Question

identify the equation for this graph.
$y = |x - 4| - 3$
$y = |x - 4| + 3$
$y = |x + 4| + 3$
$y = |x + 4| - 3$

Explanation:

Step1: Find the vertex of the graph

The vertex of an absolute - value function in the form \(y = |x - h|+k\) (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 \((4,3)\).

Step2: Recall the vertex form of absolute - value function

The general vertex form of an absolute - value function is \(y=|x - h|+k\), where \((h,k)\) is the vertex. Substituting \(h = 4\) and \(k=3\) into the formula, we get \(y=|x - 4|+3\).

We can also verify by checking the y - intercept. When \(x = 0\), for \(y=|x - 4|+3\), we have \(y=|0 - 4|+3=4 + 3=7\), which matches the y - intercept (the graph crosses the y - axis at \(y = 7\) approximately, and when we calculate \(y=|0 - 4|+3\), we get \(y = 7\)).

For \(y=|x - 4|-3\), when \(x = 0\), \(y=|0 - 4|-3=4-3 = 1\), which does not match the y - intercept.

For \(y=|x + 4|+3\), when \(x = 0\), \(y=|0 + 4|+3=4 + 3=7\), but the vertex of \(y=|x + 4|+3\) is at \((-4,3)\), which does not match the vertex \((4,3)\) of the given graph.

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

Answer:

\(y = |x - 4|+3\)