Question

How do you solve the absolute value equation `y=-2abs(5x+8)+4` and find the vertex, x intercepts and y intercept?

Answer 1

Given `y = -2abs(5x+8)+4`
If `(5x+8)<=0` then `y=-2(-5x-8)+4` which is a linear equation.
Similarly
If `(5x+8)>=0` then `y=-2(5x+8)+4` which is also a linear equation

Any vertex must exist at the point where these two condition meet;
That is when `(5x+8) = 0`
at `(x,y) =(-8/5,4)`

The y-intercept occurs when `x=0`
`y = -2abs(5(0)+8)+4 = -12`

The x-intercepts occur when `y=0`
Case 1: `(5x+8)>= 0`
`0 = -2(5x+8)+4`
`5x+8 = 2`
`x= -6/5`

Case 2: `(5x+8)<0`
#0 = -2(-5x-8)+4 #(-5x-8) = 2# #-5x = 10# #x = -2#

In summary

the critical point is at `(-8/5,4)`
the y-intercept is at `(-12)`
the x-intercepts are at `(-6/5)` and
graph{-2*abs(5x+8)+4 [-5.696, 4.17, -0.437, 4.493]}