Question

which rule describes this transformation?
$(x,y)\to(x - 5,y + 4)$
$(x,y)\to(x + 5,y - 4)$
$(x,y)\to(x + 4,y - 5)$
$(x,y)\to(x - 4,y + 5)$

Explanation:

Step1: Analyze the horizontal movement

Take a point, say \(C\). Suppose the original point (before transformation) has \(x\)-coordinate \(x_1\) and after transformation has \(x_2\). If we assume a general point \((x,y)\) in the pre - transformed figure and \((x',y')\) in the transformed figure. For the horizontal ( \(x\)-axis) movement:
If we consider the shift of a point. Let's assume a reference. If we look at the direction of the shift, if the figure moves to the right, the \(x\)-coordinate increases. If we assume a point \(C\) (pre - transformed) and its corresponding point (transformed). The rule for horizontal shift is \(x'=x + h\), where \(h>0\) for right - shift and \(h < 0\) for left - shift.

Step2: Analyze the vertical movement

For the vertical ( \(y\)-axis) movement: The rule for vertical shift is \(y'=y + k\), where \(k>0\) for up - shift and \(k < 0\) for down - shift.
Let's take a point. Suppose we consider a point in the upper - left figure (pre - transformed) and its corresponding point in the lower - right figure (transformed).
If we assume a general transformation \((x,y)\to(x',y')\).
Let's check the second option \((x,y)\to(x + 5,y-4)\)
For the \(x\)-coordinate: If we consider a point, moving from left to right (an increase in \(x\) value). For the \(y\)-coordinate: moving down (a decrease in \(y\) value)

Answer:

\((x,y)\to(x + 5,y - 4)\) (the second option)