Exponential Functions

This lesson introduces exponential functions — functions of the form y = aˣ where the variable is in the exponent. These functions model growth and decay in contexts ranging from population dynamics to radioactive decay, and they are central to the Edexcel 9MA0 specification.

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What Is an Exponential Function?

An exponential function has the form:

y = aˣ

where a is a positive constant called the base and x is the exponent (the variable). The key feature is that the variable appears as the power, not as the base.

Key Properties of y = aˣ (where a > 0, a ≠ 1)

Property	Detail
Domain	All real numbers — you can raise a to any power
Range	y > 0 — an exponential function is always positive
y-intercept	(0, 1) — because a⁰ = 1 for any valid base
Asymptote	The x-axis (y = 0) is a horizontal asymptote
Growth/Decay	If a > 1 the function increases (growth); if 0 < a < 1 the function decreases (decay)

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Graphs of y = aˣ