Question

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

Answer 1

Read below.

Explanation:

Let's think of it this way:
`abs (a)=a` and `abs (-a)=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 `2x+5>=0`, then `abs (2x+5) =2x+5`
When `2x+5<0`, then `abs (2x+5)=-2x-5`

Let's apply this to our function.
When `2x+5>=0`, then `y=x(2x+5)=>y=2x^2+5x`
When `2x+5<0`, then `y=x(-2x-5)=>y=-2x^2-5x`
We first garph these two parabolas:
desmos.com
Now, we ask ourselves,"For what values of `x` does `2x+5>=0`hold true?"
Similarly, "For what values of `x` does `2x+5<0` hold true?"

To find out the answer, we solve each inequality.
`2x+5>=0`
`2x>=-5`
`x>=-5/2`

`2x+5<0`
`2x<-5`
`x<-5/2`

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

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