Question

the figure below is reflected over y - axis and then rotated 180° clockwise. what are the coordinates of the image of point a after these transformations?

Explanation:

1. First, assume the coordinates of point \(A\) are \((x,y)\).

• Reflection over the \(y\) - axis:
• The rule for reflecting a point \((x,y)\) over the \(y\) - axis is \((x,y)\to(-x,y)\). So after reflection over the \(y\) - axis, the coordinates of point \(A\) become \((-x,y)\).
• Rotation of \(180^{\circ}\) clock - wise:
• The rule for rotating a point \((a,b)\) \(180^{\circ}\) clock - wise about the origin is \((a,b)\to(-a,-b)\).
• Since our point after reflection over the \(y\) - axis is \((-x,y)\), after rotating it \(180^{\circ}\) clock - wise, the new coordinates are \((x, - y)\).
• From the graph, assume the coordinates of point \(A\) are \((2,-7)\).
• After reflection over the \(y\) - axis: \((2,-7)\to(-2,-7)\).
• After rotating \((-2,-7)\) \(180^{\circ}\) clock - wise: \((-2,-7)\to(2,7)\).

Step1: Reflect over y - axis

The rule for reflecting a point \((x,y)\) over the \(y\) - axis is \((x,y)\to(-x,y)\). For point \(A(2,-7)\), it becomes \((-2,-7)\).

Step2: Rotate \(180^{\circ}\) clock - wise

The rule for rotating a point \((a,b)\) \(180^{\circ}\) clock - wise is \((a,b)\to(-a,-b)\). For the point \((-2,-7)\), it becomes \((2,7)\).

Answer:

\((2,7)\)