Graphing Exponential Functions

Graphing Exponential Functions

How to graph exponential functions: examples and their solutions.

Graphs

The graph of y = 2^x. If (base) > 1, then the graph of the exponential function shows exponential growth.

This is the graph of y = 2x.

It passes through (0, 1).
(blue point)

As x increases 2, 3, 4, ...
y increases 22, 23, 24 ... .

Its asymptote is the x-axis,
because as x → -∞,
y = 2x gets close to 0.
(But not equal to 0.)

Just like y = 2x,

if (base) > 1,
then the graph of an exponential function
shows exponential growth.

The graph increases faster than
any polynomial functions (y = xn).

The graph of y = (1/2)^x. If 0 < (base) < 1, then the graph of the exponential function shows exponential decay.

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.

The graphs of y = bx 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

Graph the given function. y = 4*2^x

Change y = 4⋅2x to y = 2x + 2.

Product of powers

y = 2x + 2 is the image of y = 2x
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⋅20 = 4.

The base (2) is greater than 1.
So the graph shows exponential growth.

Use these hints to draw the graph.

Example 2

Graph the given function. y = 3^-x + 1

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.