# 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 = a^{x} (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, a^{0}) = (0, 1)

Zero Exponent

The asymptote of the graph is the x-axis.

So a^{x} is always plus.

### 0 < a < 1

This is the graph of y = a^{x} (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

### Example

### Solution

y = 2^{x} is an exponential function.

So first draw (0, 1).

Put x = 1, 2, 3 into y = 2^{x}.

And draw the points.

(1, 2^{1}) = (1, 2)

(2, 2^{2}) = (2, 4)

(3, 2^{3}) = (3, 8)

Put x = -1, -2, -3 into y = 2^{x}.

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 = 2^{x}.

## Example 2

### Example

### 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}.