The Exponential Function: Growth and Decay

The exponential function $y = a^x$ looks like a single rule, but its shape depends entirely on one number: the base $a$ . Three cases cover every possibility — the base can be greater than 1, equal to 1, or between 0 and 1 — and each produces a completely different graph. One of the three is not even an exponential function at all.

Case 1 Base greater than 1 — growth

If $a > 1$ , then $y = a^x$ is an increasing exponential function.

x

increases,

y

increases as well.

The larger the base, the faster the climb.

This is what is meant by exponential growth.

Case 2 Base equal to 1 — not exponential

If $a = 1$ , raising 1 to any power always returns 1:

1^1 = 1

1^{10} = 1

1^{100} = 1

The equation collapses to $y = 1$ — a horizontal straight line, not a curve. Nothing grows and nothing decays. For this reason a base of 1 is excluded, and the definition always carries the condition $a \neq 1$ .

Case 3 Base between 0 and 1 — decay

If $0 < a < 1$ , then $y = a^x$ is a decreasing exponential function.

With

a = \frac{1}{2}

\left(\frac{1}{2}\right)^1 = \frac{1}{2}

\left(\frac{1}{2}\right)^2 = \frac{1}{4}

\left(\frac{1}{2}\right)^3 = \frac{1}{8}

With

a = \frac{1}{4}

\left(\frac{1}{4}\right)^2 = \frac{1}{16}

The larger the exponent, the smaller the value becomes. But the value never reaches zero — it only approaches it as $x \to +\infty$ . This is exponential decay, and the horizontal axis acts as an asymptote the curve gets ever closer to without touching.

Note The three cases side by side

$a > 1$ — an increasing exponential function; the curve rises to the right.

$a = 1$ — the horizontal line

y = 1

; not an exponential function, which is why

a \neq 1

is required.

$0 < a < 1$ — a decreasing exponential function, or exponential decay.

In every valid case the base must also be positive. A negative base would make results jump between positive and negative as the exponent changes, and no continuous curve could be drawn.

Example Classifying by the base

Decide whether each function grows, decays, or is neither: $y = 3^x$ , $y = \left(\frac{1}{5}\right)^x$ , $y = 1^x$ .

y = 3^x

: the base 3 is greater than 1, so the function increases.

y = \left(\frac{1}{5}\right)^x

: the base sits between 0 and 1, so the function decreases.

y = 1^x

: every power of 1 is 1, so this is the line

y = 1

⟹ growth, decay, and a horizontal line that is not exponential

Example Watching a decay shrink

Evaluate $y = \left(\frac{1}{2}\right)^x$ for $x = 1,\ 2,\ 3$ and describe what happens as $x$ keeps growing.

x = 1

\left(\frac{1}{2}\right)^1 = \frac{1}{2}

x = 2

\left(\frac{1}{2}\right)^2 = \frac{1}{4}

x = 3

\left(\frac{1}{2}\right)^3 = \frac{1}{8}

Each step halves the previous value, so the numbers keep shrinking.

⟹ the values approach 0 as x → +∞ but never actually reach it

Note Mistakes to avoid

Treating

y = 1^x

as an exponential function — it is the straight line

y = 1

Believing a decaying curve eventually touches or crosses zero.

Assuming a fractional base means the function is undefined; a base between 0 and 1 is perfectly valid.

Allowing a negative base, which breaks the continuous curve.

Confusing the base with the exponent when deciding growth or decay — it is the base that settles it.

Summary

In y = aˣ it is the base a that decides the shape of the graph.
If a > 1 the function increases — exponential growth.
If a = 1 the graph is the horizontal line y = 1, so a = 1 is excluded from the definition.
If 0 < a < 1 the function decreases — exponential decay.
A decaying curve approaches zero as x → +∞ but never reaches it, and the base must always be positive.