Question

Which single transformation that would have the same result as the two transformations (a) rotation by `180^@` about origin and (b) reflection in `y`-axis?

Answer 1

The single transformation that would have the same result as the two transformations is the one transforming `(x,y)` to `(x,-y)`, which is nothing but reflection of shape in `x`-axis .
The `x` stays the same, but the `y` changes sign.)

Explanation:

Rotation by `180^@` about the origin, transforms each point `(x,y)` on the shape to the corresponding point `(-x,-y)`
(`x and y` both change sign.)

When reflected in the `y`-axis, each point `(x,y)` transforms to
`(-x,y)`
(`x` changes sign, `y` stays the same.)

Hence, the single transformation that would have the same result as the two transformations is the one transforming `(x,y)` to `(x,-y)`,

`(x,y) rarr (-x,-y) rarr (x,-y)`

which is nothing but reflection of shape in the `x`-axis.
( `x` stays the same and `y` changes sign.)

Answer 2

`((1,0),(0,-1))` shows a reflection in the `x`-axis.

The `x` -coordinates stay the same, the signs of the `y`-coordinates change.

Explanation:

The two transformations can be described using transformation matrices.

The matrix for a rotation of 180° about the origin is

`((-1,0),(0,-1))`

This has the effect of changing the signs of the `x- and y-` coordinates of all the points.

The matrix for a reflection in the `y`-axis is

`((-1,0),(0,1))`

This has the effect of changing the signs of the `x` -coordinates of all the points, while the `y` -values stay the same.

If both transformations take place, the final result is given by:

`((-1,0),(0,-1)) xx ((-1,0),(0,1))`

`=((1,0),(0,-1))`

The effect of this matrix is to keep the `x` -coordinates the same, while changing the signs of the `y` -coordinates - indicating a reflection in the `x`-axis.