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Absolute Value Functions

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Timtasticle:
Can someone explain how these things work as dumbed down as possible? :)

kingmar:
Think of it as taking it both ways. :shock:

No seriously. Imagine a function. Then, anything below the x-axis is reflected above it. So, take |x|.

If y = x, at -1, y = -1.

If y = |x|, at -1, ordinarily y should equal -1. But as all negative y values are reflected in the x-axis, thus y = 1 instead. Instead of a straight line graph, you get a sorta V shaped graph.

The graph is defined separately - so for y = |x|, for domain (-infinity, 0] the equation is -x. For domain (0,infinity), the equation is x.

Incidentally, there is no derivative at cusps, or sharp points. Usually, it is at the x-intercepts the derivative is not defined, but check the graph just in case.  If it looks sharp, it ain't defined. Open circles.

BenBenMan:
Absolute just means, if it's negative, you make it positive. That's really all there is to it. If it's positive, then nothing happens to it, if it's negative, you make it positive.

So if your graph was y=|x^2|, then it would look exactly the same. However if it was y=|x^2 - 1|, then for where it's below the x axis, you would draw on top of the x axis:



Here the red graph is x^2-1, and the blue graph is |x^2-1|. Note that the blue graph has gone over the top of the red graph for the part of the graph on the left of the left x intercept and the right of the right x intercept.

EDIT: Beaten to it! Oh well :P

Timtasticle:
Oww I see. Thanks!

This is probably a stupid question...But how about when you have like y= 5+ |x^2-1|

Oh how I wish I had have actually done some work in methods...

kingmar:
Oh, but yours looks so much better!

A further note: When drawing, say |f(x)|, you need to draw the part that isn't reflected as well (i.e. trace over the positive part if they give you the graph).

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