Question

which transformation rule can be used to transform the figure p to figure p? (x,y)→(y,x) (x,y)→(−y,−x) (x,y)→(−x,−y) (x,y)→(y−4,x−4)

Explanation:

Step1: Analyze each transformation rule

• For the rule \((x,y)\to(y,x)\), it represents a reflection over the line \(y = x\).
• For the rule \((x,y)\to(-y,-x)\), it represents a rotation of \(180^{\circ}\) about the origin followed by a reflection.
• For the rule \((x,y)\to(-x,-y)\), it represents a rotation of \(180^{\circ}\) about the origin.
• For the rule \((x,y)\to(y - 4,x-4)\), we can check the translation.

Let's assume a point \((x,y)\) on figure \(P\). If we use the rule \((x,y)\to(y - 4,x - 4)\):
Suppose a vertex of figure \(P\) is \((x_1,y_1)\). After transformation, the new coordinates \((x_2,y_2)\) where \(x_2=y_1-4\) and \(y_2=x_1 - 4\) (which is a combination of horizontal and vertical translation).

We can also check by taking a sample point. Let's assume a point on \(P\) (for simplicity, assume a corner point). If we consider the general position - shifting the \(x\) and \(y\) coordinates as per the rule \((x,y)\to(y - 4,x-4)\) which is a translation (not a rotation or reflection as in the other rules) that can map figure \(P\) to \(P'\)

Answer:

\((x,y)\to(y - 4,x - 4)\)