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**Conic Sections Study Guide**

By David Chester

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**Types of Conic Sections**

Circle Ellipse Parabola Hyperbola

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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

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**Directory Formulas Graphing/Plotting Differences/Identifying Circle**

Ellipse Parabola Hyperbola Graphing/Plotting Horizontal Vertical Differences/Identifying

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**Formulas Circle: (x-h)2 + (y-k)2 = r2 If Center is (0,0): x2 + y2 = r2**

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 Back to Directory

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**Ellipse Formula Axis is horizontal: Axis is Vertical: a2 - b2 = c2**

Back to Directory

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**Parabola Formula Opens left or right: Opens up or Down: (y-k)2=4p(x-h)**

(x-h)2=4p(y-k) Back to Directory

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**Hyperbola Formula x2 term is positive : y2 is positive: a2 + b2 = c2**

Back to Directory

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**Graphing and Plotting Circles**

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. Back to Directory

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**Graphing and Plotting Ellipses**

Back to Directory

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**Graphing and Plotting Ellipses**

Back to Directory

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**Graphing and Plotting Parabolas**

Back to Directory

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**Graphing and Plotting Hyperbolas**

Back to Directory

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**Graphing and Plotting Hyperbolas**

Back to Directory

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**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. Discriminant Type of Conic B2 - 4AC < 0, B = 0, and A = C Circle B2 - 4AC < 0, and either B does not = 0 or A does not = C Ellipse B2 - 4AC = 0 Parabola B2 - 4AC > 0 Hyperbola Back to Directory

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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: Back to Directory

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**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: Back to Directory

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**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: Back to Directory

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**Hyperbola Traits Horizontal: x on the left side**

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: Back to Directory

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**Bibliography http://math2.org/math/algebra/conics.htm**

Major Credit to: Kevin Hopp and Sue Atkinson (Slides 9-12 directly from them)

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