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Conic Sections Study Guide. By David Chester. Types of Conic Sections. Circle. Ellipse. Parabola. Hyperbola. Solving Conics. Graphing a conic section requires recognizing the type of conic you are given

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Conic Sections Study Guide

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Conic sections study guide

Conic Sections Study Guide

By David Chester


Types of conic sections

Types of Conic Sections

Circle

Ellipse

Parabola

Hyperbola


Solving conics

Solving Conics

  • 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


Directory

Directory

  • Formulas

    • Circle

    • Ellipse

    • Parabola

    • Hyperbola

  • Graphing/Plotting

    • Circle

    • Ellipse

      • Horizontal

      • Vertical

    • Parabola

    • Hyperbola

      • Horizontal

      • Vertical

  • Differences/Identifying

    • Circle

    • Ellipse

    • Parabola

    • Hyperbola


Formulas

Formulas

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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Ellipse formula

Ellipse Formula

Axis is horizontal: Axis is Vertical:

a2 - b2 = c2

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Parabola formula

Parabola Formula

  • Opens left or right: Opens up or Down:

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

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

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Hyperbola formula

Hyperbola Formula

  • x2 term is positive : y2 is positive:

a2 + b2 = c2

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Graphing and plotting circles

Graphing and Plotting Circles

  • 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 ellipses

Graphing and Plotting Ellipses

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Graphing and plotting ellipses1

Graphing and Plotting Ellipses

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Conic sections study guide

Graphing and Plotting Parabolas

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Graphing and plotting hyperbolas

Graphing and Plotting Hyperbolas

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Graphing and plotting hyperbolas1

Graphing and Plotting Hyperbolas

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Differences identifying

Differences/Identifying

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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Circle traits

Circle Traits

  • 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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Ellipse traits

Ellipse Traits

  • 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 traits

Parabola Traits

  • 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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Hyperbola traits

Hyperbola Traits

  • 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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Bibliography

Bibliography

  • 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)


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