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# How do you graph y=xabs(2x+5)?

Jan 19, 2018

#### Explanation:

Let's think of it this way:
$\left\mid a \right\mid = a$ and $\left\mid - a \right\mid = a$

For our first case, we are just bringing out $a$ outside the absolute value.
For the second case, we are finding the opposite of whatever was inside the absolute value sign.

So we can say that:
When $2 x + 5 \ge 0$, then $\left\mid 2 x + 5 \right\mid = 2 x + 5$
When $2 x + 5 < 0$, then $\left\mid 2 x + 5 \right\mid = - 2 x - 5$

Let's apply this to our function.
When $2 x + 5 \ge 0$, then $y = x \left(2 x + 5\right) \implies y = 2 {x}^{2} + 5 x$
When $2 x + 5 < 0$, then $y = x \left(- 2 x - 5\right) \implies y = - 2 {x}^{2} - 5 x$
We first garph these two parabolas:

Now, we ask ourselves,"For what values of $x$ does $2 x + 5 \ge 0$hold true?"
Similarly, "For what values of $x$ does $2 x + 5 < 0$ hold true?"

To find out the answer, we solve each inequality.
$2 x + 5 \ge 0$
$2 x \ge - 5$
$x \ge - \frac{5}{2}$

$2 x + 5 < 0$
$2 x < - 5$
$x < - \frac{5}{2}$

This is actually telling us that for any $x$ values greater than or equal to $- \frac{5}{2}$, the blue parabola will apply. When $x$ is smaller than $- \frac{5}{2}$, then the red parabola will apply. The green graph is the graph of our function.
So we now have:

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