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Notes 8.1 Conics Sections The Parabola

I. Introduction

- A.) A conic section is the intersection of a

plane and a cone. - B.) By changing the angle and the location of

intersection, a parabola, ellipse, hyperbola,

circle, point, line, or a pair of intersecting

lines is produced.

- C.) Standard Conics
- 1.) Parabola
- 2.) Ellipse
- 3.) Hyperbola

- D.) Degenerate Conics
- 1.) Circle
- 2.) Point
- 3.) Line
- 4.) Intersecting Lines

- E.) Forming a Parabola
- When a plane intersects a double-napped cone and

is parallel to the side of the cone, a parabola

is formed.

F.) General Form Equation for All Conics

If both B and C 0, or A and B 0, the conic is

a parabola

II. The Parabola

A.) In general - A parabola is the graph of a

quadratic equation, or any equation in the form

of

B.) Def. - A PARABOLA is the set of all points in

a plane equidistant from a particular line (the

DIRECTRIX) and a particular point (the FOCUS) in

the plane.

Axis of Symmetry

Focus

Focal Width

Vertex

Focal Length

Directrix

C.) Parabolas (Vertex (0,0))

Standard Form

Focus

Directrix

Axis of Symmetry

Focal Length

Focal Width

D.) Ex. 1- Find the focus, directrix, and focal

width of the parabola y 2x2.

Focus

Directrix

Focal Width

E.) Ex. 2- Do the same for the parabola

Focus

Directrix

Focal Width

F.) Ex. 3- Find the equation of a parabola with a

directrix of x -3 and a focus of (3, 0).

G.) Parabolas (Vertex (h, k))

St. Fm.

Focus

Directrix

Ax. of Sym.

Fo. Lgth.

Fo. Wth.

H.) Ex. 4- Find the standard form equation for

the parabola with a vertex of (4, 7) and a focus

of (4, 3).

I.) Ex. 5- Find the vertex, focus, and directrix

of the parabola 0 x2 2x 3y 5.

focus

Directrix

vertex

III. Paraboloids of Revolution

A.) A PARABOLOID is a 3-dimensional solids

created by revolving a parabola about an axis.

Examples of these include headlights,

flashlights, microphones, and satellites.

- B.) Ex. 6 A searchlight is in the shape of a

paraboloid of revolution. If the light is 2 feet

across and 1 ½ feet deep, where should the bulb

be placed to maximize the amount of light

emitted?

The bulb should be placed 2 from the vertex of

the paraboloid