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

1 Answer
Jan 19, 2018

Answer:

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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