# How do you graph y = -abs(x+10)?

Apr 2, 2015

First, lets remember the absolute value operator:

$\forall c \in \mathbb{R}$ (for every element $c$ in the real numbers set)

$\mathmr{if} c \ge 0 , \left\mid c \right\mid = c$

$\mathmr{if} c < 0 , \left\mid c \right\mid = - c$

When graphing an absolute value operator, there are actually $2$ graphs. Because the absolute value operation behaves in $2$ different ways related to its input.

So we need to find the critical point of the absolute value. This means, we need to find the exact value of $x$ where the greater values will result the output as is and the smaller values will result the output as negated.

Now it is obvious that $0$ is the critical point of the absolute value operation.

To find the critical point in this problem:

$x + 10 = 0$

$x = - 10$

When $x$ is greater than $- 10$ the input will be positive. When smaller, it will be negative.

Now we are ready to graph the line.

When $x \ge - 10$, $y = - \left(x + 10\right) = - x - 10$

When $x < - 10$, $y = - \left(- 1\right) \cdot \left(x + 10\right) = x + 10$

The graph will look like this:

graph{y = - abs(x+10) [-20, 10, -5, 5]}

Why there is no positive $y$? Remember the conditions while graphing the lines. ($x \ge 10$ and $x < - 10$) When you try to plug some values of $x$ you will see that there is no chance for $y$ to be positive.