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.