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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}

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

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}

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}

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.

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