Expert: Chen Min - 1/7/2011

Question
Hi Chen, I am a maths tutor in England and I am having a lot of trouble understanding |y|=2x-3. I understand when the modulus is on the x, like y=|2x-3| , and I can see there is some kind of reflection involved, but when I do, say, |y|=(x-1)(x-2)(x-3) on Autograph, I am badly confused. If you could explain I would greatly appreciate it. Thanks a lot.

Answer
For |y|=2x-3, it's easy.
1)Draw out y=2x-3
2)Erase everything below x-axis, so the value of the remaining function is non-negative
 (Hence modulus y won't be contradicted)
3)Note that y may take a negative value while modulus y is still positive, hence make a
 'reflection' below the image in 2) such that the overall image is symmetrical w.r.t. x-axis

Similarly, you can do the same to the more complicated version |y|=(x-1)(x-2)(x-3)

The big point is not to treat x and y as rigidly independent and dependent variables but to go
with the graph. Take |y|=x-1 as an example, you see if x=0 then |y|=-1, impossible, so the graph should not have any part to the left of x=1. If x=2, |y|=1, y=1 and -1, and thus the graph should be a symmetrical one.

Hope this helps.