Absolute Value Equation with Diagonal Axis of Symmetry

Date: 01/10/2002 at 00:58:32
From: Anonymous
Subject: Absolute Value Equation with Diagonal Axis of Symmetry

What's an example of an absolute value equation where the axis of 
symmetry is a diagonal line, say y = x?

Date: 01/10/2002 at 09:11:01
From: Doctor Peterson
Subject: Re: Absolute Value Equation with Diagonal Axis of Symmetry

Hi, 

That symmetry implies that swapping the roles of x and y in the 
equation will leave it unchanged. So any equation you can write in 
which x and y play interchangeable roles will have the desired 
symmetry. For example, try

    |x + y| = |x - y|

Swapping x and y gives

    |y + x| = |y - x|

which is equivalent.

Of course, many such equations you might write will have empty graphs, 
or otherwise be uninteresting, so the hard part is to choose one that 
is of interest, and to be able to graph what you get. The one I just 
suggested is very interesting, though the graph is surprisingly 
simple. If, say, you multiply the right side by 2, or take off the 
absolute value sign on the left, it will be even more interesting. 
Have fun playing with these.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   

Date: 01/10/2002 at 12:58:57
From: Anonymous
Subject: Absolute Value Equation with Diagonal Axis of Symmetry

I'm still not getting the graphs I want. Basically, I'm looking to 
graph an absolute value graph where essentially, I'm taking y = |x| 
and "rotating it" 45 degrees, say to the right (So I now have a 
V-shaped graph with an "axis of symmetry" of y = x.)  

As you suggested, how can I get such a graph from graphing 
|x + y| = |x - y| or |y + x| = |y - x| or, as you suggest, by
"multiplying the right side by 2, or taking off the absolute value 
sign on the left."  I'm just not following this.  Thanks.

Date: 01/10/2002 at 13:33:18
From: Doctor Peterson
Subject: Re: Absolute Value Equation with Diagonal Axis of Symmetry

Hi, 

You didn't say you wanted to actually rotate y=|x|; I took your 
question to mean you wanted an equation, involving the absolute value, 
that had the desired symmetry. But in fact, I told you how to do what 
you want. Try graphing my last suggestion,

    x + y = |y - x|

and see what it looks like.

Were you not able to graph the equations I suggested? I'll show you 
how to graph the first:

    |x + y| = |x - y|

There are four cases, depending on whether each operand is positive or 
negative. Notice that x+y is positive when y > -x, so you can draw the 
line y = -x lightly and indicate that above it, x+y>0. Similarly, x-y 
is positive when y < x, so draw the line y=x lightly and indicate that 
x-y>0 below it. This divides the plane into four regions.

    x+y>0, x-y>0:

        x + y = x - y
        2y = 0
        y = 0
        So darken the x axis within this part of the plane.

    x+y>0, x-y<0:

        x + y = y - x
        2x = 0
        x = 0
        So darken the y axis within this part of the plane.

The other two regions repeat the same work. You will end up with a 
graph consisting of the x and y axes.

Now do the same with the new equation I suggested above, and compare 
what you get with y=|x|.

There are standard techniques for rotating a graph that depend on 
replacing the variables with a new set of rotated variables. To rotate 
any graph, such as y=|x|, 45 degrees to the right, you have to replace

    x with (x-y)/sqrt(2)
and
    y with (x+y)/sqrt(2)

Try doing that with y=|x|, and you will get the equation I suggested.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/