Question

the image of the point (2, -1) under a translation is (4, 2). find the coordinates of the image of the point (-2, 9) under the same translation.

Explanation:

Step1: Find the translation rule

Let the translation rule be \((x,y)\to(x + a,y + b)\).
Given \((2,-1)\to(4,2)\), then \(2 + a=4\) and \(-1 + b = 2\).
From \(2 + a=4\), we get \(a=4 - 2=2\).
From \(-1 + b=2\), we get \(b=2+1 = 3\). So the translation rule is \((x,y)\to(x + 2,y + 3)\).

Step2: Apply the translation rule to the point \((-2,9)\)

For the point \((x=-2,y = 9)\), using the rule \((x,y)\to(x + 2,y + 3)\).
The \(x\) - coordinate of the image is \(x'=-2+2 = 0\).
The \(y\) - coordinate of the image is \(y'=9 + 3=12\).

Answer:

$(0,12)$