Our goal is to determine if the following statement is sometimes, always, or never true.
The point (x,y) is reflected across the x-axis. Then the new point is reflected across the y-axis. The location of the point after both reflections is (- x, - y).
Let's remember what we know about how to find the reflections of points across the x-axis on the coordinate plane.
Keep the x-coordinate of the point the same.
Take the opposite of the y-coordinate. This is is the same as changing the sign to the opposite one.Reflection in thex-axis
( x, y) → ( x,- y)
Next, let's review how we reflect a point across the y-axis.
Take the opposite of the x-coordinate.
Keep the y-coordinate of the point the same.
Reflection in they-axis
( x, y) → (- x, y)
Let's now perform these reflections one after another.
( x, y) x-axis → ( x,- y) y-axis → (- x,- y)
The new point is (- x, - y). Notice that both coordinates have changed their sign. This suggests that the statement is always true.
The point (x,y) is reflected across the x-axis. Then the new point is reflected across the y-axis. The location of the point after both reflections is (- x, - y).
Let's work through an example. We can take (3, 4) as our point. First, we will reflect it across the x-axis by changing the sign of the y-coordinate.
Reflection in thex-axis
( 3, 4) → ( 3,- 4)
Now we reflect the point across the y-axis. For that, we change the sign of the x-coordinate.
Reflection in they-axis
( 3, - 4) → (- 3, - 4)
We see that both coordinates changed their signs!
