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Families of Polar Curves: Roses - Concept

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

One family of polar curves that we will see frequently when dealing with graphing polar equations and polar symmetry is the polar rose. It is important to be able to recognize the general equation of a polar rose, and to use that equation to interpret the symmetry and number of petals. The general equation of a polar rose is similar to that of the family of circles and cardioids.

Let's graph a rose. r=8 sine 3 theta. Now first of all let's recall that any of the sine roses are going to be symmetric about the y axis. So I want to use that because it takes a lot of points, you have to plot a lot of points to get these graphs. Notice I've already plotted many points but what I did was I I entered values for 3 theta because it's easy to evaluate sine of 3 theta and then multiply by 8. And so what I'm going to do to get theta is just divide all these values by 3 and then I'll plot those values and we'll take a look at the graph.
So 0 divide by 3 is 0. 30 divide by 3 is 10. 60 divide by 3 is 20. 30, 40, 50, 60, 70, 80 and 90. So I've plotted from 0 to 90 in increments of 10.
Now these points I've already plotted on the graph but what I want to remind you of is first of all the symmetry with respect to the y axis ad also when you're graphing a polar graph, make sure that to draw the curve in the order that the points are plotted. So we start with 0 0, and then it's 10 degrees 4 or 4 10 degrees. And we go in this direction. This is kind of important because polar curves can cross over themselves and it's really important that you know which way the curve goes at a point where it intersects itself. At this point we're back at 60 0 or 0 60, right? r=0 the angle 60 degrees. And then we're going to go to -4 70. So pointing at 70 degrees which is roughly this this direction, I need to -4 and that lands me down here, so I'm going in this direction.
Okay. So what I have here is half of my graph and if I fill in the rest of the points I'll get the rest of my graph. So this point is reflected here, this one's reflected here and this one is reflected right there so I complete this leaf. And then to finish the graph I'm going to have another leaf over here and it's going to be a reflection of this one so I need to reflect these points. This point here at 4 goes right here. This one which is at 7 goes 5, 6, 7, right about here. And then the one at 8 and back to 7 again and then back to 4 and then back to 0, so it's going to, it's coming round this way. It'll actually plot this way first and then back.
Alright, and by the time we arrive back at the origin, we're at one complete cycle and that's our 3 leaf rose. Right? This is r=8 times the sine of 3 theta. It's a 3 leaf rose, symmetric above the y axis.
If you were to graph a cosine 3 theta, you'd get a 3 leaf rose symmetric about the x axis.

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