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Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

In this lesson, we will review what a parabola is, then we will look at the formal definition of a parabola, introducing the focus and directrix of a parabola. We will look at some examples to help solidify our understanding of these concepts.

Background on Parabolas

Meet Melvin, the friendly spider!

Isn't he cute? It just so happens that Melvin is no ordinary spider - he's a mathematical spider! Okay, you're probably rolling your eyes about now, but bear with me! We're going to use Melvin to introduce a mathematical curve called a parabola.

Take another look at Melvin with some lines and curves drawn in.

Notice the red curve connecting his knees. In mathematics, this red curve is called a parabola, and it represents the graph of a quadratic equation y = ax2 + bx + c.

Parabolas have different characteristics. They look like a U or an upside down U, and the bottom or top point of that U is called the vertex of the parabola. The vertical line through the vertex of a parabola is called the axis of symmetry, and the parabola is symmetric with respect to this line.

There are two more characteristics of a parabola that we are interested in for this lesson, and those are the focus and the directrix of the parabola. We can use these characteristics to formally define a parabola.

Focus and Directrix of a Parabola

Let's get back to Melvin! As we said, Melvin is a mathematical spider, and another neat thing about him is that for each of his legs, the length of the leg from his head to his knee is equal to the length of the leg from the knee to the foot.

Because of this, Melvin's head and the line where all his feet hit the ground are two very important characteristics of the parabola connecting his knees. To introduce these characteristics, let's look at the formal definition of a parabola. The formal definition of a parabola is the set of all points that are the same distance from a single point of the parabola, called the focus, and a line of the parabola, called the directrix.

Now, let's let Melvin illustrate this. We see that the parabola connecting his knees is such that the distance from Melvin's head to the parabola is the same as the distance from the parabola to the ground. Do you see how this relates to the formal definition of a parabola? If you're thinking that Melvin's head and the ground have some kind of relationship to the focus and directrix of the parabola, you are thinking correctly!

Since each point on the parabola is the same distance from Melvin's head as it is to the ground, it must be the case that Melvin's head is the focus of the parabola, and the ground is the directrix of the parabola. Pretty neat, huh?

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Let's take a look at an example. The parabola shown has its focus and directrix labeled as well as two points on the parabola.

By the definition of a parabola, the distance between the focus and a point on the parabola should be the same as the distance between a point on the parabola and the directrix. Let's verify that this is the case!

First, let's look at the point P1 = (4, 2). All we need to do is verify that the distance from the focus F = (0, 2) to the point P1 = (4, 2) is the same as the distance from the point P1 = (4, 2) to the directrix y = -2.

We see that FP1 = 4 and that the distance from P1 to the directrix is 4. Great! Just as the definition implies, the distances are the same.

Let's do the same thing with the point P2 = (-8, 8), just to be sure. Again, we find the distance from the focus F = (0, 2) to the point P2 = (-8, 8), and we find the distance from the point P2 = (-8, 8) to the directrix y = -2, and verify that these are the same.

Awesome! Once again, we see that the distances are the same! We have that FP2 = 10, and the distance from P2 to the directrix is 10.

Lesson Summary

A parabola is a curve representing a quadratic equation y = ax2 + bx + c. It has the shape of a U or an upside down U, and the lowest or highest point of the parabola is called its vertex. The vertical line through the vertex is called the axis of symmetry of the parabola, and the parabola is symmetric with respect to this line.

The formal definition of a parabola is the set of all points that are the same distance from a single point of the parabola, called the focus, and a line of a parabola, called the directrix. Wow! It's really neat to look at parabolas in this way, and it's great that we had Melvin, the mathematical spider, to help us along!

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