Given that vertices 𝐽 negative eight, eight, 𝐾 three, negative nine, and 𝐿 negative three, five form a triangle, without graphing determine their coordinates after a reflection over the 𝑥-axis first and then over the 𝑦-axis.
So we’re given the coordinates of three points: 𝐽, 𝐾, and 𝐿. These three points are undergoing two reflections: firstly, over the 𝑥-axis and secondly, over the 𝑦-axis. We are asked to find the coordinates of the images of these three points. And we’re asked to do this without graphing, which means we’re not supposed to plot these points on a coordinate grid and then use this to help in our answer. We need to find another method of answering this question.
So let’s think about what happens to the general point with coordinates 𝑥, 𝑦 when it’s reflected over the 𝑥-axis. Well, the effect is the 𝑦-coordinate is multiplied by negative one. So the point 𝑥, 𝑦 gets mapped to the point with coordinates 𝑥, negative 𝑦. This is a general rule, which you should memorize.
But to see where it comes from, just picture the effect of reflecting in the 𝑥-axis. The 𝑥-axis is a horizontal line, which means the effect of the reflection is vertical. Points above the mirror line now appear below the mirror line and points below now appear above, which means it’s the 𝑦-coordinate that is being affected. Positive values become negative and negative values become positive. And so this is achieved by multiplying the 𝑦-coordinate by negative one.
Now, let’s think about what happens when you reflect over the 𝑦-axis. Again, for the general point with coordinates 𝑥, 𝑦, the 𝑦-axis is a vertical line, which means the effect of this reflection is horizontal. It’s on the 𝑥-coordinate. Points swap from the left to the right of the 𝑦-axis and vice versa, which means the 𝑥-values change from positive to negative or negative to positive. Therefore, this time, it’s the 𝑥-coordinate that is multiplied by negative one. Again, this effect on the 𝑥- and 𝑦-coordinates is a general rule that you should memorize.
So now we’ve seen what will happen to the 𝑥- and 𝑦-coordinates after each reflection. Let’s actually perform this reflection on the vertices 𝐽, 𝐾, and 𝐿. So we begin with the coordinates of the three points 𝐽, 𝐾, and 𝐿. The first reflection in the 𝑥-axis multiplies the 𝑦-coordinates by negative one. So in the image of the three points 𝐽, 𝐾, and 𝐿, which is 𝐽 prime, 𝐾 prime, and 𝐿 prime, the 𝑥-coordinates are the same, but the 𝑦-coordinates have been multiplied by negative one.
The second reflection is over the 𝑦-axis. And remember the effect here is on the 𝑥-coordinates. They’re multiplied by negative one. So we’re going to keep the new 𝑦-coordinates the same, but multiply the 𝑥-coordinates by negative one. These points are referred to as 𝐽 double prime, 𝐾 double prime, and 𝐿 double prime as they’re the images of 𝐽, 𝐾, and 𝐿 after two reflections. So we have the coordinates of the three points after the two reflections are 𝐽 double prime is eight, negative eight; 𝐾 double prime is negative three, nine; and 𝐿 double prime is three, negative five.
Now, once we’ve written down the effect that reflection over the 𝑥-axis and reflection over the 𝑦-axis both have individually, we could actually have performed the reflection in one step. As we noticed that both the 𝑥- and the 𝑦-coordinates are multiplied by negative one. So we could’ve written down that the overall reflection is just at the point with coordinates 𝑥, 𝑦 gets mapped to the point with coordinates negative 𝑥, negative 𝑦.
So a slightly quicker approach may be that instead of writing down the step-by-step coordinates after reflection over the 𝑥-axis and then reflection over the 𝑦-axis, we could have just performed the two reflections in one step.