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. Also, follow my store to get notifications about new products I add to my store.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 like this product, leave a review (you can get tpt credits for every review you leave - and then use those credits on future tpt purchases). Access to a sample digital bulletin board that was created by my former students.Instructions for using the digital bulletin board.Instructions for holding a brief tutorial on how to use sliders in Desmos.It will be useful for the teacher to have some experience with This is something that I do with my Precalculus students in order to pique their interest in the graphs of polar equations.Īccess to all websites needed for this activity is FREE. Students do not need to have any prior knowledge of polar graphs before doing this activity. ![]() This activity also works really well with distance learning! It can also work for normal face-to-face learning. Students make a polar graph using and then post it on a digital bulletin board which can be shared with the entire class, the entire school or the entire community!
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