For our example, let us consider the reflection on the x axis and then a translation of 1 unit to the right and 3 units up.Consider the line y=x. We will see what happens if we first apply the translation and then the reflection and then we will see what happens if we do it the other way around.
Recall that given the reflection on the x axis, we simply multiply the y coordinate by -1.
So, let us see what happens in each case.
First apply reflection and then translation.
We are given the line y=x. If we want to reflect this line on the x axis, we multiply the y coordinate by -1, so we get
[tex]\text{ -y=x}[/tex]Now, if we apply a translation of 1 unit to the right and 3 units up, we add 1 to the x and 3 to the -y. So we get
[tex]\text{ -y+3=x+1}[/tex]If we subtract 3 on btoh sides we get
[tex]\text{ -y=x+1 -3=x -2}[/tex]Finally, we multiply both sides by -1, so we get
[tex]y=\text{ -x+2}[/tex]First apply the translation and then the reflection.
We are given the line y=x. We apply the translation first, so we add 1 to the x and 3 to the y. So we get
[tex]y+3=x+1[/tex]Now, we apply the reflection, so we multiply by -1 the whole left side. So we get
[tex]\text{ -(y+3)=x}+1=\text{ -y -3}[/tex]Now we add 3 on both sides, so we get
[tex]\text{ -y =x+1+3=x+4}[/tex]Now we multiply by -1 both sides, so we get
[tex]y=\text{ -x -4}[/tex]As we can see, both equations are different
Here is a sketch of the results
