Question

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

Explanation:

Step1: Find the vertex of the graph

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

Step2: Recall the vertex form of absolute - value function

The general form of an absolute - value function is \(y=a|x - h|+k\), where \((h,k)\) is the vertex of the graph. For the parent function \(y = |x|\), the vertex is at \((0,0)\). When the vertex is \((-2,1)\), we substitute \(h=-2\) and \(k = 1\) into the vertex form. So the function becomes \(y=|x-(-2)|+1=|x + 2|+1\).

We can also verify by plugging in the vertex point \((-2,1)\) into each of the given equations:

• For \(y = |x + 2|+1\), when \(x=-2\), \(y=|-2 + 2|+1=0 + 1=1\), which matches the vertex.
• For \(y=|x + 2|-1\), when \(x = - 2\), \(y=|-2 + 2|-1=0-1=-1

eq1\).

• For \(y=|x - 2|+1\), when \(x=-2\), \(y=|-2-2|+1=4 + 1=5

eq1\).

• For \(y=|x - 2|-1\), when \(x=-2\), \(y=|-2-2|-1=4-1=3

eq1\).

Answer:

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