A point is reflected over the x-axis and then reflected over the y-axis. Will the coordinates after these two reflections be the same or different if the point is first reflected over the y-axis and then over the x-axis? Use an example to support your answer. Please help me!
(x,y) -> (x,-y) -> (-x,-y)
vs
(x,y) -> (-x,y) -> (-x,-y)
Hey, just try it
y2 = -y1
x2 = -x1
==========then
x2 = -x1
y2 = -y1 :)
The coordinates of a point after two reflections will depend on the order of the reflections.
To understand this, let's consider the point (2, 4) and apply the two reflections in the given order.
First, we reflect the point (2, 4) over the x-axis. The reflection over the x-axis changes the sign of the y-coordinate, so the reflected point will have the coordinates (2, -4).
Next, we reflect the point (2, -4) over the y-axis. The reflection over the y-axis changes the sign of the x-coordinate, so the final reflected point will have the coordinates (-2, -4).
Now, let's consider the point (2, 4) and apply the reflections in the opposite order.
If we first reflect the point (2, 4) over the y-axis, the x-coordinate changes sign but the y-coordinate remains the same. So, the point after this reflection will have the coordinates (-2, 4).
Then, if we reflect the point (-2, 4) over the x-axis, the y-coordinate changes sign but the x-coordinate remains the same. So, the final reflected point will have the coordinates (-2, -4).
Therefore, we can see that the coordinates after the reflections are different when the order is changed.
To determine whether the coordinates of a point will be the same or different when reflecting over the x-axis and then the y-axis versus reflecting over the y-axis and then the x-axis, we need to understand the effects of these reflections on the coordinates.
When a point is reflected over the x-axis, the y-coordinate is negated while the x-coordinate remains the same. For example, if we have a point (2, 3) and reflect it over the x-axis, it becomes (2, -3).
When a point is reflected over the y-axis, the x-coordinate is negated while the y-coordinate remains the same. Using the same example, if we reflect the point (2, -3) over the y-axis, it becomes (-2, -3).
Now, let's consider the order of reflection. If we first reflect the point over the y-axis and then over the x-axis, it means we apply the y-axis reflection first. Using the original point (2, 3), the y-axis reflection would result in (-2, 3). Then, reflecting this point over the x-axis, the y-coordinate gets negated, resulting in (-2, -3).
If we instead first reflect the point over the x-axis and then over the y-axis, using the original point (2, 3), the x-axis reflection would result in (2, -3). And then reflecting this point over the y-axis, the x-coordinate gets negated, resulting in (-2, -3).
From the examples above, we can see that regardless of the order of reflection, in this case, the coordinates of the point after both reflections are the same. The point is reflected to the same final position (−2, −3) in both cases.
Therefore, the coordinates will be the same whether the point is first reflected over the x-axis and then over the y-axis or first reflected over the y-axis and then over the x-axis.