10 Pull It All Together

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BIG idea Modeling

You can represent many real-world mathematical problems algebraically.

Task 1

Imagine a plane and a cone intersecting to form a parabola.

1. Explain why the plane has to intersect the axis of the cone.
2. Imagine the plane moving so that it keeps the same angle, but its point of intersection with the axis moves in the direction of the apex of the cone and eventually passes through the apex. Describe what happens to the parabola and write equations that describe how the parabola changes.

BIG idea Equivalence

You can represent any relationship in an infinite number of ways, where each representation has the same domain and the same pairing of inputs with outputs.

Task 2

You can define an ellipse using a cone, a set of points, or algebra. For each kind of definition, explain how to describe a circle as a special case of an ellipse.

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BIG idea Coordinate Geometry

You can use a coordinate system to represent and analyze geometric relationships.

Task 3

The focus of the hyperbola x 2 a 2 − y 2 b 2 = 1 fraction x squared , over eh squared end fraction . minus . fraction y squared , over b squared end fraction . equals 1  is c h c sub h  units from (0, 0). Imagine the ellipse inscribed in the central rectangle of the hyperbola. Its focus is c e c sub e  units from (0, 0). How far apart are the foci ( c e , 0 ) open , c sub e , comma 0 close  and ( c h , 0 ) ? open , c sub h , comma 0 close question mark  Give the distance in terms of a and b.

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