Mario says that are reflecting a point XY across the y axis and then dilating the reflection images by r will produce the same final image as dilating the point by r first and the reflecting the dilation image across the y axis. Use coordinate notation to show Mario is correct

matrix for reflection over y axis:

-1 +0
+0 +1

dilation matrix
r r
or
r
r

matrix multiplication is not necessarily comutative

dilate first
r r
* x y = r x, r y
then reflect
rx , ry
*
-1 +0
+0 +1
= -rx , ry

now reflect first
-1 +0 |x|
+0 +1 |y|

= -x , + y
now dilate
-rx , + r y
seems to be the same

Thank you

To determine if Mario's statement is correct, let's consider the steps involved in each transformation and compare the resulting coordinates.

Step 1: Reflecting a point XY across the y-axis
When you reflect a point across the y-axis, the x-coordinate changes sign while the y-coordinate remains the same. Thus, the reflection of a point XY across the y-axis can be represented as (-x, y).

Step 2: Dilating the reflection image by r
Dilating the reflection image by a factor of r stretches or shrinks the coordinates by a scale factor of r. Thus, the dilated reflection image would have coordinates (r(-x), ry), which simplifies to (-rx, ry).

Alternatively, we can also consider the steps in a different order:

Step 1: Dilating the point XY by r
Dilating the point XY by a factor of r stretches or shrinks the coordinates by a scale factor of r. Thus, the dilated point would have coordinates (rx, ry).

Step 2: Reflecting the dilation image across the y-axis
Reflecting the dilation image across the y-axis, with coordinates (rx, ry), would result in (-rx, ry).

Comparing the two results, we can see that the final images after both sequences of transformations are indeed the same: (-rx, ry) = (-rx, ry).

Therefore, we have confirmed that Mario is correct in stating that reflecting a point XY across the y-axis and then dilating the reflection image by r will produce the same final image as dilating the point by r first and then reflecting the dilation image across the y-axis.