Question

irma wants to tile her floor using the translation shown below. which is the rule for this translation?\\( \bigcirc\\ t_{3,2}(x,y) \\)\\( \bigcirc\\ t_{2,3}(x,y) \\)\\( \bigcirc\\ t_{3,-2}(x,y) \\)\\( \bigcirc\\ t_{-2,3}(x,y) \\)

Explanation:

To determine the translation rule, we analyze the horizontal and vertical shifts of a point \((x, y)\) from the original figure to the translated figure.

Step 1: Identify Horizontal Shift

Looking at the \(x\)-coordinates, the original figure shifts right by 3 units. A rightward shift of \(h\) units means the new \(x\)-coordinate is \(x + h\). Here, \(h = 3\), so the horizontal change is \(x \to x + 3\).

Step 2: Identify Vertical Shift

Looking at the \(y\)-coordinates, the original figure shifts down by 2 units. A downward shift of \(k\) units means the new \(y\)-coordinate is \(y - k\) (or \(y + (-k)\)). Here, \(k = 2\), so the vertical change is \(y \to y - 2\).

Step 3: Formulate the Translation Rule

A translation \(T_{h, k}(x, y)\) is defined as \((x + h, y + k)\). For a right shift of 3 (\(h = 3\)) and a down shift of 2 (\(k = -2\)), the rule becomes:
\[
T_{3, -2}(x, y) = (x + 3, y - 2)
\]

Answer:

\(T_{3, -2}(x, y)\) (assuming the option with \(T_{3, -2}(x, y)\) is the correct choice, based on the shift analysis).