3/31/2024 0 Comments Graphing polar coordinates![]() ![]() However, the test will fail on the polar form of the function. Will students be able to explain why the polar graph meets the conditions for a function on the domain from 0 to 2π? At this point in your discussion you will have a visual representation for why the vertical line test is only valid for the “auxiliary graph” in trigonometric form. Then, I used the free Screen to GIF app to re-create the animation for my blog post. I’ve created an animation using my favorite free graphing tool WINPLOT to sketch and animate the curve. This shape is commonly called a limaçon with an inner loop. When the limaçon is complete, be sure to reference the results of this behavior between 120º and 240º. Where the values for r become negative, the result is a small loop inside the larger outer loop. Here is a page of polar graph paper for you to use. Now, have students sketch the polar curve. Graphing a polar curve using a rectangular reference graph. Ask students for conjectures about how these negative values on the “auxiliary curve” will affect the polar curve. ![]() Be sure they notice where the values of r become negative. Students will have a table of (θ, y) that they can use to graph the polar curve( r, θ). The next line of questioning should naturally lead us to ask, “How can we use these values to sketch the polar graph?” If they have access to a graphing calculator, you can have your groups calculate every 30 degrees and give decimal values. You can have students work cooperatively as they create a table of values for this “auxiliary graph” using friendly angles. I want my students to know how this graph will transfer to a polar curve and what will the polar graph look like. You are ready to move on to the next phase. namely, amplitude 5, period 2π, vertical shift 2 units up, the sinusoid is negative from 120º to 240º. Once you know that students have analyzed and reviewed all of the essentials in the sinusoidal function. Since we only want to consider one period of the curve in this case, our angles give unique y-values from 0 to 2π. This will lead you to the need for a domain restriction in our example. Since we have just learned about polar coordinates and translating between rectangular and polar form, we can talk about the earlier lesson where we found there were infinite representations of polar coordinates for any point in a polar coordinate system. Students should be familiar with this line of questioning, and it will help build a sense of confidence for this new topic. It’s great to get some discussion started here about amplitude, period, vertical shift of the y-axis, and where the function is negative. Notice that all we did was replace y for r. First, have students graph the equation y = 2 + 4 cos θ in the Cartesian plane. I like to use a limaçon with an inner loop for my example. How can we help our students make a connection between the two systems? There is also a great opportunity here to show our students that the vertical line test for functions is only a valid test for equations in rectangular form. We can represent rectangular relations (nonfunctions) using polar coordinates (functions). Since many problems in calculus require us to use functions, having this tool in our toolkit will be beneficial. Now, you might ask, “Why?” By some simple investigation students can see that for every choice of θ, there is only one corresponding value of r. ![]() The best feature is that r = 1 names a function where does not. By teaching polar graphs, students learn they can write this equation using polar coordinates as r = 1, which also represents the unit circle. In fact, many of our Precalculus students will finally begin to understand that this simple circle requires two separate functions to visualize the entire circle on a calculator. Why is it important for Precalculus students to cover polar equations? Let’s consider the unit circle, as an example. If you are in that position, I would like to suggest that you cover the concepts for graphing polar equations. Spring is right around the corner and many Precalculus teachers begin to stress about what content remains. ![]()
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