Ever wonder how people are creating beautiful gifs of their graphs? Hi! It’s Halloween, and I’m loving the #mathoween graphs people are posting. Here’s one of my favorite non-circular parametric equations, a Lissajous curve. It has been stretched by a factor of 2 horizontally, and 7 vertically, and then moved left 3 and up 5.Īnd the same is true for all parametric equations. Multiply by a and b, your respective horizontal and vertical scale factors, and then shift by h and k.įor example: is a circle transformed into an ellipse. To transform parametric equations, it’s the same as transforming a single point. Or, in the case of a function y=f(x), “ if it’s inside the parentheses, multiplying by 1/2 stretches the graph horizontally, but if it’s outside, multiplying by 1/2 compresses the graph vertically.” SMH There’s none of this “ subtract 4 to move right 4” nonsense. Now, parametrics are a different story entirely. If you make it happen, please, PLEASE comment below so we can all ooh and ah at the craziness beauty of it. Oh, I’m sorry, you wanted to stretch the circle by different amounts vertically and horizontally? And then translate it? Good luck with that. Why, math, WHYYYY?Īnd although polar form makes resizing this circle a pure joy, it’s a beast to translate anywhere. I mean, if we want to shift right two, we have instead of what seems more logical. And when we translate, we have to do the OPPOSITE of what we want. Here’s an ellipse with a vertical stretch of 4 and a horizontal stretch of 5 that I then translated right 2 and down 1: Translating is fine, and changing the radius is ok, but it’s kind of a pain to do separate vertical and horizontal stretches (aka make an ellipse). But the downside is that it’s not super easy to do transformations. It’s familiar and we love the connection to the Pythagorean Theorem. Let’s stick with our circle for a few moments longer.įor a lot of us, we first learned rectangular form of the circle, and this is our happy place. We’ve already seen that for circles, it makes drawing a particular arc super as simple as changing the domain of t.įor this particular post, I’m going to focus on how delightful transforming parametric equations can be There are many reasons parametric equation are beautiful. It looks, well, unfamiliar! And learning something new takes work! Why bother? 2) Why parametrics? Here’s how it looks in the expression list:Īnd because we get to define which angles we want to graph, it’s super easy to graph whatever arc we want by just changing the domain: Desmos likes t for parametric equations, not a. Where did that t come from? Well, remember, this is math, and we can use whatever parameter names we want! (Parameter = fancy word for variable). In Desmos, instead of defining x and y in separate lines, we just write them as an coordinate pair. (And if you’re more comfortable with degrees, just switch to degree mode and use 0 to 360.) To get an entire circle, we’ll need to graph all the points from 0 to 2π radians. Instead of just a single point, we are graphing for a whole set of a values. Wanna know the parametric equations? You’re looking at them! x and y are both defined in terms of the parameter a. One of my favorite cute little features of this circle is this:Įvery point on the unit circle can be written as. Remember our good friend from precalculus, the unit circle? You’re probably familiar with its rectangular form:Īnd you might be familiar with the polar form, so beautiful in its simplicity: For now, we’ll look at one with radius 1. For a specific example (that I happen to think is a great one), let’s talk about the circle. Sounds tricky, but it’s a lot simpler than it might sound. Instead of defining an equation with x and y in terms of each other, we define our set of coordinate pairs in terms of a third parameter. If you already know what they are and you’ve been wanting to give them a whirl, then you are in luck, friend! This whole series of posts is for you. But if you’ve ever wondered a) “What are parametric equations?” or b) “What’s all the fuss? I’m perfectly happy graphing without them, thank you very much.”, this post is for you. If you’ve read my posts before it may not surprise you that I love parametric equations more than rectangular or polar equations. Hi! Welcome to a short series of posts that I hope will help get more people started with graphing using parametric equations.
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