Question

which rule represents the translation from the pre - image, \\( \triangle abc \\), to the image, \\( \triangle a ^ { \prime } b ^ { \prime } c ^ { \prime } \\)? \\( ( x, y ) \
ightarrow ( x + 7, y + 6 ) \\) \\( ( x, y ) \
ightarrow ( x + 7, y - 6 ) \\) \\( ( x, y ) \
ightarrow ( x - 6, y + 7 ) \\) \\( ( x, y ) \
ightarrow ( x + 6, y + 7 ) \\)

Explanation:

Step1: Find the change in \(x\) - coordinate

Let's assume a point \(A\) (for example). Suppose the \(x\) - coordinate of a point in \(\triangle ABC\) is \(x_1\) and the \(x\) - coordinate of the corresponding point in \(\triangle A'B'C'\) is \(x_2\).
If we consider the horizontal movement (right is positive, left is negative). From the graph, the horizontal movement is to the right. The difference \(x_2-x_1\): If we assume a reference point (say \(A\)), moving from left - hand side to right - hand side. The change in \(x\) is \(+7\) (counting the units on the \(x\) - axis).

Step2: Find the change in \(y\) - coordinate

Let's assume the \(y\) - coordinate of a point in \(\triangle ABC\) is \(y_1\) and the \(y\) - coordinate of the corresponding point in \(\triangle A'B'C'\) is \(y_2\).
If we consider the vertical movement (up is positive, down is negative). From the graph, the vertical movement is downwards. The difference \(y_2 - y_1=-6\) (counting the units on the \(y\) - axis).
In a translation rule \((x,y)\to(x + h,y + k)\), where \(h\) is the horizontal change and \(k\) is the vertical change. Here \(h = 7\) and \(k=-6\). So the rule is \((x,y)\to(x + 7,y-6)\)

Answer:

\((x,y)\to(x + 7,y-6)\)