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Graphing Radical Functions: More Examples (page 3 of 3)

• Graph

 First I'll find the domain: Again, the contents of the square-root function are a quadratic, so it may be simplest to find the domain of the radical function by looking at the graph of the quadratic inside the square root.     In this case, 16 – x2 is positive between –4 and 4, so this will be the domain of the radical. Next, I'll find some additional plot points:    Note: I used a calculator to approximate the y-values. Finally, I'll do the graph:

This graph is just what it looks like: the top half of a circle. As a matter of fact, it's the top half of the circle centered at the origin and having radius r = 4. The negative of this square-root function would give you the bottom half of the same circle.

• Graph
 First, I'll find the domain: it may be simplest to determine the domain of the radical function by looking at the graph of the quadratic argument of the function: This quadratic is positive (higher than the x-axis) before x = 0 and after x = 4. In other words, the domain of the radical is split into TWO pieces. This means that the graph of the radical function will also be in two pieces: one part on the left, stopping at x = 0, and another part on the right, starting at x = 4. There will be nothing but blank space between these two pieces. Keeping this domain restriction in mind, I'll carefully find some plot-points:
 Finally, I'll do the graph:

Here's an example of a cube-root function:

• Graph

There are no domain constraints with a cube root, because you can graph the cube root of a negative number. So you don't have to find the domain; the domain is "all x". (Note: Since you can take the fifth root, seventh root, ninth root, etc., of negative numbers, there are no domain considerations for any odd-index radical function. You only have to find the domain whenever you are dealing with even-index radical functions: a square root, a fourth root, a sixth root, etc.)

 There are no domain constraints, so I'll go straight to finding some plot points:   Note that you can find the x-values that give "neat" y-values by setting the argument of the cube root equal to a perfect cube, such as 1, 8, or 27.

Warning: Radicals graph as curved lines. Don't succumb to the temptation of trying to put a straight line through these points. Instead, use enough plot-points to clearly show the shape of the graph, and then draw the graph complete with its curves:

For radical graphs, it is well worth taking the time to find lots of plot-points for your T-chart. Then draw very neat axes and scales, and draw your line carefully. Don't just slap these graphs together, because if you do, you'll probably get many of them wrong.

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 Cite this article as: Stapel, Elizabeth. "Graphing Radical Functions: More Examples." Purplemath. Available from     https://www.purplemath.com/modules/graphrad3.htm. Accessed [Date] [Month] 2016

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