# Parabola

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## Definition

A parabola is the collection of points that are equidistant from a line, called the directrix, and a point not on the line, called the focus.

The point on both the parabola and the line segment between the focus and directrix is called the vertex.

When the vertex is at (0, 0), and the focus is at some point (a, 0), where a is some positive number, the equation for the parabola is given by: ${\displaystyle y^{2}=4ax}$ and the equation for the directrix is x = -a.

## Variations

For any parabola we can determine whether it opens up or down by looking at which variable is being squared. If y is being squared the parabola opens up or down. It opens up if the coefficient of x is positive, and down if the coefficient of x is negative. If x is being squared the polynomial opens left or right. If the coefficient of y is positive the parabola opens right, and if the coefficient of y is negative the parabola opens left.

We can also move the vertex, to a point (h, k). To find the equation for this situation, we take the equation for when the vertex is at (0, 0) and replace x, y, with (x -h), (y - k), respectively.

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