Radical functions involve the
square root of a quantity such as
where the formula for the
function has a radical. Let us start with a fundamental function:
. The
graph of this
function is shown below. Note that the domain requires that
. This is because we cannot take the
square root of a negative number.
To
graph the
function 
, we note that this is a
translation of the
graph of
2 units to the right. The domain is
. The
graph is shown below.
Similarly,
is
translated 1 unit to the left. The domain is
. The
graph is shown below.
The fundamental
graph of
is a portion of the
graph of
. If
, then
. The
graph of
is shown below. This is a
reflection of across the x-axis.
Using the principles of
translation and reflection, we can
graph various functions involving radicals. Some
examples are shown below.
#1. Let 
. This is our fundamental function 
translated 1 unit to the right and then reflected across the x-axis. The domain is 
. The graph is shown below. 
#2. Let 
. This is our fundamental function 
moved 2 units to the left and then 1 unit down. The domain still depends upon keeping what is under the radical from being negative. Thus the domain is 
. The graph is shown below. 
#3. Let 
. This is our fundamental function 
moved 3 units right, reflected across the x-axis, and then moved 2 units up. The domain is 
. The graph is shown below. 
#4. Let 
. This is still a variation of our fundamental function 
. The domain requires that 
. The graph shows the characteristic radical curve increasing from the point 
. The graph is shown below left. The graph shown below on the right compares this to 
so that we can see the effect that the coefficient 2 of x has on the graph. The coefficient of 2 forces the graph to increase more rapidly. 
