Module 7: Conics Lesson 1 Notes Parabolas Video 1 Transcript Hey everyone, this is a video for module 7 lesson 1, parabola notes, video 1. €
In the first part of the parabola notes we talked about there being 2 different standard equations for a parabola. This is because there are 2 different types of parabolas. In this video we are only going to be looking at parabolas that have a vertex on the origin. The 2 different types of standard equations that we have, are x 2 = 4 py and y 2 = 4 px . So you can see the difference here is in the first one the “x” term is squared and in the second one, the “y” term is squared. If your “x” term is squared your parabola will have a vertical axis of symmetry. So if I was graphing that, then if the “x” term is squared the axis of symmetry would run along the y=axis, that i€s a vertical axis of symmetry. We know that in this video, my vertex will always be at the origin. So what happens if this is my axis of symmetry we know its going to be symmetrical about that line. So your parabola would have to open up or down. If your “Y” term is squared the parabola has a horizontal axis of symmetry, so your axis of symmetry would be horizontal. Once again, your vertex is here, your axis of symmetry always passed through your vertex. So if it has a horizontal axis of symmetry the parabola would be open to the right or to the left. The directrix is a line associated with the parabola. If your parabola has a vertical axis of symmetry (in other words, x is the squared term) the directrix will be horizontal, and vice versa. So, if your parabola has a vertical axis of symmetry, this one does, I am going to erase this bottom, lets just say our example is going to open up. The directrix would run horizontal, so your directrix would run this way. Its going to be a line that runs horizontal and that’s going to equal to y=, you can see in your notes here if you have a vertical axis of symmetry the equation of the directrix is y= -­‐p. So what ever your value of “p” is, the directrix will have the opposite sign of that. Ok, lets talk about one that opens to the right, lets erase the one open to the left, and use that one for our example. Then your directrix would run vertically, since you have a horizontal axis of symmetry, your directrix would run vertically. So this would be x=, lets look in your notes here, its also going to be the opposite of the “p” value You have one more thing, the focus. The focus is a point associated with the parabola. The focus is inside the parabola. If your parabola has a vertical axis of symmetry, the coordinates of the focus are (0,p). So, right here we have a vertical axis of symmetry. The focus would be inside the parabola and would have the ordered pair of (0,p). So once again this is only saying when the vertex is at the origin we can say the focus is at (0,p). If you think about that, the focus here is the same distance away from the vertex as the directrix is. If your parabola opens to the right or to the left, once again your focus is going to be inside, so that is going to have an ordered pair as (p,0) So this is going to have the same sign as your “p” value, and the focus is going to be on the axis of symmetry, and you have the ordered pair of (p,0). €
So lets do an example, Find the focus and the directrix of the parabola x = 7y 2 so notice this has a y 2 term, so to put this in standard form, we are going to use the y 2 = 4 px formula, so I need to get this to look like this. So the first thing I need to do is solve for y squared. I am going to rewrite the problem down € here. To do that I divide by 7. So now I have my y squared term here by itself. So now I have to make €
it look like this. So notice that this “4p” I have to factor out essentially a 4 out of 1/7. So, write this as this. So notice this “x” is here, so “4p” in the problem would equal 1/7. So now I can solve for p, so if I divide by 4, I am going to get p= 1/7 divided by 4 over 1, which is 1/7 times ¼. Which would be 1/28. So if “4p” equals 1/7, then p=1/28. My equation would equal y 2 = 4 px so I would have a 4, my “p” is 1/28 and then I would have my “x.” €
So find the focus and directrix of the parabola, so if we have a y 2 we know we have an axis of symmetry is going to run this way (horizontal). We know our vertex is going to be at the origin, we know that the focus is going to be (p,0). I am going to switch back to that screen just to show you. So we know the coordinates of the focus is (p,0). So here is our “p” value. So our focus i€
s going to be (1/28, 0). And our directrix is going to be x= the opposite of our “p” value. So x=-­‐1/28. Now if you sketch this the directrix would be right here, the focus would be here. Since the focus has to be on the inside of the parabola you know that the parabola has to open to the right. So the focus is this (1/28,0) and the directrix is x=-­‐1/28. Ok, lets do another one. Find the focus and directrix of the parabola 8y = 5x 2 . So notice I have an “x” squared term here. So my standard form of this one would be x 2 = 4 py . So the first thing I want to do is get “x” squared by itself. I would divide €
€
by 5. So now I have to find out what “p” is. So I know that 4p=8/5. Solve for “p” there you are going to get 8/5’s times ¼, which is going to be 8/20, which would reduce to 2/5s. So our “p” value is 2/5s. So our standard form is going to be ⎛ 2 ⎞
x 2 = 4⎜ ⎟ y , notice our “p” value is 2/5. I have an “x” squared, lets just sketch the ⎝ 5 ⎠
graph to find out which way it opens. €
“x” squared would mean that my axis of symmetry would run vertical. I know here we have the vertex at the origin. My focus is going to be whatever “p” is so think about my axis of symmetry, my focus has to fall on this line, so my focus is going to be (0,2/5), which that is going to be located about right here. And our directrix would be y= the opposite of “p” which would be y=-­‐2/5. So it would go this way. So knowing those 2 pieces of information, my parabola would open up because my focus has to be on the inside. So my focus is going to be (0, p) which is (0,2/5). If you look back in your notes you can see here that if you have a vertical axis of symmetry, then your focus is (0,p), which is what we have and the directrix would be the opposite of “p.” Ok, the next problem, write the standard form of the equation of the parabola with the vertex at the origin and the given focus (5,0). Give the equation for the directrix as well. Ok so if our focus is (5,0) and I know my vertex is here. (5,0) would be here, so if my focus is here, I know my parabola has to open that way (right) because I know my focus has to be on the inside. If this happens I know I am talking about the standard form of y 2 = 4 px you can refer back to your notes or at the beginning of this lesson. But you know that this right here happened because this axis of symmetry is horizontal therefore it has to be here. Ok, if the vertex is the origin then I know my “p” value here has to be 5. So my standard form would be 4 times 5 times x would €
equal your “y” squared. So the standard form is going to be 4 times your “p” value, which is just going to be your focus value times “x.” Your directrix is going to be the opposite of the “p” value. Its going to be x=-­‐5.