Notice that as the angle changes the point at varying distance from the pole traces a curve.(Because of the way Desmos graphs, you cannot have a slider for θ the a-slider will move the line and the point on the graph. The equation in the example is You may change this to explore other graphs. It will help if you open it and follow along with the discussion below. Polar coordinates use the ordered pair ( r, θ), where r, gives the distance of the point from the pole (the origin) as a function of θ, the angle that the ray from the pole (origin) to the point makes with the polar axis, (the positive half of the x-axis). Instead of using the Cartesian approach of giving every point in the plane a “name” by giving its distance from the y-axis and the x-axis as an ordered pair ( x, y), polar coordinates name the point differently. There are also some suggestions for extending the study of polar function as the end. It will not be as much as students should understand, but I hope the basics discussed here will be a help. This blog post will discuss the basics of polar equations and their graphs. Seeing an animated version much later helped a lot. I remember not having that good an understanding myself when I entered college (where first-year calculus was a sophomore course). Getting accustomed to a new coordinate scheme and a different way of graphing is a challenge. Some classes may even omit the topic entirely. While ideally the polar coordinate system should be a major topic in pre-calculus courses, this is sometimes not the case. One teacher observed that her students do not have a very solid understanding of polar graphs when they get to calculus. A recent thread on the AP Calculus Community bulletin boards concerned polar equations.
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