Question

the graph shows g(x), which is a transformation of f(x) = |x|. write the function rule for g(x).
write your answer in the form a|x - h| + k, where a, h, and k are integers or simplified fractions.
g(x) =

Explanation:

Step1: Identify the vertex

The vertex form of an absolute - value function is \(g(x)=a|x - h|+k\), where \((h,k)\) is the vertex of the graph. From the graph, we can see that the vertex of \(g(x)\) is at the point \((4, - 2)\). So, \(h = 4\) and \(k=-2\).

Step2: Determine the value of \(a\)

We know that the parent function is \(f(x)=|x|\), which has a slope of \(1\) for \(x\geq0\) and a slope of \(- 1\) for \(x < 0\). Let's take a point on the graph of \(g(x)\) other than the vertex. Let's take the point \((6,0)\).

Substitute \(x = 6\), \(y = 0\), \(h = 4\) and \(k=-2\) into the equation \(g(x)=a|x - h|+k\):

\(0=a|6 - 4|+(-2)\)

Simplify the equation:

\(0=a|2|-2\)

Since \(|2| = 2\), we have \(0 = 2a-2\)

Add \(2\) to both sides: \(2=2a\)

Divide both sides by \(2\): \(a = 1\)

Answer:

\(g(x)=|x - 4|-2\)