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### Graphing an Equation with Absolute Value

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Date: 10/15/95 at 18:17:6
From: Tucker - Joanne
Subject: Graphing

Dear Dr. Math,

How would you approach graphing negative 2 times the
absolute value of x plus 2?

Our Algebra 2 Class
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```
Date: 10/15/95 at 21:15:49
From: Doctor Andrew
Subject: Re: Graphing

So you want to graph -2|x+2|.  Well, x = -2 is where the absolute value
kicks in, so why not graph this:

-2(x+2) for x >= -2

and

-2[-(x+2)] for x < -2

-Doctor Andrew,  The Geometry Forum
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```
Date: 01/07/98
From: Scott Mehring
Subject: Archives

I am going through your archives. Someone asks how you graph -2|x+2|.
I say the value of x is -2.  -2|x+2|=0  =>  |x+2|=0  =>  x = -2.

You say that you should graph x>=-2 and x<-2. Well, that is all x's
the whole number line. How can that be?
```

```
Date: 01/07/98 at 12:33:24
From: Doctor Sonya
Subject: Re: Archives

Dear Scott,

What might have made the answer confusing is that the math doctor who
answered it didn't go all the way and finish the problem. Let's look at
what he did do, and maybe we can find the source of your confusion.
He wrote:

"So you want to graph -2|x+2|. Well, x = -2 is where the absolute value
kicks in, so why not graph this:

-2(x+2) for x >= -2

and

-2[-(x+2)] for x < -2"

The first thing you have to do is think of -2|x+2| as a function.  When
you are asked to graph it, it's just like having to graph the function

f(x) = x-17

only a little bit more complicated with the absolute value signs there.

Then the doctor divided the function into two parts, at the point x = -2,
and found the graph of each part seperately. When x is greater than -2,
x+2 is a positive number, so it is equal to its absolute value: |x+2|.
This means that:

-2|x+2| = -2(x+2)

so you can graph the expression on the righthand side of the equals sign,
which doesn't have an absolute value in it.

When x is less than -2, x+2 is a negative number, so

|x+2| = -(x+2)

Do you see why this is true? Again, you can graph the expression without
any absolute value signs in it.

Try this out yourself with a piece of graph paper and see what you get.

Now it's true that x>=-2 and x<-2 is the whole number line, just as you
said. But these values for x are not the graph, merely the numbers we
plug into the function -2|x+2|, so it's okay for them to cover the whole
number line. In fact we want them to, because we want to be able to plug
every x into our function.

I hope this helps.  And thanks for investigating our archives. If
anything else seems fishy to you, let us know.

-Doctor Sonya,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
High School Basic Algebra

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