Question

what is the equation of the function graphed? \\( g(x) = -3|x + 6| \\) \\( g(x) = -3|x| + 6 \\) \\( g(x) = 3|x - 6| \\) \\( g(x) = -3|x - 6| \\)

Explanation:

Step1: Identify the vertex of the 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 function. From the graph, we can see that the vertex of the function \(G(x)\) is at \((6,0)\). So \(h = 6\) and \(k = 0\).

Step2: Determine the direction and the vertical stretch/compression

The graph opens downwards, which means that the coefficient \(a\) of the absolute - value function is negative. Also, we can check the slope of one of the branches. Let's take a point on the right - hand branch. For example, when \(x = 9\), let's assume the value of \(y\). If we consider the function \(y=-3|x - 6|\), when \(x = 9\), \(y=-3|9 - 6|=-3\times3=-9\); when \(x = 12\), \(y=-3|12 - 6|=-3\times6=-18\). The general form with \(h = 6\), \(k = 0\) and \(a=-3\) gives us the function \(G(x)=-3|x - 6|\).

Step3: Eliminate the other options

• For \(G(x)=-3|x + 6|\), the vertex would be at \((-6,0)\), which does not match the vertex from the graph.
• For \(G(x)=-3|x|+6\), the vertex is at \((0,6)\), which does not match the vertex from the graph.
• For \(G(x)=3|x - 6|\), the graph would open upwards (since \(a = 3>0\)), but our graph opens downwards, so this option is incorrect.

Answer:

\(G(x)=-3|x - 6|\)