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.

Examples
For each of the following functions, state the domain and describe the graph.

M Ransom

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