How do we define a parabola?

 

A curve that is symmetrical with regards to a line of symmetry and its focal point lies on that line is a parabola. To make the definition very clear, we need to understand each technical term in it.

Focus – This is a fixed point on a line of symmetry where any reflection from the parabola curve converges.

Directrix – This is a straight line on the opposite side of the parabolic curve, a certain distance from the vertex.

How does an equation of a parabola look like?

Parabolas have unique equations that describe the curve. The general equation of a parabola takes the form

    \[y^{2}=4 a x\]

where the vertex of the parabola is at the center. If the vertex is at an arbitrary point (h, k), then the equation becomes:

    \[(y-k)^{2}=4 a(x-h)\]

How do I derive the equation of a parabola with vertex at the origin?

The derivation in this tutorial will assume a parabola with its vertex at the origin and opens upwards. With it, any parabolic equation is easy to deduce.

Look at the diagram below:

From the definition of parabola, p is equal to the distance -p with reference to the focal point.

Therefore:

    \[F A=A B\]

Using Pythagoras Theorem:

    \[F A=\sqrt{(x-0)^{2}+(y-p)^{2}}\]

       and      

    \[A B=\sqrt{(x-x)^{2}+(y-(-p))^{2}}\]

Equating the two:

    \[\sqrt{(x-0)^{2}+(y-p)^{2}}=\sqrt{(x-x)^{2}+(y-(-p))^{2}}\]

    \[(x-0)^{2}+(y-p)^{2}=(x-x)^{2}+(y-(-p))^{2}\]

    \[x^{2}+(y-p)^{2}=0^{2}+(y+p)^{2}\]

    \[x^{2}-2 p y=2 p y\]

    \[x^{2}=4 p y\]

    \[y=\frac{1}{4 p} x^{2}\]

That is the general equation of a parabola whose vertex is at the (0, 0) point. It can take four different forms depending on whether the parabola opens to the top, bottom, right, or left as, shown.

       

   

Equation of a parabola with vertex at (h, k)

By translation, we can modify the equation of a parabola with vertex at the origin to place it anywhere on the graph. In general, the equation takes the form

    \[(y-k)^{2}=4 a(x-h)\]

where h and k are the coordinates of the vertex.

For example: If the equation of a parabola is

    \[(y-k)^{2}=4 a(x-h)\]

and (h, k) = (2, -5), then the graph curve undergoes displacement as shown:

Remarks

From the graphs and different forms of parabolic equations, you can easily tell whether the parabola opens upward, downwards, left, or right. If you have a parabolic equation, then you can find the focus, directrix, and vertex. In our next tutorial, we will discuss how you can get the coordinates of all the critical points on a parabolic curve. Check it out!

About the Author

This lesson was prepared by Robert O. He holds a Bachelor of Engineering (B.Eng.) degree in Electrical and electronics engineering. He is a career teacher and headed the department of languages and assumed various leadership roles. He writes for Full Potential Learning Academy.