1. Conic Sections Study Guide
By David Chester

2. Types of Conic Sections
Circle
Ellipse
Parabola
Hyperbola

3. 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

4. Directory Formulas Graphing/Plotting Differences/Identifying Circle
Ellipse
Parabola
Hyperbola
Graphing/Plotting
Horizontal
Vertical
Differences/Identifying

5. 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
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6. Ellipse Formula Axis is horizontal: Axis is Vertical: a2 - b2 = c2
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7. 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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8. Hyperbola Formula x2 term is positive : y2 is positive: a2 + b2 = c2
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9. 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.
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10. Graphing and Plotting Ellipses
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11. Graphing and Plotting Ellipses
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12. Graphing and Plotting Parabolas
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13. Graphing and Plotting Hyperbolas
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14. Graphing and Plotting Hyperbolas
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15. 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
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16. 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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17. 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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18. 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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19. 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:
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20. Bibliography http://math2.org/math/algebra/conics.htm
Major Credit to: Kevin Hopp and Sue Atkinson (Slides 9-12 directly from them)