A parabola is formed by the intersection of

A parabola is formed by the intersection of a plane with a cone when the cone intersects parallel to the slant height of the cone.

On a cartesian plane, the set of points that describe a parabola is defined using a point called the FOCUS and a line called the DIRECTRIX. The distance from the focus to the vertex or from the directrix to the vertex is ‘a’. This value plays a role in The distance of a given point defining the equation of on the parabola from the focus the parabola. is equal to the distance of that FOCUS same point to the directrix. When that point is the vertex (the tip of the parabola) that distance has a special significance. It defines an important parameter for the parabola known as ‘a’. DIRECTRIX

Parabolas are the shapes that define projectile motion (the path that a ball takes when it is hit or thrown into the air). Ball

Parabolas show up in the architecture of bridges.

The parabolic shape is used when constructing mirrors for huge telescopes, satellite dishes and highly sensitive listening devices.

focus If this parabola is a satellite dish or a telescopic mirror, the receiver is located at the focus. Parallel rays strike the parabola and all reflect to the focus If it is the reflective surface of a lamp, then the light-bulb is located at the focus of the mirror and the arrows point in the other direction.

The equation for a parabola with a vertex at the origin can have one of two formats depending on whether it opens vertically or horizontally. y 2 = 4 ax ‘a’ is positive ‘a’ is negative x 2 = 4 ay ‘a’ is positive

To construct graphs of a parabola, we first must decide whether the parabola opens horizontally or vertically and that depends on the format of the equation: y 2 = 4 ax OR x 2 = 4 ay. Equation: Value of “a”: Direction of Opening: Vertex: Focus: Axis of Symmetry: Directrix: y 2 = 12 x 4 a = 12 a = 3 Horizontal: Right x 2 = 20 y 4 a = 20 a = 5 Vertical: Up V (0, 0) F (3, 0) V (0, 0) F (0, 5) s: y = 0 s: x = 0 D: x = -3 D: y = -5

Other Points: If x = 3, If y = 5, Then y 2 = 12(3) Then x 2 = 20(5) y 2 = 36 D: x=-3 x 2 = 100 y = 6 y = -6 x = 10 x = -10 (3, 6) (3, -6) (10, 5) (-10, 5) F (0, 5) s: y = 0 V (0, 0) F (3, 0) V (0, 0) D: y=-5 s: x=0

y 2 = -10 x 4 a = -10 a = -2. 5 Opens left V(0, 0) F(-2. 5, 0) s: y = 0 D: x = 2. 5 Other Points If x = -2. 5 then y 2 = -10(-2. 5) y 2 = 25 y = -5 (-2. 5, 5) (-2. 5, -5)