topic badge

12.04 Parabolas


Definition of a parabola

A parabola is one of the special shapes you can make when you slice a cone on an angle and is known as one of the conic sections.


The most basic parabolas are of the form $y=x^2$y=x2 and $x=y^2$x=y2. We have looked at parabolas of the form $y=x^2$y=x2 previously and the square root function which is half of the parabola of the form $x=y^2$x=y2

Definition of a parabola

A parabola is an arch-shaped curve such that any point on the curve is equal distance from a fixed point (called the focus) and a fixed line (called the directrix).

So all the points that are both equidistant from the focus and directrix can be joined together to form a parabola. We call all these points the locus of points.

Focus and directrix

Let's take a look at the graph of the parabola below. You can see I've included the location of the focus and directrix.

Now, if I've placed the focus and directrix in the right location, distance A and distance B should be equal (because remember that the distance from any point on the curve to both the focus and directrix will be the same).

It's easy to see that distance $A$A = $5$5 units

To calculate distance $B$B we will use Pythagorean Theorem

distance $B$B $=$= $\sqrt{4^2+3^2}$42+32
  $=$= $\sqrt{25}$25
  $=$= $5$5

And so we can see that both distances are the same!

Practice question

question 1

Consider the parabola $x^2=16y$x2=16y.

  1. What are the coordinates of its vertex?

  2. What is the equation of its axis of symmetry?

  3. Solve for $a$a, the focal length of the parabola.

  4. What are the coordinates of its focus?

  5. What is the equation of its directrix?


Graphing parabolas

As with any type of graph, it is helpful to have a standard form. We have four possible orientations for a parabola, opening up, opening down, opening right and opening left. Each of these can be identified from the standard form.

Parabola general form

For an real number $a$a, with $a>0$a>0,

Opening up: $\left(x-h\right)^2=4a(y-k)$(xh)2=4a(yk)
Opening down: $\left(x-h\right)^2=-4a(y-k)$(xh)2=4a(yk)
Opening right: $\left(y-h\right)^2=4a(x-h)$(yh)2=4a(xh)
Opening left: $\left(y-h\right)^2=-4a(x-h)$(yh)2=4a(x