Question

triangle rst is the image of triangle rst under a translation followed by a reflection across the line y = -x.
write the rules for the translation and reflection.
translation: (x, y) ↦ (\boxed{\quad}, \boxed{\quad})
reflection: (x, y) ↦ (\boxed{\quad}, \boxed{\quad})

Explanation:

Step1: Find the translation rule

First, assume the translation rule is \((x,y)\to(x + a,y + b)\).
Let's take a point, say \(R(4,-2)\) and its image \(R'(1,4)\) (before reflection).
For the \(x\) - coordinate: \(4+a = 1\), so \(a=1 - 4=-3\).
For the \(y\) - coordinate: \(-2 + b=4\), so \(b = 4+ 2=6\).
The translation rule is \((x,y)\to(x-3,y + 6)\).

Step2: Recall the reflection rule across \(y=-x\)

The rule for reflection across the line \(y=-x\) is \((x,y)\to(-y,-x)\).
We can verify this. Let's take a point after translation. Suppose a general point \((x',y')=(x - 3,y + 6)\). After reflection across \(y=-x\), the new point \((x'',y'')=(-y',-x')\).
For example, if we take \(T(3,-5)\), after translation \(T'(3-3,-5 + 6)=(0,1)\). After reflection across \(y=-x\), using the rule \((x,y)\to(-y,-x)\), we get \((-1,0)\) (but we can also check from the graph - the process of reflection formula is a standard geometric transformation formula).

Answer:

Translation: \((x,y)\to(x - 3,y + 6)\)
Reflection: \((x,y)\to(-y,-x)\)