Reflecting Points on a Coordinate Plane
A reflection is when a graphical line or shape is mirrored to another side of a specific line or axis. When reflecting points, an easy way to remember how to do it successfully it to change the sign of the non-reflecting axis values. For example, if a question were to ask a student to reflect the point (-2,3) across the x-axis, all you need to do is change the sign of the opposite axis value, which in this case would be the y-value. So the y-value of this point is 3. This is a positive value. If you change the sign to negative you get -3. So the new point is (-2,-3), which is a reflection of the original point across the x-axis.
When we think of the term reflection, we most likely think of looking in a mirror or a still body of water. This idea is related to what happens when we reflect figures on the coordinate plane. Like I mentioned, there are two main types of reflections: reflections over a line and reflections on a point. Let’s start by looking at reflections over a line.
We can reflect a figure in the coordinate plane over any line in the coordinate plane. Here’s a triangle reflected over the line x=7.
The triangle on the left is our original figure, or preimage, named Triangle ABC. The triangle on the right is our reflected figure, or image, named Triangle A’B’C’. Prime notation designates the figure that is the image. Point A on the preimage corresponds to point A’ on the image, and so forth.
Now there are a few things to notice here:
Firstly, the preimage and image are congruent, but “flipped”.
You’ll also notice that all corresponding points on the preimage and image are the same distance from the line of reflection but in the opposite direction.
And lastly, the line of reflection bisects all segments connecting corresponding points of the preimage and image.
Now, here is a trapezoid being reflected over three different lines. Notice how the three properties we just discussed hold true, but also notice how, to actually create a reflection, it often involves simply counting the distances between the line of reflection and the points, or adding and subtracting coordinates.
In some cases, the line of reflection may be on the edge – or even inside – the figure.
Three common ways to reflect figures are over the x-axis, y-axis, and the line y=x. Something to notice as we look at these is that the signs of the coordinates change.
For example, when a figure is reflected over the x-axis, notice the y-coordinates change sign.
When a figure is reflected over the y-axis, notice the x-coordinates change sign.
When a figure is reflected over the line y=x, notice that the coordinates change order.
We can also reflect a figure in the coordinate plane in any point on the coordinate plane. Here’s a dart reflected in the point (12,10).
The point the dart is reflected over is called, you guessed it, the point of reflection.
The properties of reflections in points are very similar to those of reflections over lines:
The preimage and image are congruent, but the image is a 180-degree rotation of the preimage.
All corresponding points on the preimage and image are the same, but opposite, distance from the point of reflection.
And, the point of reflection is the midpoint of all segments connecting corresponding points of the preimage and image.
Sounds just like reflection over a line, right?
Here is a kite being reflected over several different points. The point of reflection can be on the edge – or even inside – the figure. Notice how our properties still hold true.
And finally, let’s look at one last reflection. One often-used reflection is one about the origin (0,0). When a figure is reflected about the origin, the signs of all coordinates change.
That’s all for this video. Thanks for watching, and happy studying!