Hey, everyone. We know that the general shape of a lemniscate is this sort of infinity symbol or propeller shape with 2 petals. So in order to graph a lemniscate, if we just figured out where those 2 petals were, we'd be good to go. Now we can actually do this rather easily by just figuring out where our first petal is the same way we did for roses and then simply reflecting that petal over the pole. So let's not waste any time here and jump right into graphing our lemniscate. Now remember that the equation of a Lemniscate is always going to be of the form r2=±a2×cos(2θ) or a2×sin(2θ). Now here, the equation that we're specifically tasked with graphing is r2=4×sin(2θ). Now it's easy to tell that this is the equation of a lemniscate because it's the only polar equation that we're working with here that has a squared value of r.
So let's go ahead and jump into step number 1 and figure out where that very first petal is. Now, the first thing that we want to do here is look at our value of a. Now looking at my equation here, I have r2=4×sin(2θ). And something that's really important to remember here is that this 4 is not a. It's actually a2. So in order to get a, we need to take the square root of that 4 in order to get 2. So that tells us that our r value for that first petal is going to be 2. Now, in order to determine theta, we need to look at the sine of our equation and also whether we have a cosine or sine function. Here we have positive 4 sine of 2 theta. So that tells us we're dealing with a positive a2×sinθ, which means that theta will be equal to π4 for this first petal. I can go ahead and graph this first petal at 2π14, which will end up being right here.
Now all that's left to do is reflect this petal over the pole. Remember, my pole is what we think of as our origin in rectangular coordinates. So if I reflect this point over that pole, I'm going to end up down here for that second petal. Now all we have to do is connect these with a smooth and continuous curve, remembering that the shape of our lemniscate is a sort of infinity symbol. So let's go ahead and connect our points here. Coming out from the pole, I'm going to go out to that first point and then out to that second point. And here I have the graph of my lemniscate. Remember that if you're asked to get more precise or if you just want to get more precise, you can always calculate more points by plugging in values of theta to your equation. But now that we know how to graph a lemniscate, let's continue practicing together. Thanks for watching, and I'll see you in the next one.