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An absolute value equation graphs like a "V".

These equations come in several forms.

The simplest is in this form: .                      The standard form looks like   or  .  These are discussed below.

First, the "a" tells you whether the graph opens up ( + ) or opens down ( - ).  The discussion below will deal with a positive "a" but just flip upside down if "a" is negative.

The parent graph for absolute value (first degree) equations is .  This graph is a V with its minimum at (0,0).

• From looking at the parent graph, you can see that it could be shifted up or down from (0,0).  This would leave it centered on the x-axis, meaning it is symmetric about the y-axis, but moved up or down along the vertical axis.

This shift up or down would come from a value outside the absolute value signs.   By order of operations, this means that the last thing calculated is the addition of "k".  If k is positive the shift is up.  If k is negative the shift is down.

examples:

This is the parent graph shifted up 5 units.

or Either of these gives the parent graph shifted down three units.

• You could also see that the parent graph could be shifted right or left.   This shift is seen inside the absolute value signs.  It might be confusing to figure out which way the graph shifts.  Notice the expression inside the absolute value signs is written "x - h" and remember that "x + h" is the result of "x - ^-h".

  When the h is 4 then an x value of 4 gives a zero ( 4 - 4 ) inside the signs.  

  When the h is ^-4 then an x value of ^-4 gives a zero ( ^-4 - ^-4 )

examples:

This is the parent graph shifted right 4 units.

This is the parent graph shifted left 3 units.

• Also, the graph could be shifted both horizontally and vertically

  This is the parent graph shifted left one unit and down three units.

• The "a" in the equation tells you whether the graph is "fat" or "skinny" .  If a is 1 then it is neither, it is the same as the parent graph . If a is greater than one (a > 1) then the graph is "skinny" and if a is between zero and one ( 0 < a < 1 ), then the graph is "fat."

  on the left, this graph is thinner than the parent graph and on the right, this graph is wider than the parent graph