Question

identify the equation for this graph.
options:
y = |x + 3| + 1
y = |x + 3| - 1
y = |x - 3| + 1
y = |x - 3| - 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 rewritten as \(y=|x-(-h)|+(-k)\)) is at the point \((h,k)\). From the graph, we can see that the vertex is at \((- 3,1)\).

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 \((-3,1)\), then \(h=-3\) and \(k = 1\). Substituting these values into the general form, we get \(y=|x-(-3)|+1=|x + 3|+1\).

We can also verify by checking the y - intercept. Let \(x = 0\) in the function \(y=|x + 3|+1\). Then \(y=|0 + 3|+1=3 + 1=4\), which matches the y - intercept (the graph crosses the y - axis at \(y = 4\)) shown in the graph.

For the other options:

• For \(y=|x + 3|-1\), when \(x = 0\), \(y=|0 + 3|-1=3-1 = 2\), which does not match the y - intercept of 4.
• For \(y=|x - 3|+1\), when \(x = 0\), \(y=|0 - 3|+1=3 + 1=4\), but the vertex of \(y=|x - 3|+1\) is at \((3,1)\), which does not match the vertex \((-3,1)\) of the given graph.
• For \(y=|x - 3|-1\), when \(x = 0\), \(y=|0 - 3|-1=3-1 = 2\), and the vertex is at \((3,-1)\), which does not match the graph.

Answer:

\(y = |x + 3|+1\)