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

1 Answer
Alan P.
May 28, 2015

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]}

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