Parabola

A parabola is the set of all points T (x, y) in a plane that are equidistant from the directrix (v) and from the point F (focal point).

The geometric definition of a parabola can be written as:

The distance between the focal point of the parabola (F) and the directrix is called the parameter of the parabola p.

The characteristics of a parable are most easily identified with the help of a sketch:

the marked characteristics on the sketch are:

A parabola is a second-order curve, so the general equation for second-order curves also applies to it:

where and the equation is transformed into:

In fact, as we will find out later that it is a quadratic equation. This can also be seen by placing in the general equation of the second-order curve; we get nothing but quadratic equation:

A parabola with a vertex whose coordinate is the origin is a parabola with vertices at the point A (0,0).

Let's look at a sketch of this parabola:

As we said in the introduction, a parabola is a set of points that satisfies the equation:

From the graph we can understand by comparing that applies:

for we use the simple formula for distance between points:

From what has been said, we can derive the equation of the parabola.

We derive the equation of parabola:

If the first parabola is mirrored over the y axis, we get the second form of the parabola. Let's look at a sketch of this parabola:

When mirroring across the y axis, x turns into its negative value: . We write the equation of the mirrored parabola:

We remove the brackets and get the equation of the parabola:

If we rotate the first parabola by an angle , we get a new form of the parabola. In this case, the parabola is also a graph of a quadratic function.

We derive the equation of the parabola in this position with the help of the introductory definition of the parabola, which we already know from the introduction:

From the graph we can understand that applies:

for we use the simple formula for distance between points:

From what has been said, we can derive the equation of the parabola.

We derive the parabola equation:

If we rotate the first parabola by an angle , we get a new form of the parabola. Even in this case, the parabola is a graph of a quadratic function.

Note that in this case the parabola is the same as if the third form of the parabola were mirrored across the y axis or if . We write the equation of the mirrored parabola:

We remove the bracket and get the equation of the parabola:

In case we move the parabola, every for and every for :

we get a parabola in a position away from the origin.

The characteristics are also marked on the sketch are: