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?

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.)

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.