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Identify Characteristics of Parabolas


When we think of the word parabola, we usually associate it with the function we're very familiar with, the quadratic function. 

Quadratic function recap

Quadratic functions can be written in each of the following forms:

  • $y=ax^2+bx+c$y=ax2+bx+c This is known as the general form of y intercept form
  • $y=a\left(x+d\right)\left(x+e\right)$y=a(x+d)(x+e) This is known as the factored of x intercept form
  • $y=a\left(x+k\right)^2+v$y=a(x+k)2+v This is known as the turning point form

Each of these names gives us a clue to various features we can identify on a parabola.

The intercepts

All parabolas of quadratic functions have a $y$y intercept, however the number of $x$x intercepts can vary, ranging from none to two.

The turning point

As the name suggest, every parabola has a point at which the graph turns and changes direction.

The nature of the turning point of a quadratic function can either be described as:

  • a minimum turning point, indicating that the turning point is at the lowest point on the graph
  • a maximum turning point, indicating that the turning point is at the highest point on the graph.

Line of symmetry

The turning point not only gives us information on the highest or lowest point on the graph, but it also informs us of where the line of symmetry exists for the parabola.

If a parabola has a turning point at $h,k$h,k then the line of symmetry is given by $x=h$x=h.

The graph of relation $y^2=x$y2=x is also a parabola

The name parabola is not only used for the graph of a quadratic function, but also for the relation $y^2=x$y2=x. You will look at this relation in more detail soon.

So what features make a parabola?

If the graph of a quadratic function and the relation $y^2=x$y2=x are both called parabolas, are there more?

And if there are, then they must all have certain characteristics in common to each be given the same name.

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.

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.

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 = $5$5 units

To calculate distance B we will use Pythagoras' Theorem

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

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

Worked Example

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?

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