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

  • Graph y = sqrt(16 - x^2)

    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.





    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 y = sqrt(x^2 - 4x)
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    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 y = cbrt(x - 5)

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:

      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 Accessed


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