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# Rotated Conic Sections - Day 1 of 2

Lesson 6 of 10

## Objective: SWBAT analyze graphs and points on a rotated coordinate plane.

*50 minutes*

#### Launch

*5 min*

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.

#### Resources

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#### Explore

*30 min*

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.

#### Resources

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#### Summarize

*15 min*

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.

#### Resources

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- UNIT 1: Functioning with Functions
- UNIT 2: Polynomial and Rational Functions
- UNIT 3: Exponential and Logarithmic Functions
- UNIT 4: Trigonometric Functions
- UNIT 5: Trigonometric Relationships
- UNIT 6: Additional Trigonometry Topics
- UNIT 7: Midterm Review and Exam
- UNIT 8: Matrices and Systems
- UNIT 9: Sequences and Series
- UNIT 10: Conic Sections
- UNIT 11: Parametric Equations and Polar Coordinates
- UNIT 12: Math in 3D
- UNIT 13: Limits and Derivatives

- LESSON 1: Conics Sections with Princess the Dog
- LESSON 2: Exploring Ellipses
- LESSON 3: Hashing Out Hyperbolas
- LESSON 4: Pondering Parabolas
- LESSON 5: Classifying Conics
- LESSON 6: Rotated Conic Sections - Day 1 of 2
- LESSON 7: Rotated Conic Sections - Day 2 of 2
- LESSON 8: Unit Review: Conic Sections
- LESSON 9: Review Game: Lingo
- LESSON 10: Unit Assessment: Conic Sections