Parabolas

A Parabola is the set of all points that are equidistant from a fixed point called the focus and a fixed line called the directrix. The focus and the directrix are key elements of the parabola and determine its shape and position

Test the definition yourself, by moving the point on the parabola, notice how the distances to the focus point and to the directrix are always the same.

There are two types of parabolas, horizontals and verticals.

The formula of a horizontal parabola centered at the origin $(0,0)$ is

where $p>0$ will determine the shape of the parabola:

• Directrix at $x = -p$.
• Focus at $(p,0)$.

For example, the following parabola has formula:

And what happens if we change the center?

If we move the parabola horizontally or vertically, we need to aply the corresponding transformations to the variables $x$ and $y$.

• Horizontal shift of $h$ units: Change the value of $x$ for $x-h$.

• Vertical shift of $k$ units: Change the value of $y$ for $y-k$.

Together with the variables $x$ and $y$, the other elements affected by the translation are the focus point and the directrix, these are now $(h+p,k)$, $x=h-p$ for horizontal parabolas and $(h,k+p)$, $y=k-p$ for vertical parabolas.

Perfect! Now let's perform this transformations on the formula of the parabola.

The formula of a horizontal parabola centered in $(h,k)$ is:

let's see what this looks like. The fomula:

gives us the following parabola:

To sum up:

we've learnt how to plot a general horizontal parabola centered at $(h,k)$, with directrix $x = h-p$ and focus point $(h+p,k)$.