Question

identify the equation for this graph.
$y = |x - 1| - 2$
$y = |x + 1| - 2$
$y = |x + 1| + 2$
$y = |x - 1| + 2$

Explanation:

Step1: Find the vertex of the graph

The vertex of the absolute - value graph \(y = |x - h|+k\) (or \(y=|x + h|-k\) which can be rewritten as \(y = |x-(-h)|+(-k)\)) is at the point \((h,k)\). From the given graph, we can see that the vertex is at the point \((- 1,-2)\).

Step2: Recall the vertex form of the absolute - value function

The general form of an absolute - value function is \(y=|x - h|+k\), where \((h,k)\) is the vertex of the graph. If the vertex is \((h,k)=(-1,-2)\), then substituting \(h=-1\) and \(k = - 2\) into the general form, we get \(y=|x-(-1)|+(-2)=|x + 1|-2\).

We can also verify by plugging in some points. For example, when \(x=-1\), \(y=| - 1 + 1|-2=0 - 2=-2\) (which matches the vertex). When \(x = 0\), \(y=|0 + 1|-2=1-2=-1\)? Wait, no, from the graph, when \(x = 0\), \(y=-1\)? Wait, no, looking at the graph, when \(x = 0\), the \(y\) - coordinate of the point on the graph is \(-1\)? Wait, no, the graph has a vertex at \((-1,-2)\), and when \(x = 0\), let's check the function \(y=|x + 1|-2\): when \(x = 0\), \(y=|0 + 1|-2=1 - 2=-1\). When \(x = 1\), \(y=|1 + 1|-2=2 - 2 = 0\), and from the graph, when \(x = 3\), \(y=|3 + 1|-2=4 - 2=2\), which also matches the graph.

Answer:

\(y = |x + 1|-2\)