Problem

Examine the graph of the equation

for symmetry about the x-axis, y-axis, and about the origin. Sketch the graph.

Solution

A note first about symmetries: we do not refer to this equation as being odd, or even, despite its qualities of symmetry, because it is not a function.

Symmetry about the x-axis

The graph will have symmetry about the x-axis if, whenever the point (a, b) is on the graph, (a, –b) is on the graph.

since

so (a, –b) is on the graph.

Symmetry about the y-axis

The graph will have symmetry about the y-axis if, whenever the point (a, b) is on the graph, (–a, b) is on the graph.

since

so (–a, b) is on the graph.

Symmetry about the origin

The graph will have symmetry about the origin if, whenever the point (a, b) is on the graph, (–a, –b) is on the graph.

since

and

so (–a, –b) is on the graph.

Sketching the graph

Knowing the symmetries of the equation helps in sketching its graph. Because the graph of the equation has all three symmetries we tested, we need only sketch the portion in the first quadrant (for x ³ 0 and y ³ 0), then reflect this to fill in the other three quadrants.

We are able to define a function (which the original equation is not) by restricting the domain to nonnegative values of x.

In fact, the domain is restricted further to x ³ 2, to avoid square roots of negative numbers.

Now copy this piece of the graph by reflecting across the x axis, y axis, and around the origin.