Sunday, November 25, 2012

Equation of a Circle

Once you have worked with functions for a while, inevitably you will begin to wonder "what about circles?"  You have explored all sorts of different equations and their graphs.  Perhaps you've even gone so far as to rotate graphs sideways and learn how to manipulate those equations as well.  But despite these seemingly more involved concepts, you've yet to come across circles.  For such an apparently simple shape, why have these not been included in such rudimentary graphing lessons.

Well, circles are a little different from what you've done so far.  For starters, circles technically aren't functions.  This may surprise you at first, but recall the vertical line test.  Would a circle pass such a test?  Of course not, because passing a straight vertical line through any point on a circle (except for the tangent points on the sides) would also intersect the circle on its opposite side.  Try it and you will see!  So, if a circle is not a function, then how do we handle describing their equations?  That is a little different, but really not much harder than your typical parabola graph.  Follow along and I will explain the equation of a circle in more detail.

For starters, let me just show you the equation for a simple circle.  Consider a circle with a center at the origin (0, 0) with a radius of 1.  We call this a unit circle.  Here is what the equation looks like:

It looks simple, right?  This equation can also be modified using similar concepts to how you would manipulate a function, such as a parabola.  There is a simple change to make to the equation that causes the graph of the function to translate left or right, and a second similar change you can make that results in a vertical translation of the graph.  In this case, depending on where you want to shift your circle, that is the variable you modify.

So, if you want to center your circle at (2, 0), which is a horizontal translation of 2 units to the right, you would change the above formula to include an (x-2)2 term.  Think of it as "if x=2, the whole x term becomes 0.  So 2 is its root."  Same thing applies to vertical translation.  If you change the y term to be (y-5)2, then you can see that the whole y term becomes 0 when y=5, so this means that the circle is centered at a height of y=5.  If we combine these two translations, we shift the circle to have a center of (2, 5), and the final equation looks like this:

Hopefully you can see how easy it can be to locate the center of a circle based on its equation, and how equally simple it can be to determine the equation of a circle just by a visual inspection.  However, to fully describe a circle, there is still something missing.  We haven't looked at what the "1" means.

To fully describe a circle mathematically, the only things that you need to know about it are the coordinates of its center, as well as its radius.  You may think that you should need more information, but think of it as the mathematical equivalent of a geometry set's compass.  To draw a perfect circle with a compass, all you do is put the point down at the center, set the radius, and spin the pencil around the paper.  You only needed those two pieces of information to be able to construct a circle.

Continuing with our example, the radius of our circle is described by the 1 term.  Technically, you can think of it as a 12 term, which provides for our circle to have a radius of 1 (or a unit circle).  To change our radius to 5, we would change the 1 to a 25, because just like the x and y terms, the radius term is also squared.  If we wanted a radius of 7, we would put the term as 49.  In general, this term is r2.

If we put all of these components together, we can come up with the general form for the equation of a circle.  We can include terms that allow for horizontal and vertical translation, as well as whatever radius we want.  Let's call the horizontal shift term "a" and the vertical shift term "b", and the radius "r".  In this case, here is the general form of the equation of a circle:

The equation describes a perfect circle, and doesn't allow for any stretching or compressing along either of the axes.  All that needs to be done is determine the circle's center point and radius, and you can easily fill in the relevant values.

As an example, consider the circle that has radius of 10, and is centered at (-5, 7).  Here is its equation:

Here is the opposite form of the question: what is the center and radius of the circle described by the following equation:

Quick analysis of this by comparison to the general form allows you to find that the center point is (-12, -2) and has a radius of 12.

Hopefully you can see that graphing circles is a little bit different from graphing more familiar functions, though still similar enough that the math concepts we've learned so far can easily be adapted to apply here as well.  In my next post, I would like to explain a little about just "why" the equation of a circle looks the way it does.  For teachers, this is also a good exercise to have students explore when first learning how to graph circles.  

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