[G.GPE.2] Expressing Geometric Properties With Equations #2
Objective
Common Core Text:
[G.GPE.2] Derive the equation of a parabola given a focus and directrix.
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Explanation
Introduction
Derivation of Basic
Translating an equation, translating a parabola
Deriving an equation for a specific parabola (given vertex, then given focus and directrix)
Application: "Bionic Ear"
Parabolas are probably the 2nd most important curve in mathematics (the most important being a straight line). Many relationships form parabolas when you graph them (acceleration vs. time, energy vs. velocity), so it will be nice to learn more about them. It's also the trajectory of a thrown object. Parabolas can be used to make satelite dishes and "bionic ear" spy guns. Wikipedia gives lots of other examples here.
First, let's look at the definition of a parabola
A parabola is a set of all points that are equidistant from a given point (called the focus) and a given line (called the directrix)
This video shows that the points on a parabola are the same distance from the directrix and the focus. Notice that the distance from the directrix must be perpendicular.
Here we have
focus C, located on the y-axis p units above the x-axis
a directrix, located p units below the x-axis
a random point q on a parabola, with variable coordinates (x, y)
Now, we know one thing about parabolas, and that's that the points on the parabola must be equidistant from the directrix and the focus.
First, we'll find the distance from point q to the directrix. I'll zoom in close and cut out the rest.
Just like circles, we can make an equation for a parabola. Let's see how we can do this.
Here we can see that the distance from the directrix to point q is made of 2 parts.
Top part: Starting at q, we go down to the x-axis. That distance is y, since the ordered pair for point q is (x, y).
Bottom part: Now from the x-axis, we go down till we reach the directrix. Since the directrix is at y = -p, this distance will be p.
Therefore, the distance from point q to the directrix is y + p
Now for the distance from point q to the focus.
This one is a little harder, because the distance from the focus to point q is diagonal. Whenever you want to find the length of a diagonal line on a coordinate plane, that means using the Pythagorean Theorem.
Vertical Leg: this goes from the height of point q (which is y) to the height of the focus (which is p)
Therefore the length of the vertical leg of the triangle is p - y
Horizontal Leg: this is the horizontal distance from the y-axis to point q, which is a length of x
Hypotenuse: this is the distance we care about. By using the Pythagorean Theorem with the two legs, we get
a2 + b2 = c2
(p - y)2 + x2 = CQ2
Therefore, the distance from point q to the focus is
Now we have our two distances
The distance from point q to the directrix
y + p
And the distance from point q to the focus
y + p =
Square both sides to get rid of the square root
(y + p)2 = (p - y)2 + x2
Expand
y2 + 2py + p2 = p2 - 2py + y2 + x2
Subtract y2 and p2 from both sides
2py = -2py + x2
Add 2py to both sides
4py = x2
Divide both sides by 4p
Remember, the only thing we know about parabolas is that these two distances are equal, because that's the definition of a parabola. Well, if the distances are equal, we can set them equal to each other.
y = x2
And this is our basic equation for a parabola.
This equation is nice, but as you can see, the only variable we can use to change the shape of our parabola is p. Remember, p is the distance from the x-axis to the focus above and the directrix below.
The following gadget shows what happens as you change the value of p. Click and drag the focus (point C) to change the value of p. Try going under the x-axis, too.
Here's a video of me doing it
As you can see, it only works if we want the vertex (point V, the tip) of our parabola to be on the origin. But some parabolas look like this.
And there's no way we can get that with y = x2
How can we write the equation of a parabola with the vertex at (h, k)? Well, all I'm doing by moving the parabola is a translation. Remember those? Now recall, how do you change an equation to translate it?
To translate a shape, subtract the distance you want to translate from the corresponding variable.
Let's do it. I want to translate the x-coordinate of my vertex from 0 (on the origin) to h (any x value we want). As you can see in the picture, this means the vertex moves a distance of h along the x-axis. To change the equation so this translation will occur, simply replace x with (x - h). We also want to translate the y-coordinate of the vertex by k. Therefore, replace y with (y - k).
y - k = (x - h)2
This is the equation for any parabola (well, any parabola that opens up or down).
Try using this applet to see how the equation changes. Click and drag the vertex, then the focus.
Here's a video of me doing it.
If the parabola opens sideways, simply switch the (y - k) and the (x - h)
x - h = (y - k)2
Try using this applet to see how the equation changes. Click and drag the vertex, then the focus.
Just make sure the y and k are always together, and the x and h are always together.
Unfortunately, there are a lot of ways to write these equations, all of which are the same. The thing to know is that a =
For up and down parabolas
y - k = (x - h)2
4p(y - k) = (x - h)2
y = (x - h)2 - k
y - k = a(x - h)2
y = a(x - h)2 - k
For sideways parabolas
x - h = (y - k)2
4p(x - h) = (y - k)2
x = (y - k)2 - h
x - h = a(y - k)2
x = a(y - k)2 - h
Deriving equations for specific parabolas
Here are some questions you might be asked
Example 1
Write the equation for a parabola with vertex at (2, -3) and focus at (2, -2)
First thing we need to figure out right away is which direction will the parabola open. Up, down, left, or right? To do this, just figure out which side of the vertex the focus is on. If the focus is on the left side of the vertex the parabola will open left, if the focus is above the vertex the parabola will open up, etc. Just remember, the parabola must always open around the focus.
