Graphing Radical Functions: More Examples

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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.

y = 16 - x^2

Next, I'll find some additional plot points:

Note: I used a calculator to approximate the y-values.

T-chart

Finally, I'll do the graph:

y = sqrr(16 - x^2)

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
Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

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:

y = x^2 - 4x

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:

T-chart, from calculator

Finally, I'll do the graph:

sqrt(x^2 - 4x)

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.

T-chart

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:

y = cbrt(x - 5)

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
http://www.purplemath.com/modules/graphrad3.htm. Accessed

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