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Conic Sections Study Guide

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Conic Sections Study Guide

By David Chester

Circle

Ellipse

Parabola

Hyperbola

- Graphing a conic section requires recognizing the type of conic you are given
- To identify the correct form look at key traits of the conic that distinguish it from others
- Once you know what type of conic it is you can start graphing by applying the points and properties starting from the center/vertex

- Formulas
- Circle
- Ellipse
- Parabola
- Hyperbola

- Graphing/Plotting
- Circle
- Ellipse
- Horizontal
- Vertical

- Parabola
- Hyperbola
- Horizontal
- Vertical

- Differences/Identifying
- Circle
- Ellipse
- Parabola
- Hyperbola

General Equation for conics:

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

- Circle:

(x-h)2 + (y-k)2 = r2

If Center is (0,0):

x2 + y2 = r2

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Axis is horizontal: Axis is Vertical:

a2 - b2 = c2

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- Opens left or right: Opens up or Down:

(y-k)2=4p(x-h)

(x-h)2=4p(y-k)

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- x2 term is positive : y2 is positive:

a2 + b2 = c2

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- Circle:

- To Graph a Circle:
- Write equation in standard form.
- Place a point for the center (h, k)
- Move “r” units right, left, up and down from center.
- Connect points that are “r” units away from center with smooth curve.

r

p

Definition of a Circle

A circle is the set of all points in a plane that are equidistant from a fixed point, called the center of the circle. The distance r between the center and any point P on the circle is called the radius.

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Graphing and Plotting Parabolas

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Generally:

Using the General Second Degree Equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 and the properties you can determine the type of conic, more specific ways to identify are on the next few slides.

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- Circles x, y, and r are terms will always be squared or be squares, this does not guarantee perfect squares
- Circles are generally simple formulas as they do not have an a, b, c, or p

Examples:

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- A key point of an ellipse is that you add to equal 1
- In an ellipse a and b term switch with horizontal versus vertical
- a>b
- Horizontal: a on the left side
- Vertical: a on right side
- a2 - b2 = c2

Examples:

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- Parabola is unique because it has a p in its equation
- Only one term is squared
- The x and y switch place with left & right versus up & down
- Up & Down: x on the left
- Left & Right: x on the right

Examples:

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- A key point for a hyperbola is that you subtract in order to equal 1
- In a hyperbola the x and y terms switch in a horizontal versus a vertical
- Horizontal: x on the left side
- Vertical: x on right side
- a2 + b2 = c2

Examples:

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- http://math2.org/math/algebra/conics.htm
- http://mathforum.org/dr.math/faq/formulas/faq.analygeom_2.html#twoconicsections
- http://www.clausentech.com/lchs/dclausen/algebra2/formulas/Ch9/Ch9_Conic_Sections_etc_Formulas.doc
- Major Credit to: Kevin Hopp and Sue Atkinson (Slides 9-12 directly from them)