Rotated Conic Sections - Day 1 of 2

We have been working with the standard form for hyperbolas, so I find it really throws students off if you give them the graph of y = 1/x and ask if it is a hyperbola. The equation y = 1/x looks nothing like the equation we have been using for a hyperbola, but the graph is looks just like the other hyperbolas that we have worked with.

I start by giving students this worksheet and have them think about questions #1-3. They will discuss with their table group and then we will share out with the class. At this point I just want them to recognize that the graph is a hyperbola that has been tilted or rotated 45 degrees. They may not understand why the equation is so different, but we will get there.

Yesterday we used Desmos to show that when the B value of the general form of a conic equation (Ax² + Bxy + Cy² + Dx + Ey + F = 0) is not zero, the conic will be rotated. Some of my students will remember and I use that as our starting point for the discussion. The equation y = 1/x is not in general form, but we convert it and students see that there is a non-zero B value.

Next I will draw a new coordinate plane that has been rotated 45 degrees, and I explain the convention of naming the axes x’ and y’. This is sometimes a difficult conceptual leap, but students now have to come up with an equation for the hyperbola on the x’y’ coordinate system. We begin by writing the standard equation of the hyperbola and I usually walk them through this with some guiding questions. Here are some sample questions I will ask as we go through the process. 

1. What is the center of the hyperbola on the x’y’ coordinate plane?
2. What are the coordinates of the vertices on the x’y’ coordinate plane?
3. What is the a value of this hyperbola?
4. How can we find the b-value?
5. Where are the asymptotes of the hyperbola located?

After we get this example completed, we look at question #5 to think about what type of conic it will be. At this point students will know that it will be rotated, so we need to come up with a way to graph it. I explain our process in the video below.

After we a majority of students have graphed #5 on their calculator, I explain that we will work more on this tomorrow. Today we have figured out that when an xy-term is present, the conic section is rotated, and we have grappled with isolating y in order to graph. I tell students that tomorrow we will work on finding the equation in the x'y' plane if we know the equation in the xy-plane. 

Finally I assign #6 and #7 as homework problems; students can work on those for the rest of class.