Question

select the rule that models the translation shown. (x + 6, y - 4) (x - 4, y + 6) (x + 4, y - 6) (x - 6, y + 4)

Explanation:

Step1: Determine the horizontal translation

Let's assume a point \(R\) on the original triangle. If we observe the horizontal movement from \(R\) to \(R'\), we can count the units. Suppose the \(x\) - coordinate of \(R\) is \(x_1\) and of \(R'\) is \(x_2\). By looking at the graph (counting the grid units), if we start from the left - hand triangle (original) to the right - hand triangle (translated), we find that the horizontal shift is \(x_2=x_1 + 6\) (moving 6 units to the right). In general, for a point \((x,y)\), the horizontal part of the translation rule is \(x\to x + 6\).

Step2: Determine the vertical translation

Now, for the vertical movement. Suppose the \(y\) - coordinate of \(R\) is \(y_1\) and of \(R'\) is \(y_2\). By counting the grid units, we see that the vertical shift is \(y_2=y_1-4\) (moving 4 units down). In general, for a point \((x,y)\), the vertical part of the translation rule is \(y\to y - 4\).

Combining the horizontal and vertical translations, the rule for the translation of a point \((x,y)\) is \((x,y)\to(x + 6,y - 4)\)

Answer:

\((x + 6,y - 4)\) (the orange option)