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: Recall the transformation rules

The parent function is \( f(x)=\vert x\vert \). The general form of a transformation of \( y = a\vert x - h\vert + k \), where \( a \) is the vertical stretch or compression, \( h \) is the horizontal shift, and \( k \) is the vertical shift.

Step2: Determine the values of \( a\), \(h\), and \(k\)

From the graph, we can see that the vertex of the transformed graph is at \((- 2,1)\), so \(h=-2\) and \(k = 1\).
Let's take a point on the parent function \(y=\vert x\vert\), say \((0,0)\). On the transformed graph, when \(x = 0\), we can assume two - point method. Let's pick two points on the transformed graph. Suppose we consider the "V - shape" points.
Another way: we know that for the parent function \(y = \vert x\vert\), and the transformed function \(y=a\vert x-( - 2)\vert+1=a\vert x + 2\vert+1\).
Let's assume a non - vertex point. For \(y=\vert x\vert\), when \(x = 1\), \(y = 1\). For the transformed graph, assume a point. Let's use the fact that the slope of the right - hand side of \(y=\vert x\vert\) is \(1\). For the transformed graph, if we consider the right - hand side line. Let's take two points on the right - hand side of the transformed graph. Suppose two points \((-2,1)\) and \((2,3)\). The slope of the line \(m=\frac{3 - 1}{2-( - 2)}=\frac{2}{4}=\frac{1}{2}\).
For \(y=a\vert x + 2\vert+1\), when \(x\geq - 2\), \(y=a(x + 2)+1=ax+2a + 1\), and the slope \(a=\frac{1}{2}\)

Answer:

\(y=\frac{1}{2}\vert x + 2\vert+1\)