Geometry › How to graph an exponential function
Define a function as follows:
Give the vertical aysmptote of the graph of .
The graph of does not have a vertical asymptote.
Since any number, positive or negative, can appear as an exponent, the domain of the function is the set of all real numbers; in other words,
is defined for all real values of
. It is therefore impossible for the graph to have a vertical asymptote.
Define a function as follows:
Give the horizontal aysmptote of the graph of .
The horizontal asymptote of an exponential function can be found by noting that a positive number raised to any power must be positive. Therefore, and
for all real values of
. The graph will never crosst the line of the equatin
, so this is the horizontal asymptote.
Evaluate .
The system has no solution.
Rewrite the system as
and substitute and
for
and
, respectively, to form the system
Add both sides:
.
Now backsolve:
Now substitute back:
and
Give the -intercept of the graph of the function
Round to the nearest tenth, if applicable.
The graph has no -interceptx
The -intercept is
, where
:
The -intercept is
.
An important part of graphing an exponential function is to find its -intercept and concavity.
Find the -intercept for
and determine if the graph is concave up or concave down.
.
Give the -intercept of the graph of the function
Round to the nearest hundredth, if applicable.
The graph has no -intercept
The -intercept is
:
is the
-intercept.
Define a function as follows:
Give the -intercept of the graph of
.
The -coordinate ofthe
-intercept of the graph of
is 0, and its
-coordinate is
:
The -intercept is the point
.
Give the domain of the function .
The set of all real numbers
Let . This function is defined for any real number
, so the domain of
is the set of all real numbers. In terms of
,
Since is defined for all real
, so is
; it follows that
is as well. The correct domain is the set of all real numbers.
Give the -intercept(s) of the graph of the equation
The graph has no -intercept.
Set and solve for
:
Define a function as follows:
Give the -intercept of the graph of
.
Since the -intercept is the point at which the graph of
intersects the
-axis, the
-coordinate is 0, and the
-coordinate is
:
,
The -intercept is the point
.