If we imagine plotting the vertex and the focus the question gave us, we'd see that the focus is above the vertex. Why do we care? Because now we know to use the equation for parabolas that open up and down
y - k = (x - h)2
Now that we know which equation to use, we just need to figure out our three values: p, h, and k
p is the distance between the focus (or directrix) and the vertex.
If we plot our two points (or imagine), we can see that the focus is 1 space above the vertex. Because this means the parabola will open upward, p will be positive 1.
(up or right are positive, down or left is negative is negative).
h is the x-coordinate of the vertex
We are given that the vertex is at (2, -3), so h will be 2
k is the y-coordinate of the vertex
We are given that the vertex is at (2, -3), so k will be -3
Plugging these values in
y - (-3) = (x - 2)2
Now we can simplify a bit
y + 3 = (x - 2)2
And we have our answer. You can even use this applet to check
Example 2
Write the equation for a parabola with focus at (-2, 1) and directrix at x = 2.
This time they didn't tell us the vertex, so we need to find it. Not too hard. We remember our most basic rule of parabolas: all points on the parabola are equidistant from the focus and the directrix. The vertex is no exception. Moreover, it's the easiest point of all to figure out. Since the directrix goes up and down, we need to figure out how far to the side the focus is. We'll the x-value of the focus is -2, and the x-value of the directrix is 2, so they are 4 spaces apart. Now, what's half way between -2 and 2? 0 is. If you can't imagine this, you can always take the average of the two numbers
(-2 + 2)/2 = 0
So now we know that the x-coordinate of the vertex will be 0. The y coordinate must be the same as the focus, the the y-coordinate of the vertex will be 1. Therefore, our vertex will be (0, 1)
If we imagine plotting the vertex and the focus, we'd see that the focus is to the left of the vertex. Why do we care? Because now we know to use the equation for parabolas that open sideways.
x - h = (y - k)2
Now that we know which equation to use, we just need to figure out our three values: p, h, and k
p is the distance between the focus (or directrix) and the vertex.
If we plot our two points (or imagine), we can see that the focus is 2 spaces to the left of the vertex. Because this means the parabola will open left, p will be -2.
(up or right are positive, down or left is negative is negative).
h is the x-coordinate of the vertex
We found that the vertex is at (0, 1), so h will be 0
k is the y-coordinate of the vertex
We found that the vertex is at (0, 1), so k will be 1
Plugging these values in
x - 0 = (y - 1)2
Now we can simplify a bit
x = (y - 1)2
And we have our answer. You can even use this applet to check
Application: "Bionic Ear"
What is this girl doing? Is that a 3D parabola (aka paraboloid) on the end of her device? And a microphone at the focus? It is.
As if parabolas weren't already useful enough, they have another interesting property. Anything that hits the side of the parabola will bounce off towards the focus.
Look at the picture below to see how all the little red arrows get bounced to the focus of the parabola. Don't worry about all the other stuff.
(Technically, the lines need to be coming at the parabola perpendicular to the directrix, like they all are in this picture, But if the lines are coming from far enough away, they'll all be almost perpendicular, and no math is perfect in the real world anyways).
Now, what is this girl bouncing into the microphone? Sound waves. In fact, she's listening to people talk from really far away. Normally people can't hear conversations from far away, because not enough sound waves reach their ears. But this "Bionic Ear" gathers a big dish of sound waves together right into the microphone, allowing people to hear things that would normally be too far away.
Want to buy one?
http://www.amazon.com/Bionic-Ear-And-Booster-Set/dp/B0012N6GZ2
Or you can make your own.
Example
You want to design your own "Bionic Ear". You've assembled the microphone and the headset together, and put a mini telescope like in the picture. Now you just need to build the parabola. For this you'll use a 3D printer.
But you'll need to give the printer the equation of the parabola. No problem.
There's no coordinate system given, so you get to make your own. This is great, because that means you can put the vertex right on the origin, so that your vertex will be (0, 0). It also, means you get to decide which way the the parabola opens. The machine needs the equation of an upward facing parabola, so you decide that your focus will be above the parabola.
All that's left is to decide where your focus will be. And since your focus will be the mic, you just have to decide how far you want the mic to stick out. Well, you saw this picture here and decided you want to make one like it.
You estimate that this mic sticks out about 2 inches from the vertex, so you're going to do the same. This means that p will be 2.
Now we're ready to plug everything in.
y - k = (x - h)2
Substituting
y - 0 = (x - 0)2
Simplifying
And we've got it. Just type it into the 3D printer, print your paraboloid, and prepare to spy on whoever you want.
What else could we reflect?
This drawing shows how Archimedes (200 BC) supposedly used a parabola of mirrors to light ships on fire when they attacked Syracuse. According to Wikipedia:
A test of the Archimedes heat ray was carried out in 1973 by the Greek scientist Ioannis Sakkas. The experiment took place at the Skaramagas naval base outside Athens. On this occasion 70 mirrors were used, each with a copper coating and a size of around five by three feet (1.5 by 1 m). The mirrors were pointed at a plywood mock-up of a Roman warship at a distance of around 160 feet (50 m). When the mirrors were focused accurately, the ship burst into flames within a few seconds.
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Practice your skills