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 Depdendent Variable

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 Dependent Variable

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# The parabola

While we have studied the parabola earlier, we have not looked at it as a set of points lying in a plane. Using the two stacked cones, the parabola is another conic section, involving one of the cones.

The parabola is the set of points that from a point P to a fixed point F, equals the distance from P to a fixed line l.

One other condition is also true, and that is PD = PF

Â· The point F is the focus, and the line l is a fixed line called the directrix.

Â· The vertex of the parabola lays Â½ between the directrix and the focus.

o The directrix

Â· The parabola will never cross the directrix, and hence always opens away from this line

Example 1

A parabola has a focus at F(0,3) and a directrix Y = -3. Find the equation of the parabola.

Solution:

The middle from the focus and the directrix is at the origin. We know that any point on the parabola will be of the form P( x, y ). Any point on the directrix will be of the form D( x, 3). Since we know PD = PF, we can use this information to create the equation.

In general, if the focus is F(0, p) and the directrix is y = -p then the equation of the parabola is

 The standard form of the equation of a parabola with vertex V(h, k) and a horizontal directrix is The vertex is V(0, 0). If p > 0, the parabola opens up. If p < 0, the parabola opens down. The focus is F(0, p). The equation of the directrix is y = - p. The equation of the axis of symmetry is the y-axis. The standard form of the equation of a parabola with vertex V(h, k) and a vertical directrix is The vertex is V(0, 0). If p > 0, the parabola opens right. If p > 0, the parabola opens left. The focus is F(p, 0). The equation of the directrix is x = - p. The equation of the axis of symmetry is the x-axis.

For parabolas in non-standard position:

 The standard form of the equation of a parabola with vertex V(h, k) and a horizontal directrix is The vertex is V(h, k). If p > 0, the parabola opens up. If p < 0, the parabola opens down. The focus is F(h, k + p). The equation of the directrix is y = k - p. The equation of the axis of symmetry is x = h. The standard form of the equation of a parabola with vertex V(h, k) and a vertical directrix is The vertex is V(h, k). If p > 0, the parabola opens right. If p > 0, the parabola opens left. The focus is F(h + p, k). The equation of the directrix is x = h - p. The equation of the axis of symmetry is y = k.