# Graphing Exponential Functions

How to graph exponential functions: examples and their solutions.

## Graphs

This is the graph of *y* = 2^{x}.

It passes through (0, 1).

(blue point)

As *x* increases 2, 3, 4, ...*y* increases 2^{2}, 2^{3}, 2^{4} ... .

Its asymptote is the *x*-axis,

because as *x* → -∞,*y* = 2^{x} gets close to 0.

(But not equal to 0.)

Just like *y* = 2^{x},

if (base) > 1,

then the graph of an exponential function

shows exponential growth.

The graph increases faster than

any polynomial functions (*y* = *x*^{n}).

This is the graph of *y* = (1/2)^{x}.

It passes through (0, 1).

(blue point)

As *x* increases 2, 3, 4, ...*y* decreases (1/2)^{2}, (1/2)^{3}, (1/2)^{4} ... .

Its asymptote is the *x*-axis,

because as *x* → ∞,*y* = (1/2)^{x} gets close to 0.

(But not equal to 0.)

Just like *y* = (1/2)^{x},

if 0 < (base) < 1,

then the graph of an exponential function

shows exponential decay.

The graphs of *y* = *b*^{x} and *y* = (1/*b*)^{x}

show the reflection in the *y*-axis.

(*b* > 0, *b* ≠ 1)

Both graphs pass through (0, 1).

(blue point)

And both graphs' asymptotes are the same.

(*x*-axis, *y* = 0)

## Example 1

Change *y* = 4⋅2^{x} to *y* = 2^{x + 2}.

Product of powers

*y* = 2^{x + 2} is the image of *y* = 2^{x}

under the translation (*x*, *y*) → (*x* - 2, *y*).

Translation of a function

So the 'blue point' is (0 - 2, 1).

And the asymptote is *y* = 0. (= *x*-axis)

The *y*-intercept is 4⋅2^{0} = 4.

The base (2) is greater than 1.

So the graph shows exponential growth.

Use these hints to draw the graph.

## Example 2

Change *y* = 3^{-x} + 1 to *y* = (1/3)^{x} + 1.

*y* = (1/3)^{x} + 1 is the image of *y* = (1/3)^{x}

under the translation (*x*, *y*) → (*x*, *y* + 1).

Translation of a function

So the 'blue point' is (0, 1 + 1).

And the asymptote is *y* = 0 + 1.

The base (1/3) is between 0 and 1.

So the graph shows exponential decay.

Use these hints to draw the graph.