# Exponential Function: Graph

How to graph the given exponential function: basic graphs, 2 examples, and their solutions.

## Graph

### a > 1

This is the graph of y = ax (a > 1).

If a > 1,
then the graph goes upward.
It shows exponential growth.

It has two properties:

The graph passes through (0, 1).
(0, a0) = (0, 1)

Zero Exponent

The asymptote of the graph is the x-axis.
So ax is always plus.

### 0 < a < 1

This is the graph of y = ax (0 < a < 1).

If 0 < a < 1,
then the graph goes downward,
toward the x-axis.
It shows exponential decay.

It also has the same two properties:

The graph passes through (0, 1).
The asymptote of the graph is the x-axis.

## Example 1

### Solution

y = 2x is an exponential function.

So first draw (0, 1).

Put x = 1, 2, 3 into y = 2x.
And draw the points.

(1, 21) = (1, 2)
(2, 22) = (2, 4)
(3, 23) = (3, 8)

Put x = -1, -2, -3 into y = 2x.
And draw the points.

(1, 2-1) = (1, 1/2)
(2, 2-2) = (2, 1/4)
(3, 2-3) = (3, 1/8)

Negative Exponent

Connect the points
and draw the graph.

The asymptote of an exponential function
is the x-axis.
So, as x goes to -∞,
the graph gets close to the x-axis.
(not below the x-axis.)

So this is the graph of y = 2x.

## Example 2

### Solution

y = (1/3)x is an exponential function.

So first draw (0, 1).

Put x = 1, 2 into y = (1/3)x.
And draw the points.

(1, (1/3)1) = (1, 1/3)
(2, (1/3)2) = (2, 1/9)

Put x = -1, -2 into y = (1/3)x.
And draw the points.

(-1, (1/3)-1) = (-1, (3/1)1) = (-1, 3)
(-2, (1/3)-2) = (-2, (3/1)2) = (-2, 9)

Connect the points
and draw the graph.

The asymptote of an exponential function
is the x-axis.
So, as x goes to ∞,
the graph gets close to the x-axis.
(not below the x-axis.)

So this is the graph of y = (1/3)x.