Question

How do you figure out if an absolute value function like |y|=|x| is symmetric to the x-axis, y-axis, or the origin? 

There is 2 posibilities

either y=x   or y=-x

so

\(|y|=|x|\\ \text{is equivalent to}\\ y=x\;\;\;\cup\;\;\;y=-x\)

To me it is obvious that this is symmetrical about both the x and the y axis

And it has point symmetry of degree 4 about (0,0)

Maybe to you it is not obvious..

You could draw it and see

OR

If it is symmetrical about the y axis then  for any point x=a and x=-a, the y value/s will be the same.

That is true, both times the y value can be a or -a

If it is symmetrical about the y axis then  for any point y=b and y=-b, the x value/s will be the same.

That is true, both times the x value can be b or -b

SO it is symmetrical about both the x and the y axes.

Here is the graph :)

Thank you Melody.