Math ProjectParabola

Degenerate Case

•Limiting case in which a class of objects changes its nature so as to belong to another simpler class

•ex. A point is a degenerate circle with radius one

•ex. A line is a degenerate form of parabola

Application

•Parabolas have a property that if they are made of a material that reflects light, the light enters the parabola traveling parallel to it axis of symmetry and is reflected to the focus.

•Ex. Head light reflectors, telescopes, bridges, search lights, and satellite dishes.

Standard form

•Common equation

•ex. Y=Ax²+Bx+C

Geometry and Algebra II Form

•Standard- Y=Ax²+Bx+C•Vertex- Y=a(x-h)²+k•Intercept- Y=a(x-h)(x-k)

Key Words

•Square function•Parabola•Intercept•Vertex•Axis of symmetry•Minimum/Maximum•Focus•Directrix•Focal distance

Relationships

•Vertex form- y=a(x-h)²-k•a›0 opens up/ a‹o opens down•h=horizontal shift•k=vertical shift•a=vertical stretch or compression

Rotated form

•y²=4ax•(y-k)²=4a(x-h)

Conic form

•4a(y-k)=(x-h)²•x²=4ay

Eccentricity

•E= c/a•A=distance from center to vertex•C=distance from center to focus•Eccentricity=1 for a parabola

Labeled Graph

Construction & Origin•The parabola is defined as the locus of a

point which moves so that it is always the same distance from a fixed point (called the focus) and a given line (called the directrix).

• [The word locus means the set of points satisfying a given condition. See some background in Distance from a Point to a Line.]

•In the following graph, •The focus of the parabola is at (0,p)  .

Construction & Origin

•The directrix is the line y=−p  . •The focal distance is |p|  (Distance from

the origin to the focus, and from the origin to the directrix. We take absolute value because distance is positive.)

•The point (x, y) represents any point on the curve.

•The distance d from any point (x, y) to the focus (0,p)  is the same as the distance from (x, y) to the directrix.

Construction & OriginConic section: ParabolaAll of the graphs in this chapter are examples of conic sections. This means we can obtain each shape by slicing a cone at different angles.How can we obtain a parabola from slicing a cone?We start with a double cone (2 right circular cones placed apex to apex):                       If we slice a cone parallel to the slant edge of the cone, the resulting shape is a parabola, as shown.