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The four types of conic sections are parabolas, ellipse, circles and hyperbolas.
Conic Sections
A conic section is…..
______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Graph the following equations. Name the conic. Define the domain and range of each graph. Identify the
lines of symmetry.
a. 푥 + 푦 = 64
b. 9푥 + 16푦 = 144
Name:___________________________
Domain:_________________________
Range:___________________________
Lines of symmetry:________________________
Name:___________________________
Domain:_________________________
Range:___________________________
Lines of symmetry:________________________
c. 푥 −푦 = 16
d. 푥 − 6푥 − 푦 = −11
Name:___________________________
Domain:_________________________
Range:___________________________
Lines of symmetry:________________________
Name:___________________________
Domain:_________________________
Range:___________________________
Lines of symmetry:________________________
Parabolas
Parabola: _________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
Focus of a
parabola:__________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
Directrix:__________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
Focal
length:____________________________________________________________________________________
__________________________________________________________________________________________
Parabolas can be either vertical or horizontal
Equation of a Vertical Parabola
The standard form of the equation of a parabola with vertex (h,k) and an axis of symmetry x = h is:
If a>0 then the parabola opens ____________________.
If a<0 then the parabola opens ____________________.
The equation of a vertical parabola with the vertex at the origin can be found by using the geometric
definition. If the focus is the point (0,c), the directrix is the line with the equation y = - c
Using the distance formula:
Note that the equation has the expected quadratic form 푦 = 푎푥 for a vertical parabolawith vertex (0,0). The coefficient 푎 =
determines both the focus (0,c) and the directrix y = -c. This is the key to shifting between algebraic and geometric representations of
a parabola.
Example 1: Find the equation of a parabola with the vertex at the origin and the focus at (0,2).
Example 2: Find the focus and directrix of a parabola with equation 풚 = − ퟏퟏퟔ풙ퟐ.
Equation of a Horizontal Parabola
The standard form of the equation of a parabola with vertex (h,k) and an axis of symmetry x = h is:
If a>0 then the parabola opens ______________________.
If a<0 then the parabola opens ______________________.
Example 3: Find the equation of a parabola with the vertex at the origin and the directrix at x = 1.5?
Example 4: What are the vertex, focus and directrix of a parabola with the equation 풙 = ퟎ.ퟕퟓ풚ퟐ?
Example 5: The parabolic solar reflector has a depth of 3 feet at the center. How far from the vertex is the
focus?
Example 6: Find the vertex, focus and directrix of the parabola with the equation 풚 = 풙ퟐ + ퟖ풙 + ퟏퟖ?
Circles
Circle:_____________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
Center of a circle:___________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
Radius:____________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
Equation of a Circle
The standard form of the equation of a circle with a center (h,k) and radius r is:
Example 7: Write the equation of a circle with center (-4,3) and radius 4.
Example 8: Write the equation of the circle 풙ퟐ + 풚ퟐ = ퟗ that has been translated 4 units to the left and 3
units up.
Example 9: Find the center and the radius of a circle with the given equation
lll
Example 10: Graph the circle with center (2, -3) and radius 5
Use the information provided to write the standard form equation of each circle.
9) 10)
11) 12)
13) 14)
16)
18)
15)
17)
19) 20)
Ellipses
Ellipse:____________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
Foci of an Ellipse:___________________________________________________________________________
__________________________________________________________________________________________
Major Axis:________________________________________________________________________________
__________________________________________________________________________________________
Center of the ellipse:________________________________________________________________________
_________________________________________________________________________________________
Minor Axis:________________________________________________________________________________
__________________________________________________________________________________________
Vertices:___________________________________________________________________________________
__________________________________________________________________________________________
Co-vertices:________________________________________________________________________________
__________________________________________________________________________________________
The standard form of the equation of an ellipse with center (h,k) is:
(풙 − 풉)ퟐ
풂ퟐ + (풚 − 풌)ퟐ
풃ퟐ = ퟏ
Example 11: Write the equation for an ellipse in standard form centered at the origin with vertex (-6,0) and
co-vertex (0,3).
Example 12: What are the foci of the ellipse with the equation ? Graph the foci and the
ellipse.
Example 13:
Example 14:
Identify the center, vertices, co-vertices, foci, length of the major and minor axes.
Identify the center, vertices, co-vertices, foci, length of the major and minor axes and graph the ellipse.
Use the information provided to write the standard form equation of each ellipse.
Hyperbolas
The standard form of the equation of a parabola with vertex (h,k)
(풙 − 풉)ퟐ
풂ퟐ −(풚 − 풌)ퟐ
풃ퟐ = ퟏ
Focus of the hyperbola:______________________________________________________________________ __________________________________________________________________________________________
Vertex:______________________________________________________________________________________________________________________________________________________________________________
Transverse axis:_____________________________________________________________________________ __________________________________________________________________________________________
Axis of symmetry:___________________________________________________________________________ __________________________________________________________________________________________
Center of the hyperbola:_____________________________________________________________________ __________________________________________________________________________________________
Hyperbola:_________________________________________________________
__________________________________________________________________
__________________________________________________________________
Example 15: A hyperbola centered at (0,0) has vertices (+4,0) and one focus (5,0) write the standard form
equation of the hyperbola and graph it.
Example 16: What are the vertices, foci and asymptotes of the hyperbola with equation
Graph the hyperbola.
Identify the vertices, foci, and direction of opening of each.
Identify the vertices and foci of each. Then sketch the graph.
Use the information provided to write the standard form equation of each hyperbola.