Four Quadrants and Transformations

Extending the axes into negative numbers cuts the plane into four quadrants, but the rule has not changed: across first, then up. Going left counts as a negative across, and going down as a negative up. So minus three comma two means three to the left and two up, while three comma minus two means three to the right and two down. Points to the left of the vertical axis have a negative first number, and points below the horizontal axis have a negative second.

Reflecting a point in the vertical axis moves it sideways only, so its height is unchanged and just the first number changes sign: three comma two becomes minus three comma two. Reflecting in the horizontal axis moves it up or down only, so the second number changes sign instead: three comma two becomes three comma minus two. Only one number changes each time, and which one depends on which mirror.

A half turn about the origin sends a point to the place diagonally opposite, the same distance out, which changes the sign of both numbers: three comma two becomes minus three comma minus two. It is the same as reflecting in one axis and then the other. And whatever turn you make, the origin itself never moves, because it is the point you are turning about.

A quarter turn anticlockwise about the origin swaps the two numbers and then makes the new first one negative: a comma b becomes minus b comma a. So three comma two goes to minus two comma three, and one comma four goes to minus four comma one. Four quarter turns bring the point back where it started. If the grid is in front of you, physically turning the page a quarter turn is a perfectly good way to check the answer.

Three rules, and they are easy to tell apart by how much changes. A reflection changes one sign. A half turn changes both. A quarter turn swaps the numbers and then changes one sign. When in doubt, plot the point and look at the picture, because these are questions where a quick sketch beats remembering a rule. Quiz and exercises now.