Intro to Transformations | Translations, Reflections, & Rotations

Welcome to Math with Mr. J. In this video, I'm going to go through an introduction to transformations. Specifically, we're going to look at rigid transformations, also called congruent transformations. Simply put, we're moving figures, so changing their position or orientation without changing their size or shape. Our three transformations are translations, reflections, and rotations. By the end of this video, we'll have a basic understanding of all three of these. Let's jump into our first transformation, a translation. Now simply put, a translation is a slide. We can translate or slide a figure in any direction, up, down, left, right, or diagonally. For example, let's take a look at a triangle. Let's slide this triangle to the right, so our triangle here has been translated to the right. Notice that the orientation, how the triangle is facing, stays the exact same. We just slid it over. With translations, the orientation will stay the same. The figure is just Slid to a new position. And that's it. That's a translation. Take a look at the original triangle on the left. I made it slightly transparent to make the difference clear between the original triangle and the translated triangle. The original figure is what we call the pre-image, and then the transformed figure is what we call the image, so something to keep in mind. Now again, we can translate this in any direction. Left. Right, up, down, or diagonally. So for example, here is up, and here is diagonally. Diagonally. Let's move on to our next transformation, a reflection. A reflection is a flip that creates a mirror image of a figure. Now when it comes to reflections, the figure flips across what we call a line of reflection. Also called a mirror line. So for example, let's use our triangle again. And here's our line of reflection. Imagine folding a piece of paper right along that line. The original triangle and its reflection will line up exactly. So let's reflect this. You can see our pre-image and image right here. A horizontal line of. Reflection, like this, causes the figure to flip vertically, so up or down. Now, can a line of reflection go in other directions? Absolutely. A line of reflection can be horizontal, vertical, or diagonal. Let's actually look at a vertical line of reflection. So here's our original triangle, and here is a vertical line of reflection. A vertical line makes the figure flip horizontally, so side to side. Now let's reflect this triangle across that vertical line. Let's move on to our last transformation, a rotation. Now a rotation is a turn. A rotation is a transformation where we turn a figure around a fixed point. We call this point the center of rotation or point. of rotation. So for example let's use our triangle again. Now we need a center of rotation. So basically what are we going to rotate this triangle around? Now a center of rotation can be anywhere. It can be outside the figure, inside the figure, on a vertex, or even along an edge. Let's put our center of rotation on that right vertex. So right here. We're going to rotate this triangle around this point. Now we can rotate clockwise or counterclockwise. Let's rotate clockwise here. So here is an example of a rotation. We can see our pre-image and our image. Here's another rotation and here's one more. Now let's look at an example where the center of rotation is outside of the figure. Here's another triangle, and we'll place our center of rotation right out here. Watch what happens as we rotate this counterclockwise through several positions, all the way back to the start. Notice that as the triangle rotates, the center of rotation stays fixed. It doesn't move. Also, take a look how the triangle stays the same distance from that center of rotation as it rotates. At no point does the triangle move closer or further away from that center of rotation. Let's do one more example of a rotation, and this time we're going to use a rectangle. Let's put the center of rotation right in the middle, so right here. Now let's rotate this rectangle clockwise. Watch what happens as it rotates through several positions, and eventually returns to its starting position. Notice how the rectangle turns around that fixed center point. The center of rotation stays in the same place throughout the entire transformation. So there's a few examples of rotations with different centers of rotation. Let's actually go back and put up our original triangle. So there you have it. There's an intro to translations, reflections, and rotations. I actually use these three transformations on a daily basis, whether I'm editing videos, creating backgrounds for my videos, or designing thumbnails. I'm constantly transforming figures, numbers, words, and other graphics. I might need to translate text to an exact position on a thumbnail, reflect a graphic, to get it facing the opposite direction, or rotate something to the exact position I need. These transformations are used in all kinds of different fields and applications, and having a basic understanding of how they work is incredibly useful. I hope that helped. Thanks so much for watching. Until next time, peace.