Question

what is the image point of (2, -6) after the transformation ( r_{270^circ} circ r_{y\text{-axis}} )?

Explanation:

Step1: Apply reflection over y - axis

The rule for reflection over the \(y\) - axis (\(r_{y - \text{axis}}\)) is \((x,y)\to(-x,y)\). For the point \((2,-6)\), after reflection over the \(y\) - axis, we get \((- 2,-6)\).

Step2: Apply 270 - degree rotation

The rule for a \(270^{\circ}\) rotation counter - clockwise (\(R_{270^{\circ}}\)) is \((x,y)\to(y,-x)\). We will apply this rule to the point \((-2,-6)\) obtained from the previous step.
Let \(x=-2\) and \(y = - 6\). Using the rotation rule \((x,y)\to(y,-x)\), we substitute the values: \(y=-6\) and \(-x=-(-2) = 2\). So the image after \(R_{270^{\circ}}\) is \((-6,2)\).

Answer:

\((-6,2)\)