

(0, 1)

Tip — a⁰ = 1 for any base, so every exponential curve y = aˣ goes through (0, 1).

Growth and decay

If $a > 1$ the function $\textbf{grows}$ ; if $0 < a < 1$ it $\textbf{decays}$ (falls towards zero). A decay curve is a reflection of a growth curve in the $y$ -axis.

Example 1Does

y = 0.5^x

grow or decay?

1Base

0.5

is between 0 and 1.

Answerdecay

Formula recap

y = a^{x}

Exponential function (variable in the power).

a^{0} = 1 \Rightarrow \text{through } (0,1)

Shared point.

a > 1 \text{ grows}, \ 0 < a < 1 \text{ decays}

Growth vs decay.

Common mistakes to avoid

Confusing aˣ with xᵃ.

In an exponential the variable is the power; in a power function it is the base.

Drawing the curve touching the x-axis.

y = aˣ approaches but never reaches y = 0.

Key takeaways

Exponential functions have the variable in the power: y = aˣ.
All pass through (0, 1) with a horizontal asymptote y = 0.
a > 1 grows; 0 < a < 1 decays.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Check your understanding

Work through questions on Exponential Functions. Use the feedback to decide what needs another look.

1 past-paper question · 4 marks

Practise this topic

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