How do you find absolute value equation from graph?

The graph will be shaped like a #V# or an upside down #V#

The vertex is the point #(h, k)#, so look at the graph to determine the coordinates of the vertex. (It is the point of the #V#.)

The graph will have straight lines on both sides of the vertex. To the right of #x=h#, the slope of the line will be what we need for #a# in the equation. (To the left of #x=h#, the slope will be #-a#.

So after you have found the vertex #(h,k)# find another point on the graph to the right of the vertex. Call it #(x_2, y_2)#.

Find the slope of the line through the points #(h,k)# and #(x_2, y_2)#. That is #a#

#a = (y_2-k)/(x_2-h)#

Here are two examples:

Example 1

graph{y = 5/2abs(x-3) + 2 [-3.9, 16.1, -0.856, 9.145]}

(Use your mouse: wheel to scroll in or out and click, hold and drag the graph around as needed.)

The vertex is at #(3,2)# so the equation looks like

#y = aabs(x-3)+2#

To find #a#, find a pont on the graph to the right of the vertex. I'll use #(5,7)#:

#a# is the slope:

#a = (7-2)/(5-3)#

So #a = 5/2#

The equation is:

#y = 5/2abs(x-3)+2#

If you want to get rid of the fraction, multiply both sides by #2#, to get:

#2y = 5abs(x-3)+2#

Example 2

graph{y = -2abs(x-4)+1 [-1.25, 11.24, -3.97, 2.276]}

Find the vertex:

.
.

.

The vertex is #(4,1)#.

Find #a#.
First find a point to the right of the vertex, then #a# = the slope of the line throught the two points.

.

.

.I'll use the point #(6, -3)#

#a = (-3-1)/(6-4) = (-4)/2 = -2#

The equation is

#y = -2abs(x-4)+1#