Question

writing a rule for a translation
which rule describes the translation?
$(x,y)\to(x - 8,y - 3)$
$(x,y)\to(x - 3,y + 8)$
$(x,y)\to(x + 8,y - 3)$
$(x,y)\to(x + 3,y + 8)$

Explanation:

Step1: Analyze the horizontal translation

Let's take a point, say \(B\). Suppose 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 assume \(B\) is at \(x=- 5\) and \(B'\) is at \(x = 3\), then the change in the \(x -\)coordinate is \(x_2-x_1=3-(-5)=8\). A positive change in the \(x -\)coordinate means a shift to the right. So the rule for the \(x -\)coordinate is \(x\to x + 8\).

Step2: Analyze the vertical translation

Count the number of units moved vertically. Suppose the \(y -\)coordinate of \(B\) is \(y_1\) (say \(y_1 = 2\)) and the \(y -\)coordinate of \(B'\) is \(y_2\) (say \(y_2=-1\)). Then the change in the \(y -\)coordinate is \(y_2 - y_1=-1 - 2=-3\). A negative change in the \(y -\)coordinate means a shift downwards. So the rule for the \(y -\)coordinate is \(y\to y-3\).

Answer:

\((x,y)\to(x + 8,y-3)\) (the third option)