Question

p→p(2, - 3) for (x,y)→(x - 2,y) and reflection line y = 3. find the coordinates of p. the coordinates of p are (type an ordered pair.)

Explanation:

Step1: Reverse the translation

The translation is \((x,y)\to(x - 2,y)\). To reverse it, we use \((x,y)\to(x+2,y)\).
If \(P'\) has coordinates \((2,-3)\) after translation, before translation (let the point be \(P_1=(x_1,y_1)\)), we have \(x_1=2 + 2\), \(y_1=-3\). So \(P_1=(4,-3)\).

Step2: Reverse the reflection

The formula for reflecting a point \((x,y)\) over the line \(y = k\) is \((x,y)\to(x,2k - y)\). Here \(k = 3\), and we know the point after reflection is \((4,-3)\). Let the original point (before reflection) be \(P=(x,y)\).
Using the formula \(y=2\times3-(-3)\), \(x = 4\).
Since \(2k-y=-3\) (where \(k = 3\)), then \(6 - y=-3\), so \(y=6 + 3=9\).

Answer:

\((4,9)\)