page 1 of 2
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) = 2x and g(x) = 5ƒ3x are exponential functions. We
can graph exponential functions. Here is the graph of f (x) = 2x:
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) = 2x+5 - 3:
We can stretch and
shrink the graph vertically by
multiplying the output by a constant--see
Stretches. For example, f (x) = 3ƒ2x is stretched vertically by a factor of 3:
We can also graph exponential functions with other bases, such as f (x) = 3x
and f (x) = 4x. We can think of these graphs as differing from the graph of
f (x) = 2x by a horizontal stretch or shrink: when we multiply the input of
f (x) = 2x by 2, we get f (x) = 22x = (22)x = 4x. Thus, the graph of
f (x) = 4x is shrunk horizontally by a factor of 2 from f (x) = 2x:
The graph of f (x) = ax does not always differ from f (x) = 2x 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.
Take a Study Break!