# Minor clarification on modulus graphs

• Jan 18th 2010, 06:10 AM
Punch
Minor clarification on modulus graphs
If the equation of a curve is $y=(x-4)^2+(-1)$. It shows that the curve has a minimum point of (4,-1).

Now, if I have a modulus of this equation, that is $|(x-4)^2+(-1)|$, the turning point would be (4,1). Is the turning point a maximum point or minimum point now?
• Jan 18th 2010, 06:25 AM
HallsofIvy
Quote:

Originally Posted by Punch
If the equation of a curve is $y=(x-4)^2+(-1)$. It shows that the curve has a minimum point of (4,-1).

Now, if I have a modulus of this equation, that is $|(x-4)^2+(-1)|$, the turning point would be (4,1). Is the turning point a maximum point or minimum point now?

Suppose x is close to 4, say x= 3.9 or x= 4.1. If x= 3.9 $|(x-4)^2- 1|= |(3.9- 4)^2- 1$[tex]= |(-.1)^2- 1|= |.01- 1|= |-.99|= .99 which is less than 1. If x= 4.1, $|(x-4)^2- 1|= |(4.1-4)^2- 1|$ $= |(.1)^2- 1|= |.01- 1|= |-.99|= .99$ which is also less than 1. Now, do YOU think (4,1) is a maximum or minimum point?

There are, in fact, three turning points for that graph, (3, 0), (4, 1), and (5, 0).
• Jan 18th 2010, 06:29 AM
Punch
Well, I am just speaking in general, looks like you are going really in-depth. The turning point I am referring to is (4,1), just unclear about whether it would be called the maximum point or minimum point...
• Jan 18th 2010, 06:54 AM
Soroban
Hello, Punch!

Quote:

The equation of a curve is $y\:=\:(x-4)^2+(-1)$.
It shows that the curve has a minimum point of (4,-1).

Now, if I have a modulus of this equation, that is: $y \:=\:|(x-4)^2+(-1)|$,
. . the turning point would be (4,1).

Is the turning point a maximum point or minimum point now?

Visualize their graphs . . .

The graph of the parabola looks like this:
Code:

```      |       |  *                *       |       |       |  *              *       |     --+----*-------------*-----       |    *          *       |      *      *       |          *       |```

There is an absolute minimum at (4, -1).
There is no maximum.

With absolute values, any point below the $x$-axis
. . is reflected upward.

The graph of the modulus function is:

Code:

```      |       |  *                *       |       |          *       |  *  *      *  *       |    *          *     --+----*-------------*-----       |    3            5       |```

It has absolute minimums at (3,0) and (5,0).
. . and a relative (local) maximum at (4,1).

• Jan 19th 2010, 04:25 AM
Punch
Quote:

Originally Posted by Soroban
Hello, Punch!

Visualize their graphs . . .

The graph of the parabola looks like this:
Code:

```      |       |  *                *       |       |       |  *              *       |     --+----*-------------*-----       |    *          *       |      *      *       |          *       |```
There is an absolute minimum at (4, -1).
There is no maximum.

With absolute values, any point below the $x$-axis
. . is reflected upward.

The graph of the modulus function is:

Code:

```      |       |  *                *       |       |          *       |  *  *      *  *       |    *          *     --+----*-------------*-----       |    3            5       |```
It has absolute minimums at (3,0) and (5,0).
. . and a relative (local) maximum at (4,1).

Thanks Soroban, this was what I was asking for!