SWBAT analyze graphs and points on a rotated coordinate plane.

y = (1/x) is a hyperbola, but why is its equation so different than other hyperbolas?

5 minutes

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.

30 minutes

Yesterday we used Desmos to show that when the *B* value of the general form of a conic equation (*Ax*^{2} + *Bxy* + *Cy*^{2} + *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.

- What is the center of the hyperbola on the x’y’ coordinate plane?
- What are the coordinates of the vertices on the x’y’ coordinate plane?
- What is the
*a*value of this hyperbola? - How can we find the
*b*-value? - 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.

15 minutes

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

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