Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Parabola, all the points equal distance from a fixed point focus and a line directrix. So the distance from the vertex to the focus, we're going to call some distance P and we're going to have our parabola with any given point XY where the distance from the point XY to the focus is going to be equal to the distance of the point XY to the directrix. When we talk about a distance to a line, we always are talking about the shortest distance or the perpendicular. So when we look at this graph, if we think about letting our vertex be at the origin 00, we've gone to the right P amount. So our focus is at the point P0. We've gone to the left negative P amount, same distance. So this line here would be X equal negative P. So every single point along the vertical line would have negative P for its X value. So then if we choose some generic point XY on our parabola, draw back a horizontal line, the point where it intersects the directrix has got to be negative PY. So our distance from our point P to the directrix X - -P ^2 + y - y ^2 has got to equal the distance from the point to the focus X -, P ^2 + y - 0 ^2. I'm going to square each side, foil out the X + P ^2, so X ^2 + 2 PX plus P ^2. Foil out the right side X ^2 -, 2 PX plus P ^2 + y ^2. So the X squareds and the P squareds on both sides will cancel. If we take all the X's to one side and the YS to the other, we get 4 PX equal y ^2. Once again, P is the distance from the vertex to the focus. So the generic formulas y -, K quantity squared equal four P * X -, H The vertex is HK. The horizontal axis of symmetry, the focus H + P, K. The directrix X equal H -, P If P is greater than 0, we're going to open to the right. If P is less than 0, we're opening to the left. The same formula, but we switched what was squared, so X -, H ^2 equal 4 PY minus K, the vertex being HK. Now the axis of symmetry is vertical. The focus is H, K + P The directrix Y equal K minus PP is greater than 0, it opens up. If P is less than 0, it opens down a new term. The lattice ****** and graphing of parabolas. The lattice ****** of a parabola is a line segment that passes through its focus, is parallel to its directrix, and has its end point on the parabola. So below you can see the length of the lattice ******. For the graphs of y ^2 equal 4 PX and X ^2 equal 4 PY. The distance has got to be 4, the absolute value of four P If we look at some examples, let's look at y ^2 equaling -12 X. So the first thing is that the four P is going to equal -12 or P = -3 because it's AY squared. We know that we're going to be opening off to the side because it was a -12, we're going to be opening to the left. So our vertex is at 00. Our focus, we're moving right and left, so we're going to move in around the curve -3. So our focus is going to be -3, zero. Our directrix. Directrix are always a line. It's going to be X equal, the opposite of -3 or the negative of -3. So our directrix is +3. If we look at this next example, we have 4P equaling -8. So our P equal -2. Our vertex this time is at -2 positive 1, so we're going to go -2 positive one. We're going to be opening down because the squared term this time is our X ^2. So now we're going to know that our focus has got to be two N or two lower. The focus is always inside the curve. So if we're going to go down to, we'd have a focus of negative two 1 -, 2 is -1 our directrix is going to be to the other direction. So our directrix in this case is going to be Y equaling 1 - -2 or Y equal 3. If we have one that's not already completed, we're going to complete the square. So we're going to keep our Y terms on one side in this example and everything else to the other. What do we have to do to complete the square? We need to take half of the -2 and square it. And if I add 1 to one side, we're going to add 1 to the other. So the left side turns into y -, 1 quantity squared. And the right side is 8X. So our 4P is 8. Our P is 2. So our vertex is going to be 0 because there was no number added or subtracted with the X, 1. So if we're at 01, the Y is the one that's squared. So we're opening side to side and it's a positive 8. So we're going to be opening to the right. We know that our P is 2, so our focus is going to be two and to the right. So our focus is going to be two, one. Our directrix is going to be two in the opposite direction or X equaling -2 what if we're given pieces and parts and we're asked to come up with the rest? So if we know our focus is 0 to 20 and we know that our directrix is -20 we know that halfway between the focus and the directrix is our vertices. So in this case, if I have a +20 and a -20, that tells me my vertices has got to be at 00. If we think about this pictorially, if we're at zero 20, we're way up here somewhere, and if we're at Y equal -20 our directrix is down here. So we can see that this has got to be a parabola. We've got to always curve around our focus. So it's going to be going up and it's going to be an X ^2. So we're going to have y -, 0 in this case equaling X -, 0 ^2. Now, what we haven't figured out is what this 4P is, but the four P is always the distance between the focus and the center is the distance for P actually from the focus to the center. So P is 20. So 4 P is going to be 80. So we're going to get 80 Y equaling X ^2. If we look at this next one, we're given the vertices of five -2 and we're given the focus of seven -2. So our distance between the vertices and the focus is 2. We know that our parabola has got to go through the vertex and wrap around the focus. So this one, we're going to be having our Y being our squared. So we're going to have y -, a negative 2 ^2 equaling. We've got to figure out that four PX -5. The P here is going to be two because the P is always the distance between the vertice and the foci, the focus. So if P equal to, then 4P has got to be 8. So we're going to have an 8 right there. Looking at a couple more examples, if we're given the focus and the directrix here, we're given the focus as 7 positive seven -1 and we're given the directrix as Y equal -9. So we need to figure out the midpoint between here. Well, if we're in between, we know that the X value is going to be 7 and halfway between -1 and -9 is going to be -5. So we're going to have the parabola going through the vertex and curving around the focus. So we're going to have Y minus a -5 equaling X -, 7 because we're going up and down or because we're going up, we know that the X is going to be the squared term. So what we need to do now is we need to figure out the four P what goes in front of that y + 5. Well, the distance here is a distance of four between the -1 and the -5. So if P equal 4, four P = 16, so 16 times the quantity y + 5 equal the quantity X -, 7 ^2. Thank you and have a wonderful day.