Question

the rule as a mapping for the translation of a rectangle is $(x,y)\to(x - 2,y + 7)$. which describes this translation?
a translation of 2 units down and 7 units to the right
a translation of 2 units down and 7 units to the left
a translation of 2 units to the right and 7 units up
a translation of 2 units to the left and 7 units up

Explanation:

Step1: Analyze the \(x\) - coordinate change

For a point \((x,y)\) in the coordinate plane, if the transformation is \((x,y)\to(x - 2,y+7)\).
The change in the \(x\) - coordinate is given by \(\Delta x=(x - 2)-x=- 2\).
A negative change in the \(x\) - coordinate means a shift to the left. So, a change of \(-2\) in the \(x\) - coordinate implies a translation of 2 units to the left.

Step2: Analyze the \(y\) - coordinate change

The change in the \(y\) - coordinate is \(\Delta y=(y + 7)-y = 7\).
A positive change in the \(y\) - coordinate means a shift upwards. So, a change of \(+7\) in the \(y\) - coordinate implies a translation of 7 units up.

So, the translation \((x,y)\to(x - 2,y + 7)\) is a translation of 2 units to the left and 7 units up.

Answer:

D. a translation of 2 units to the left and 7 units up