![]() Oh! We can use that! r^2 = x^2+y^2 = (1-2*(y^2 / (x^2 + y^2)))^2, drop it into desmos and bam! Cartesian expression for a polar curve! Maybe you can clean that up a bit and simplify things, but that's just flavor: the core relationship between x and y is fully captured by this expression, and it's the same relationship captured by r=cos(2θ) Well, if it were an r^2, it would have to be equal to (1-2*(y^2 / (x^2 + y^2)))^2. Wait! We do! That square on the right hand side is to the full (y/r) quotient, so that's a r^2 in the denominator! Whoopie, we can capture the same r=cos(2θ) relationship by writing r=1-2*(y^2 / (x^2 + y^2))! Gah, still have an r on the left side. We could convert an r^2 into x^2+y^2, but we don't have an r^2. There's still r in a couple places, though. (*I didn't actually remember - I have it written down on my desk because I use it just infrequently enough to not have it memorized). ![]() ![]() Okay but does that help us re-write r=cos(2θ) with only x and y variables? You bet! Lots of different approaches possible here, but I happen to remember* that cos(2θ)=1-2sin(θ)^2. Oh! And since r is the distance from (0,0) to (x,y), we can say r^2=x^2+y^2! Similarly, we can use the height of the triangle to write down sin(θ)=y/r, and tan(θ)=y/x. So how can we keep the r=cos(2θ) relationship, but express it with x and y variables?Įasy! The base of this triangle is definitely x units long, which means cos(θ)=x/r. In a cartesian graph, you want the variables to be x: horizontal signed distance and y: vertical signed distance. For a polar graph, those variables are r: signed distance from the origin and θ: angle from horizontal. Every graph comes from a relationship between your variables. I see in line 8 you had the polar equation for a 4-leaved rose: r=cos(2θ). Here's a teaser: īut put that in the back of your mind for a moment, because I want to think about converting between polar and cartesian coordinates. ![]() That said, you probably learned about shifting (cartesian) curves in some earlier class! If you have a function f(x)=x^2 and you want to plot y=f(x+2), how should you move the original function? What about the graph for y+3=f(x)? (It may be helpful to think of it as y=f(x)-3) There's a lot of beautiful connections ready for you when you think about "adding or subtracting from the x- and y-coordinates" for graphs. Here are some tutorials on how to use :ĪND, my final bit of advice is to join Twitter and follow Desmos.Hi! Looks like this was a challenge screen in your activity? My best guess is your teacher is thinking about this lesson as an early part of your investigation into polar equations, or maybe into plotting polar equations in Desmos specifically - they probably aren't expecting you to be fully proficient with the full mathematics here just yet. If you’re a teacher, you really should check out Play around with anything you can in Desmos! Graphing should be (and can be) fun! Here are some great art examples Desmos has curated. Once you have a reasonable working knowledge of Desmos, practice by making a mathy graph, or try making a picture! You can see my very first (ever) attempt at a picture followed by the improved version I made with just a little more time and effort. ![]() Here are some great math examples Desmos has curated. Once you feel comfortable with the basics, branch out into any of the other topics on that look interesting to you! I recommend regressions, restrictions, and lists. Not sure where to start? How about with some of these basics: If you like reading manuals, here’s a pdf user guide, and if you like having a paper version, it’s only 13 pages long so it’s easy to print! To learn as much as you can about learn how Desmos works, the best thing you can do for yourself is get on over to to watch video tutorials and try some interactive tours. But if you’d like a more structured approach, here’s my advice! The Desmos team has worked hard to make it as user friendly as possible. Many people begin learning about Desmos without any official help, just trying it out. Hi there! If you’re new to Desmos, you’re in for a treat! It’s everything you ever wanted a graphing calculator to be. ![]()
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