Exponential Functions
Exponential Functions
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) = 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) = a x 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.
The graph of an exponential function can also be reflected over the x -axis or the y -axis, and rotated around the origin, as in Heading .
The general form of an exponential function is f (x) = cƒa x-h + k , where a is a positive constant and a≠1 . a is called the base. The graph has a horizontal asymptote of y = k and passes through the point (h, c + k) .
The domain of f (x) is and the range of f (x) is .
