Given the graph of an absolute value function, the general equation in the form \(y=a|x-h|+k\), can be found by locating the vertex \((h,k)\) and dilation factor and orientation, a.

How can we find the dilation factor from the graph? Starting at the vertex, find the rise when you run 1 to the right.

In this graph the vertex is \((-3,-1.5)\) and a is -2. So the equation is \(y=-2|x+3|-1.5\).

The general function \(y=a|x-h|+k\) can be rewritten as a piecewise function as

\[y=\begin{cases}

a(x-h)+k, & x-h\geq0 \\

-a(x-h)+k, & x+h<0

\end{cases}\]

So, to rewrite \(y=-2|x+3|-1.5\) as a piecewise function

\[y=\begin{cases}

-2(x+3)-1.5, & x+3\geq0 \\

2(x+3)-1.5, & x+3<0

\end{cases}\]

You may use the graph to confirm.

Given that information we can see find the equations for the 1st 3 graphs:

First graph: \(a = 3\) and \(k = 2\); \(y = 3|x| + 2\)

Second graph: \(a = -3\) and \(k = -2\) ; \(y = -3|x| -2\)

Third graph: \(a = 3\) and \(h = -2\); \(y = 3|x+ 2|\)

Fourth graph is a piecewise function but not an absolute value function

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