Exponential functions and e

Functions with the variable in the power, y = a^x — their graph shape, the asymptote and intercept — and the special exponential e^x whose gradient equals itself.

Pure · Exponentials & logarithms

Exponential functions and e

Functions with the variable in the power, y = a^x — their graph shape, the asymptote and intercept — and the special exponential e^x whose gradient equals itself.

Why it works

An exponential function has the variable in the exponent:

y = a^x

with a fixed base

a > 0

. That is the opposite of a power function

y = x^a

, where the variable is in the base — and the difference completely changes the shape.

Every curve

y = a^x

shares three features, which is what makes them easy to sketch:

It passes through $(0, 1)$ , because $a^0 = 1$ for any base.
It never touches the $x$ -axis. The line $y = 0$ is a **horizontal

asymptote*: $a^x > 0$ for all $x$ , so the curve only approaches* the axis.

For $a > 1$ it is increasing (growth); for $0 < a < 1$ it is decreasing

(decay). Since

a^{-x} = \left(\tfrac1a\right)^x

, the graphs of

y = a^x

and

y = a^{-x}

are reflections of each other in the $y$ -axis.

The number $e$ . Among all these curves there is one whose gradient at every point equals the $y$ -value at that point. The base that does this is

e \approx 2.718

, and

y = e^x

is the exponential function. So the curve

y = e^x

rises at a rate equal to its own height — the engine behind every growth and decay model. (More generally the gradient of

y = e^{kx}

k\,e^{kx}

.)

Transformations work exactly as for any function (so the asymptote moves too):

$y = e^x + c$ shifts up by $c$ — the asymptote becomes $y = c$ .
$y = e^{x - a}$ shifts right by $a$ ; $y = e^{-x}$ reflects in the $y$ -axis (decay).
$y = A e^{kx}$ stretches vertically by $A$ , and still passes through $(0, A)$ .