Graphing Exponential Functions

Idea

Every exponential graph is one of two shapes. Which one you get is decided by a single thing: whether the base is above or below $1$ .

You never need a table of twenty values. Three points and one line fix the whole curve.

Rule

through $(0, 1)$ , because $a^0 = 1$

the $x$ -axis is an asymptote, because $a^x > 0$

domain: every real $x$

range: $y > 0$

How it is used

1. Two shapes, and the base picks one

Base	Shape	Reading left to right
$a > 1$	growth	rises, slowly then steeply
$0 < a < 1$	decay	falls, steeply then slowly

2. Three points fix the curve

Take the powers $-1$ , $0$ and $1$ . They give the same three points for any base.

\left(-1, \frac{1}{a}\right)

(0, 1)

(1, a)

Plot those, follow the asymptote on one side and climb on the other, and the sketch is done.

Exponential and logarithm

Try this. Compare bases 2 and 0.5: growth becomes decay. Move the input to 1; the logarithm is zero for either base.

BasePositive input x0.5

Base 3 · Blue: y = 3ˣ. At x = 0.5, y = 1.732.

3. The asymptote is a floor, not a limit on how far it goes

The curve approaches the $x$ -axis on one side and never meets it, because $a^x$ is never $0$ . On the other side it grows without any ceiling at all.

The two sides are not symmetric, and that is the shape's most recognisable feature.

4. Decay is growth, reflected

A base below $1$ is a reciprocal, and a reciprocal is a negative power.

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

Replacing $x$ by $-x$ reflects a graph in the $y$ -axis. So $y = \left(\frac{1}{2}\right)^x$ is exactly $y = 2^x$ held up to a mirror.

5. Adding a constant moves the asymptote

y = a^x + k

Every point rises by $k$ , and the asymptote rises with them, to $y = k$ . The shape does not change.

Worked example

Sketch $y = 3^x$ , then state its range.

The base is above $1$ , so this is growth: it rises left to right.
Take the three standard points.

\left(-1, \frac{1}{3}\right)

(0, 1)

(1, 3)

To the left the curve flattens towards the $x$ -axis without touching it. To the right it climbs steeply.
Every value it takes is positive, and it reaches every positive value exactly once.

\text{range} = (0, \infty)

Explore

Drag the base slider from below $1$ to above it and watch the curve flip between the two shapes.

Whatever base you choose, the curve passes through $(0, 1)$ and stays above the $x$ -axis. Those two facts are what make the sketch quick.