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 of the absolute - value function \(g(x)=a|x - h|+k\) is at the point \((h,k)\). From the graph, we can see that the vertex of \(g(x)\) is at \((0,0)\). So, \(h = 0\) and \(k=0\).

Step2: Determine the value of \(a\)

We can use a point on the graph to find the value of \(a\). Let's take the point \((2,1)\) (we can also take other points like \((4,2)\), \((6,3)\) etc.). The general form of the function is \(g(x)=a|x - 0|+0=a|x|\). Substitute \(x = 2\) and \(g(x)=1\) into the function:

$$1=a|2|$$

Since \(|2| = 2\), we have the equation \(1 = 2a\). Solving for \(a\), we divide both sides of the equation by 2: \(a=\frac{1}{2}\).

Answer:

\(g(x)=\frac{1}{2}|x - 0|+0=\frac{1}{2}|x|\) or in the form \(a|x - h|+k\) where \(a=\frac{1}{2}\), \(h = 0\) and \(k = 0\), so \(g(x)=\frac{1}{2}|x|\)