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Exponential Functions

Exponents can be variables. Variable exponents obey all the properties of
exponents listed in Properties of Exponents.

An exponential function is a function that contains a variable exponent.
For example, *f* (*x*) = 2^{x} and *g*(*x*) = 5ƒ3^{x} are exponential functions. We
can graph exponential functions. Here is the graph of *f* (*x*) = 2^{x}:

Figure %: *f* (*x*) = 2^{x}

The graph has a horizontal asymptote at

*y* = 0, because

2^{x} > 0 for all

*x*.
It passes through the point

(0, 1).

We can translate this graph. For example,
we can shift the graph down 3 units and left 5 units. Here is the graph of *f* (*x*) = 2^{x+5} - 3:

Figure %: *f* (*x*) = 2^{x+5} - 3

This graph has a horizontal asymptote at

*y* = - 3 and passes through the point

(- 5, - 2).

We can stretch and
shrink the graph vertically by
multiplying the output by a constant--see
Stretches. For example, *f* (*x*) = 3ƒ2^{x} is stretched vertically by a factor of 3:

Figure %: *f* (*x*) = 3ƒ2^{x}

This graph has a horizontal asymptote at

*y* = 0 and passes through the point

(0, 3).

We can also graph exponential functions with other bases, such as *f* (*x*) = 3^{x}
and *f* (*x*) = 4^{x}. We can think of these graphs as differing from the graph of
*f* (*x*) = 2^{x} by a horizontal stretch or shrink: when we multiply the input of
*f* (*x*) = 2^{x} by 2, we get *f* (*x*) = 2^{2x} = (2^{2})^{x} = 4^{x}. Thus, the graph of
*f* (*x*) = 4^{x} is shrunk horizontally by a factor of 2 from *f* (*x*) = 2^{x}:

Figure %: *f* (*x*) = 4^{x}

This graph has a horizontal asymptote at

*y* = 0 and passes through the point

(0, 1).

The graph of *f* (*x*) = *a*^{x} does not always differ from *f* (*x*) = 2^{x} by a rational
factor. Thus, it is useful to think of each base individually, and to think of
a different base as a horizontal stretch for comparison purposes only.