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Graphing Radical Functions
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.
Example
What is your answer?
 
Example
What is your answer?
 
Example
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M Ransom

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