#### Transcript of Circles and Parabolas

**Parabolas**

**Alice W., Meen, Nick, Woody**

**Conics**

Conic (Conic section) is the intersection of a plane and a double-napped cone.

Parabola = the set of all points (x,y) in a plane that are equidistant from a fixed line, the directrix, and a fixed point, the focus, not on the line.

Standard Form Equation

General Form Equation

Vertical axis: y = ax^2 + bx + c

Horizontal axis: x = ay^2 + by + c

Parabolas in real life

Interesting Properties

Picture 1: Parabola receiver- when a person whisper at the focus of one of the parabola reflector, it could be hear on the other parabola reflector.

Degenerate form

The degenerate form of parabola is a straight line!

why?

because in order to make a parabola, we have to have a cutting plane angle equal to generator angle.

Thus, we will get a straight line if we cut through vertex.

Interesting Sites

http://www.pleacher.com/mp/mlessons/calculus/appparab.html

http://www.mathwarehouse.com/geometry/parabola/standard-and-vertex-form.php

http://www.purplemath.com/modules/parabola.htm

Directrix - The line that is perpendicular to the axis of symmetry.

- directrix = (y-p)

Degenerate conic - The resulting figure of when the plane does pass through the vertex.

Vertex = the midpoint between the focus and the directrix.

Parabola

Axis of the parabola = the line passing through the focus and the vertex.

Vertical axis; directrix (x-h)^2 = 4p(y-k)

Horizontal axis; directrix (y-k)^2 = 4p(x-h)

Focus = the fixed point inside of a parabola

Picture 2: parabola reflector has the property that when the light source is placed at the focus point, the light ray will reflect from the mirror as ray parallel to the axis.

- (h,k) is the vertex

- The focus lies on the axis p units (directed distance) from the vertex

- h determines the horizontal shift

- k determines the vertical shift

- "a" represents if the parabola open upward or downward

- x = -b/2a

axis of symmetry

Reflective property of a parabola

- The tangent line to a parabola at a point P makes equal angles with the following two lines

1. The line passing through P and the focus

2. The axis of parabola

Parabolas can be found in many places in our daily lives, we just need to notice them.

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