Question

writing a rule for a translation
which rule describes the translation?
$t_{-3, -2}(x, y)$
$t_{-3, 2}(x, y)$
$t_{3, -2}(x, y)$
$t_{3, 2}(x, y)$

Explanation:

Step1: Determine the horizontal translation

Let's assume a point \( (x,y) \) on the original figure. For example, take point \( B\). Suppose the \(x -\)coordinate of \(B\) is \(x_1\) and the \(x -\)coordinate of \(B'\) is \(x_2\).
If we observe the horizontal movement (left - right), moving from left to right is a positive \(x -\)direction. Count the number of units. If we assume \(B\) is at \(x=- 1\) (approximate) and \(B'\) is at \(x = 2\) (approximate), then \(x_2=x_1 + 3\). In general, for a point \((x,y)\), the new \(x -\)coordinate \(x'=x + 3\) (since \(x'=x+h\) and \(h = 3\) for horizontal translation).

Step2: Determine the vertical translation

Now, consider the vertical movement (up - down). Moving down is a negative \(y -\)direction. Suppose the \(y -\)coordinate of \(B\) is \(y_1\) and the \(y -\)coordinate of \(B'\) is \(y_2\). If \(y_1 = 2\) (approximate) and \(y_2=0\) (approximate), then \(y_2=y_1-2\). In general, for a point \((x,y)\), the new \(y -\)coordinate \(y'=y - 2\) (since \(y'=y + k\) and \(k=-2\) for vertical translation).
The translation rule \(T_{h,k}(x,y)=(x + h,y + k)\). Here \(h = 3\) and \(k=-2\).

Answer:

\(T_{3,-2}(x,y)\)