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?

Feb 10, 2017

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

Explanation:

Rotation by ${180}^{\circ}$ about the origin, transforms each point $\left(x , y\right)$ on the shape to the corresponding point $\left(- x , - y\right)$
($x \mathmr{and} y$ both change sign.)

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

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

$\left(x , y\right) \rightarrow \left(- x , - y\right) \rightarrow \left(x , - y\right)$

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

Feb 10, 2017

$\left(\begin{matrix}1 & 0 \\ 0 & - 1\end{matrix}\right)$ 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

$\left(\begin{matrix}- 1 & 0 \\ 0 & - 1\end{matrix}\right)$

This has the effect of changing the signs of the $x - \mathmr{and} y -$ coordinates of all the points.

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

$\left(\begin{matrix}- 1 & 0 \\ 0 & 1\end{matrix}\right)$

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:

$\left(\begin{matrix}- 1 & 0 \\ 0 & - 1\end{matrix}\right) \times \left(\begin{matrix}- 1 & 0 \\ 0 & 1\end{matrix}\right)$

$= \left(\begin{matrix}1 & 0 \\ 0 & - 1\end{matrix}\right)$

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.