Question

which rule describes the translation? (x,y)→(x - 8,y - 8) (x,y)→(x - 3,y + 8) (x,y)→(x + 8,y - 3) (x,y)→(x + 3,y + 8)

Explanation:

Step1: Analyze the horizontal translation

Take a point (e.g., point \(B\)). Assume the \(x\) - coordinate of \(B\) is \(x_1\) and the \(x\) - coordinate of \(B'\) is \(x_2\). Count the number of units moved horizontally. If we move from the left - hand figure to the right - hand figure, we can see that the horizontal movement is to the right (positive \(x\) - direction). By counting the grid units, if we assume a starting \(x\) value (say \(x=- 5\) for a left - hand vertex) and the corresponding \(x\) value of the translated vertex (say \(x = 3\)), the change in \(x\) is \(\Delta x=x_2-x_1=3-(-5)=8\). So the rule for the \(x\) - coordinate is \(x\to x + 8\).

Step2: Analyze the vertical translation

Take the same point (e.g., point \(B\)). Assume the \(y\) - coordinate of \(B\) is \(y_1\) and the \(y\) - coordinate of \(B'\) is \(y_2\). Count the number of units moved vertically. The movement is downwards (negative \(y\) - direction). If we assume a starting \(y\) value (say \(y = 2\) for a left - hand vertex) and the corresponding \(y\) value of the translated vertex (say \(y=-1\)), the change in \(y\) is \(\Delta y=y_2 - y_1=-1 - 2=-3\). So the rule for the \(y\) - coordinate is \(y\to y-3\).

Combining the horizontal (\(x\) - direction) and vertical (\(y\) - direction) translations, the translation rule is \((x,y)\to(x + 8,y-3)\).

Answer:

\((x,y)\to(x + 8,y-3)\)