Hey, everyone. We can graph a limacon just like we graphed a cardioid. By determining the symmetry of our graph and finding and plotting points at our quadrantal angles, we're first going to take one additional step and determine whether our limacon has a dimple or an inner loop, which we can do easily based on our a and b values given in our equation. So let's jump right into graphing our limacon here. Remember that the equation of a limacon will always be of the form r=a±bcos⁡θ or r=a±bsin⁡θ, just like a cardioid, except here, a is either greater than b or less than b. Now here, the equation that we're tasked with graphing is r=3-2sin⁡θ. Now I know that this is going to be the graph of a limaçon because I have subtraction happening here, and I have an a value of 3 and a b value of 2, so a is greater than b.

Now let's look at our first step here. We just said that a is greater than b. This tells us that our limaçon will have a dimple rather than an inner loop. So, that tells me that my graph will be shaped something like this, and it's just up to us to get more precise with these other steps. Now in determining whether our graph has a dimple or inner loop, we also learn some more information about our graph, whether or not it has a zero. Since here our graph only has a dimple, it will not have a zero, meaning that our graph will not pass through the pole.

So, let's continue on with step 2 and determine the symmetry of our graph. Now here our equation contains a sine function, so that tells us that our graph will be symmetric about the line θ=π2, which again is something that I want to keep in mind as I move on to my next step, where we're actually going to find and plot points at our quadrantal angles. Here at our quadrantal angles, I know that I'm going to plug in these values of θ into my equation. So here for that first value of θ=0, I'm going to take 3-2sin⁡0 and the sine of 0 is simply 0 so this will end up giving me a value of 3. So I can plot that first point at 3θ which will be right here.

Then plugging in π2 to my equation, 3-2timessin⁡π2, will give me a value of 1. So I can plot this second point at 1,π2 right here. Now for my final point: plugging in 3π/2 gives me 3-2timessin⁡3π/2 giving me a value of 5. So this final point here is going to be located at 53π2, which will be right here.

Now all that's left to do here is connect all of these points with a smooth and continuous curve and we know the general shape of the graph of a limaçon and we know, of course, that this limaçon has a dimple. So in connecting these points, I want to make sure that I reflect those. Now here I'm going to go ahead and connect these points with a dimple at the top of my graph here and my graph of this limaçon will end up looking something like this. Now remember, if you're asked to get more precise or if you just want to get more precise, you can always plot more points and that's totally fine. But now that we have fully graphed this limaçonnessome, let's continue getting some practice. Thanks for watching, and I'll see you in the next one.