Challenge Transformation Problem

[Baron]

Given that the coordinate (a,b) is on the function y = F(x). What will the coordinates be when the function is transformed to
y = F(abs x) ?

I know what the y= F(abs x) does. Everything that is in quadrant 1
will be reflected over the y-axis into quadrant 2. Everything that is
in quadrant 4 will be reflected over the y-axis into quadrant 3. The
graph will have vertical symmetry over the y-axis.

I also know that if coordinates (a,b) are on the graph, y = F(x) and a
is positive, the coordinates will stay (a,b). I just don't know how
the coordinates (a,b) will change if a is negative.

I am also aware that the graph y = F(abs x) will have vertical symmetry over the line y-axis

 

[tkhunny]

Just how are you numbering quadrants? "Absolute Value" turns negative to positives. Are you SURE you are going the right direction?

Positive values of 'a' are unaffected by |a|. Not so for a < 0.

 

[Baron]

I number the quadrants counterclockwise starting with the quadrant on the top right corner as 1.

The transformation is y = f(x) --> y = f ( |x|) not y = f(x) --> y = | f(x) |

The difference between the two above functions is that for the first function, x will always be positive. (Therefore symmetry over y-axis). For the second function, y will always be positive.

 

[tkhunny]

Let's try again.

If a > 0, f(|a|) = f(a)
If a < 0, f(|a|) = f(-a) -- Remember that since a < 0, then -a > 0.

This is a reflection across the y-axis for a < 0. That's Quadrant II to Quadrant I and III to IV. Those things already in Quadrants I and 4 don't move.

 

[Baron]

When you say "This is a reflection across the y-axis for a < 0." Is that implying, when 'a'' is negative the coordinates (a,b) will become (|a|,b) so the 'b' coordinate does not change?

That is not true. Let, for example f(x) = (x + 3)^2. Then f(|x|) will be equal to (|x|+3)^2.
For f(x) = (x + 3)^2, the coordinates (-5, 4) are on the graph.
However for (|x|+3)^2. , the coordinates (5, 4) are not on the graph

So there isn't a reflection across the y-axis for a<0

 

[tkhunny]

Very good. I was inadvertently assuming some very odd symmetry on the y-axis. That is incorrect, as you point out, unless "very odd symmetry " means "no symmetry whatsoever". You're not quite off the hook, though, since you stated in your original post that the transformation "will have vertical symmetry over the y-axis". Care to explain that? You are very close.

It remains incorrect to say that quadrants I and IV are reflected to II and III. The points from quadrants I and IV go nowhere.

What needs to be said is that for a < 0, the locus of points for f(a) is mapped to a reflection of Quadrants I and IV. Of course, this presents quite a bit of difficulty when a > 0 is NOT in the Domain of f. Try f(x) = ln(-x).

 

[Baron]

What I meant by "vertical symmetry over the y-axis" is easier to explain when I use an example.

Let f(x) = (x + 3)^2. Then f(|x|) will be equal to (|x|+3)^2.
Make a table of values for f(x) = (x + 3)^2

(-3, 0)
(-2, 1)
(-1, 4)
(0, 9)
(1, 16)
(2, 25)
(3, 36)

For f(|x|) = (|x|+3)^2. The coordinates
(-3, 0) become (-3, 36)
(-2, 1) --> (-2, 25)
(-1, 4) --> (-1, 16)
(0, 9) --> (0, 9)
(1, 16) --> (1, 16)
(2, 25) --> (2, 25)
(3, 36) --> (3, 36)

If you graph these points, the translated graph ( f(|x|) = (|x|+3)^2) will have vertical symmetry over the y-axis.
So everything in Quadrant I and IV are reflected over the y-axis, but also the original points where the x-coordinate is already positive don't change. And everything that was originally in Quadrant II and III disappear.

Saying all that, I still don't know how the coordinates (a,b) will change, since it depends on whether "a" is positive or negative

 

[tkhunny]

Think in these very hard. They are not the same.

"So everything in Quadrant I and IV is reflected over the y-axis"
"for a < 0, f(a) is mapped to a point in the reflection of Quadrants I and IV"

The difficulty is that f(a) and f(-a) are not particularly related. It would be trivial if they were.

 

[Baron]

I guess the phrase "quadrants I and IV are reflected to II and III" were a bad choice of words because as you said those coordinates don't change. What I meant but didn't know how to phrase was that those coordinates were "mapped to a point in the reflection of Quadrants I and IV" This will still result in vertical symmetry over the y-axis.

I understand that f(a) and f(-a) are not really related. That's why I'm having problems finding a general formula for the transformation as it differs over whether the x coordinate is positive or negative.