The four types of conic sections are parabolas, ellipse, circles and hyperbolas.

Conic Sections

A conic section is…..

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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: _________________________________________________________________________________

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__________________________________________________________________________________________

__________________________________________________________________________________________

Focus of a

parabola:__________________________________________________________________________________

__________________________________________________________________________________________

__________________________________________________________________________________________

Directrix:__________________________________________________________________________________

__________________________________________________________________________________________

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Focal

length:____________________________________________________________________________________

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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.