# Thread: Minor clarification on modulus graphs

1. ## Minor clarification on modulus graphs

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

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

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

Now, if I have a modulus of this equation, that is $\displaystyle |(x-4)^2+(-1)|$, 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 $\displaystyle |(x-4)^2- 1|= |(3.9- 4)^2- 1$[tex]= |(-.1)^2- 1|= |.01- 1|= |-.99|= .99 which is less than 1. If x= 4.1, $\displaystyle |(x-4)^2- 1|= |(4.1-4)^2- 1|$$\displaystyle = |(.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).

3. 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...

4. Hello, Punch!

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

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

5. 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 $\displaystyle x$-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!