Graphing Radical (Square-Root) Functions

Since "graphing radical functions" usually means "graphing functions involving square roots", most of the examples in this lesson will deal with square roots.

What is the basic square-root graph?

The basic square-root function is , and its graph looks like this:

Since there can't be a negative inside the radical, then the domain is x ≥ 0. The graph begins at (0, 0) and then heads to the right, growing slowly upward and x speeds rightward. Nearly all square-root functions' graphs will be some version of the above.

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• Graph

How do you graph square-root functions?

In sum, the steps for graphing radical (that is, square root) functions are these:

1. Find the domain of the function: set the insides of the radical "greater than or equal to" zero, and solve for the allowable x-values.
2. Make a T-chart to hold your plot points.
3. Pick x-values within the domain (including the "or equal to" endpoint of the domain), and simplify to get the corresponding y-values. (It can be helpful to find x-values that will give you a perfect square inside the radical; these will be the nicest points to plot.)
4. Make a neat axis system (yes, using a ruler to make straight lines and to mark of consistent scales).
5. Plot your points on the axis system.
6. Draw a curved line through the points, and remember to mark clearly the endpoint (or start point, depending on your perspective) so that it's clear to the grader that you know what you're doing.

How do you draw a radical graph?

When drawing a radical graph, keep the following in mind:

• You should expect radical functions to graph as curved lines. Do not try to put a straight line through the points that you've plotted.
• You should also expect radical graphs to be much wider (that is, stretched out, side-to-side) than they are tall. Be sure to use adequate space on your paper for a good graph.

You saw, in the above example, how I was careful to pick x-values for my T-chart that gave nice neat y-values. This is not required, nor is this even always reasonably feasible. But it can make graphing much simpler so it can be worth the time to be sure of your plot points, especially if you've got a graphing calculator that can do a table of values for you.