What he did wrong is that he did not flip all of the points. since S is 1 point up, it would then reflect 1 point down. point P is 3 up and the reflection should be 3 down. Q should be 3 down as well and R is 1 down below the x axis. The Y does not move for all of the points.
A linear equation has 1 for all exponents on the placeholders aka the equation is first degree (aka the hightest exponent on placeholders=1) so it is linear