Question

write a rule to describe the transformation below.
answer attempt 5 out of 6
rotation
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about the origin
over

Explanation:

Step1: Find coordinates of a point

Take a point, say \(W\). Let's assume original coordinates of \(W\) (black - colored) are \((x,y)\). Suppose \(W\) has coordinates \((- 1,4)\). After transformation (red - colored \(W'\)), its coordinates are \((-4, - 1)\).

Step2: Use rotation formula

The rotation formula for a \(90^{\circ}\) clockwise rotation about the origin is \((x,y)\to(y,-x)\), for a \(90^{\circ}\) counter - clockwise rotation about the origin is \((x,y)\to(-y,x)\), and for a \(180^{\circ}\) rotation about the origin is \((x,y)\to(-x,-y)\).
If we use the \(180^{\circ}\) rotation formula:
For a general point \((x,y)\), after \(180^{\circ}\) rotation about the origin, the new point \((x',y')\) is given by \(x'=-x\) and \(y'=-y\).
Let's check another point. Suppose \(P\) (black - colored) has coordinates \((-2,-2)\). After transformation (red - colored \(J\)), if we assume it's a \(180^{\circ}\) rotation, then using \((x,y)\to(-x,-y)\), for \(x = - 2,y=-2\), we get \(x'=-(-2) = 2\) and \(y'=-(-2)=2\). But if we consider the mid - point of a segment joining a pre - image and image point. The mid - point \(M\) of two points \((x_1,y_1)\) (pre - image) and \((x_2,y_2)\) (image) is \(M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\). For a \(180^{\circ}\) rotation about the origin, the mid - point of \((x,y)\) and \((-x,-y)\) is \((0,0)\) (the origin).
Let's take two corresponding points \(W(-1,4)\) and \(W'(-4, - 1)\). The mid - point of \(W\) and \(W'\) is \((\frac{-1-4}{2},\frac{4 - 1}{2})=(-\frac{5}{2},\frac{3}{2})
eq(0,0)\).
Let's check for a reflection. The reflection over the \(x\) - axis is \((x,y)\to(x,-y)\), over the \(y\) - axis is \((x,y)\to(-x,y)\).
Let's check the reflection over the line \(y=-x\). The formula for reflection over the line \(y =-x\) is \((x,y)\to(-y,-x)\).
If we take a point \(W(-1,4)\), after reflection over \(y=-x\), we substitute \(x=-1\) and \(y = 4\) into \((x,y)\to(-y,-x)\). We get \((-4,1)\) (incorrect).
Let's check the reflection over the line \(y=x\). The formula for reflection over the line \(y=x\) is \((x,y)\to(y,x)\). If \(W(-1,4)\), then \((4,-1)\) (incorrect).
Let's check the rotation.
Take a point \(S\) (black - colored). Suppose \(S(-1,1)\). After transformation (red - colored \(S'\)), if we assume a \(180^{\circ}\) rotation about the origin. Using \((x,y)\to(-x,-y)\), for \(x=-1,y = 1\), we get \((1,-1)\). The mid - point of \((-1,1)\) and \((1,-1)\) is \((0,0)\) (the origin).

Answer:

The transformation is a \(180^{\circ}\) rotation about the origin. So the rule is \((x,y)\to(-x,-y)\)