Topic 4.2 Analyze the General Form Equation
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The General Form
Ax2+Cy2+Dx+Ey+F=0
This equation is called the General form for a conic section.
By analyzing the General Form we can deduce a few things about the conic section
Shape: Determined by looking at A and C
A=C
Circle
AC>0
Ellipse
AC<0
Hyperbola A or C =0 Parabola 2
The General Form
Ax2+Cy2+Dx+Ey+F=0
Position: Determined by looking at D and E
If D and E are both zero then the conic section is centered about the origin. If D is a value other than zero then the conic section has been shifted horizontally.
If E is a value other than zero then the conic section has been shifted vertically.
Thus:
by changing D we can change the horizontal position
by changing E we can change the vertical position
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The General Form
Ax2+Cy2+Dx+Ey+F=0
Size: Determined by F
By changing the value of F the size of the conic section changes
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Standard form of a circle
The standard form equation for the circle is
The center of the circle is located at (h, k)
The radius of the circle is r
Example 1. Sketch the given conic section
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Example 2
y
Given the following graph of a circle determine it’s standard form equation 10
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­10­9 ­8 ­7 ­6 ­5 ­4 ­3 ­2 ­1 0 1 2 3 4 5 6 7 8 9 10
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Standard form of an ellipse
The standard form equation for the ellipse is
The center of the ellipse is located at (h, k)
The horizontal axis has a length of 2a
The vertical axis has a length of 2b
The longest axis is called the major axis
The shortest axis is called the minor axis. 7
Example
Sketch the given conic section
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Example 2
y
Given the following graph of an ellipse
determine it’s standard form equation 10
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­10­9 ­8 ­7 ­6 ­5 ­4 ­3 ­2 ­1­10 1 2 3 4 5 6 7 8 9 10
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Standard Form of a Hyperbola
The standard form equation for the hyperbola is
The Center of the hyperbola is where the asymptotes intersect. ﴾h, k﴿
The slopes of the asymptotes have a value of If the equation then the hyperbola opens horizontally and the vertices are 2a units apart. If the equation then the hyperbola opens vertically and the vertices are 2b units apart. 10
Example 1
Sketch the following hyperbola
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Example 2
y
Given the following hyperbola determine
it’s equation in standard form.
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­10­9 ­8 ­7 ­6 ­5 ­4 ­3 ­2 ­1 0 1 2 3 4 5 6 7 8 9 10
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Standard form of a parabola
The standard form equation for the parabola is
Vertical Parabolas
Horizontal Parabolas
The vertex is located at (h, k)
The vertex is located at (h, k)
The parabola opens up if a>0
The parabola opens right if a>0
The parabola opens down if a<0
The parabola opens left if a<0
The parabola has a vertical stretch by a factor of “a” about the x­axis
The parabola has a horizontal stretch by a factor of “a” about the y­axis
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Example 1
Sketch the following conic section
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Example 2
y
Determine the standard form equation for the given graph
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­10­9 ­8 ­7 ­6 ­5 ­4 ­3 ­2 ­1 0 1 2 3 4 5 6 7 8 9 10
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