Question

the given figure shows a transformation of the graph of f(x)=|x|. write the equation for the transformed graph. the equation is y = \square. (type an expression using x as the variable. do not simplify.)

Explanation:

Step1: Identify the vertex of the transformed graph

The original function \( f(x) = |x| \) has its vertex at \((0,0)\). From the graph, the vertex of the transformed absolute - value graph is at \((-3, - 3)\).

Step2: Recall the transformation rules for absolute - value functions

The general form of a transformed absolute - value function is \( y=a|x - h|+k \), where \((h,k)\) is the vertex of the graph. For the parent function \( f(x)=|x| \) (\(a = 1\), \(h = 0\), \(k = 0\)), when we shift the graph horizontally by \(h\) units and vertically by \(k\) units, the new function is \( y=|x - h|+k \).

If the vertex is \((h,k)=(-3,-3)\), then we substitute \(h=-3\) and \(k = - 3\) into the general form. So \( y=|x-(-3)|+(-3)\), which simplifies to \( y = |x + 3|-3\).

Answer:

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