Question Video: Understanding the Effect of Reflection in the 𝑦-Axis on a Point given Its Coordinates Mathematics

Video Transcript

If a point with coordinates 𝑥, 𝑦 is reflected in the 𝑦-axis, determine the coordinates of the image.

We can start this question about reflections in the 𝑦-axis by recalling that if we draw a usual coordinate grid, the 𝑦-axis is the vertical line. When we are reflecting in the 𝑦-axis, that means that the mirror line of the reflection is this line. We need to consider how a general point with coordinates 𝑥, 𝑦 changes when it is reflected in the 𝑦-axis. And there is in fact a general rule that will immediately give us an answer to this question. But let’s think first about how this rule might come about.

We can take any point on the grid, say the coordinates four, one, and reflect it in the 𝑦-axis. Four, one is at a perpendicular distance of four units from the mirror line of the 𝑦-axis. So the image of this point will also be at a perpendicular distance of four units away from the 𝑦-axis on the opposite side. This image will be at the coordinates negative four, one.

Let’s try a reflection in the 𝑦-axis again, but this time with the coordinates one, negative two. It’s at a perpendicular distance of one unit from the 𝑦-axis. So the image is one unit away on the opposite side. And its coordinates will be negative one, negative two. So far, we’ve considered two points with positive 𝑥-coordinates and reflected them onto the left side of the grid.

But can we plot a point with a negative 𝑥-coordinate and reflect it onto the right side of the coordinate grid? Well, let’s take the coordinates negative six, negative four. It doesn’t matter that this point has an 𝑥-coordinate of negative six, because we know that we can still work out that this point is at a perpendicular distance of six units from the 𝑦-axis. So the image will also be six units in the opposite direction from the mirror line at the coordinates six, negative four.

So what pattern can we deduce from the reflections in the 𝑦-axis with all these coordinates and their images? Firstly, we can note that the 𝑦 coordinate stays the same in each of the starting coordinates and their images. And it doesn’t matter if they’re positive or negative; they stay the same. That’s because when we reflect in the 𝑦-axis, our reflected points won’t move up or down. They will be the same height, or the same distance away, from the 𝑥-axis, no matter whether that’s above or below the 𝑥-axis.

But what about the 𝑥-coordinates? In our first coordinate pair, the 𝑥-coordinate of four became negative four in the image. The same is true for the next coordinate. The 𝑥-coordinate of one became negative one in the image. And if we look at the final pair, the 𝑥-coordinate, which was originally negative six, became a positive six in the image. In each case, the 𝑥-coordinate has changed sign. So if we wanted to reflect a general point 𝑥, 𝑦 in the 𝑦-axis, the 𝑦-coordinate stays the same and the 𝑥-coordinate changes sign.

This negative 𝑥 will allow the 𝑥-coordinate to become negative in the image if the original 𝑥-coordinate had a positive value and positive if the original 𝑥-coordinate had a negative sign. And these coordinates negative 𝑥, 𝑦 will be the answer to the question, because it is the image of the coordinates 𝑥, 𝑦 when reflected in the 𝑦-axis.

And as mentioned at the beginning of this video, this is a general property that we can recall and use. We say that for a general point 𝑃 with coordinates 𝑥, 𝑦, a reflection in the 𝑦-axis maps point 𝑃 to point 𝑃 prime with coordinates negative 𝑥, 𝑦. And this is the same result that we have discovered for ourselves.