91Ó°ÊÓ

A reflection is a type of geometric transformation that flips a figure over a line, creating a mirror image. In the exercise, we reflected across the line \(y=x\) and across the \(x\)-axis. When reflecting over the line \(y = x\), the coordinates of a point \((x, y)\) become \((y, x)\). This means that each point on the figure is flipped across the line, swapping the x and y values. For a reflection across the \(x\)-axis, the y-coordinate of each point changes its sign. This is represented as \((x, y) \rightarrow (x, -y)\). These transformations alter the positions of points without changing their shapes.

Rotation is a geometric transformation that turns a figure around a fixed point, usually the origin. In the exercise, we dealt with a \(270^{\text{\circ}}\) counterclockwise rotation and a \(180^{\text{\circ}}\) rotation. A counterclockwise \(270^{\text{\circ}}\) rotation transforms a point \((x, y)\) to \((y, -x)\). For a \(180^{\text{\circ}}\) rotation, the point \((x, y)\) becomes \((-x, -y)\). These rotations do not change the size or shape of the figure; they only change its orientation. Understanding these basic transformations helps to comprehend more complex combinations.

A composite transformation involves performing multiple transformations in sequence. In the given exercise, we combined reflection and rotation transformations. For part (a), the reflection across \(y = x\) followed by a \(270^{\text{\circ}}\) counterclockwise rotation resulted in a composite transformation equivalent to a reflection across the \(x\)-axis, \((x, y) \rightarrow (x, -y)\). For part (b), a \(180^{\text{\circ}}\) rotation followed by a reflection across the \(x\)-axis resulted in a composite transformation equivalent to a reflection across the \(y\)-axis, \((x, y) \rightarrow (-x, y)\). Composite transformations simplify complex movements by combining basic transformations into one equation or rule.

Coordinate transformation involves changing the coordinates of points to map the figures from one place to another in a coordinate plane. In our exercise, reflection and rotation affected the coordinates of points. After reflecting across \(y = x\), a point \((x, y)\) changes to \((y, x)\). Applying a \(270^{\text{\circ}}\) counterclockwise rotation to \((y, x)\) results in \((x, -y)\). Likewise, for part (b), rotating \(180^{\text{\circ}}\) changes a point \((x, y)\) to \((-x, -y)\) and reflecting it across the \(x\)-axis results in \((-x, y)\). Understanding how the coordinates transform under different operations is crucial for mastering geometric transformations.