Question

in the graph below pre - image \\( \triangle mnp \\) is translated to \\( \delta m ^ { \prime } n ^ { \prime } p ^ { \prime } \\).
which set of coordinates models the transformation of \\( \triangle mnp \\) to \\( \delta m ^ { \prime } n ^ { \prime } p ^ { \prime } \\)?
\\( ( x, y ) \
ightarrow ( x + 2, y - 9 ) \\)
\\( ( x, y ) \
ightarrow ( x - 2, y - 9 ) \\)
\\( ( x, y ) \
ightarrow ( x - 2, y + 9 ) \\)
\\( ( x, y ) \
ightarrow ( x + 2, y + 9 ) \\)

Explanation:

Step1: Analyze the horizontal translation

Let's assume a point \( (x,y) \) in \( \triangle MNP \). To find the horizontal change, we can look at the \( x \)-coordinates. If we move from left - to - right (a positive \( x \)-direction), for example, if a point \( x\) becomes \(x + 2\), it means a translation of 2 units to the right. If it becomes \(x-2\), it means a translation of 2 units to the left.

Step2: Analyze the vertical translation

For the \( y \)-coordinates, if \(y\) becomes \(y - 9\), it means a translation of 9 units down (since subtracting from the \( y \)-coordinate moves the point in the negative \( y \)-direction). If \(y\) becomes \(y + 9\), it means a translation of 9 units up.

Since the translation is to the right (positive \( x \)-direction) and down (negative \( y \)-direction), the transformation rule is \( (x,y)\to(x + 2,y-9) \)

Answer:

\((x,y)\to(x + 2,y - 9)\) (the first option)