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# Focus & directrix of a parabola from equation

CCSS.Math:

## Video transcript

this right here is an equation for a parabola and the goal of this video is to find an alternate order to explore an alternate method for finding the focus and directrix of this parabola from the equation so the first thing I like to do is solve explicitly for y I don't know my brain just processes things better that way so let's get this 20 3 or 4 to the right hand side so let's add 23 over 4 to both sides and then we'll get Y is equal to negative 1/3 times X minus 1 squared plus 23 over 4 now let's remind ourselves what we've learned about folk folk I and and directrix is I think is how to say it so the focus if the focus of a parabola is at the point a comma B and the directrix the directrix directrix is the line y equals K we've shown in other videos with a little bit of hairy algebra that the equation of the parabola in the form like this is going to be Y is equal to 1 over 2 times B minus K and this B minus K is then the difference between this y-coordinate and this Y value I guess you could say times X minus 1 squared plus B plus K I'm sorry not X minus 1 I'm going to confuse with this X minus a squared X minus a squared plus B plus K over over 2 the focus is a comma B then the direction is y equals K and this is going to be the equation of the parabola well we've already seen a technique where look we can see the different parts we can see that ok this X minus 1 squared actually let me do this in a different color this X minus 1 squared corresponds to the X minus a squared and so one corresponds to a so just like that we know that a is going to be equal to 1 and actually let me just write that down a is equal to 1 in this example right over here and then you could see you could see that the negative 1/3 over here corresponds to the 1 over 2 B minus K and you would see that the 23 over 4 corresponds to the B plus K over 2 now the the first technique that we explored we said okay let's set negative 1/3 to this thing right over here solve for B minus K we're not solving for B or K we're solving for the expression B minus K so you get B minus K equals something and then you could use 23 over four and this to solve for B plus K so you get B plus K equals something and then you have two equations two unknowns you can solve for B and K what I want to do in this video is explore a different method that really uses our knowledge of the vertex of a parabola to be able to figure out where the focus and the directrix is going to be so let's think about the vertex of this parabola right over here remember the vertex if the parabola is upward opening like this the vertex is this is this minimum point if it is downward opening it's going to be this maximum point and so when you look over here you see that you have a negative 1/3 in front of the X minus 1 squared so this quantity over here is either going to be 0 or negative so it's always going it's not going to add to 23:04 it's either going to add nothing or take away from it so this thing is going to hit a maximum point when this thing is 0 when this thing is 0 and other and it's just going to go down from there and when this thing is 0 Y is going to be equal to 23 over 4 so our vertex is going to be that maximum point our vertex is going to be well when does this equal 0 well when x equals 1 when x equals 1 you get 1 minus 1 squared so 0 squared times negative 1/3 this is 0 so when X is equal to 1 we're at our maximum Y value of 23 over 4 which is 5 and 3/4 it's actually let me let me write that as a knock I'll leave it just like that's our vertex 23 or 4 and it is a downward-opening it is a downward-opening parabola so actually let me start to draw this so let me get some axes here so we have to go all the way up to 5 and 3/4 so let's make this our that's our Y this is our y-axis this is the x-axis let's the x-axis and see we're going to see we go to one let's call that one let's call that two and then I want to get let's see if I go to 5 and 3/4 let's go up to see 1 2 3 4 5 6 7 we can able them 1 2 3 4 5 6 and 7 and so our vertex our vertex is right over here 1 comma 23 over 4 so it's 5 and 3/4 so it's going to be right around right around there and as we said since we have a negative value in front of this X minus 1 squared term I guess we could call it this is going to be a downward-opening parabola this is going to be a maximum point so our actual parabola is going to look it's going to look something it's going to look something like this it's going to look something something like this and we could we could obviously I'm hand drawing it so it's not not going to be exactly perfect but hopefully you get you get the general idea of what the parabola is going to what the parabola is going to look like I'm actually let me let me just do part of it I actually don't know that much information about the parabola just yet I'm just going to draw it like that so we don't know just yet where the where the directrix and focus is but we do know a few things the focus is going to sit on the same I guess you could say the same x-value as the vertex so if we draw this is x equals 1 if x equals 1 we know from our experience with focuses foci Kitt's and that they want to sit on the same axis as as the vertex so the focus might be right over here and then the directrix is going to be an equal distant on the other side equidistant on the other side so the directrix might be something like this might be right over here and once again I haven't figured it out yet but what we know is that because this point the vertex sits on the parabola by definition has to be equidistant from the focus and from the focus and the directrix so this distance has to be the same as this distance right over here and what's another way of thinking about this entire distance remember this coordinate right over here is a comma B and this is the line y is equal to K this is y equals K so what's what's this distance in yellow what's this difference in Y going to be well you could call that in this case the directrix is above the focus so you could say that this would be K minus B or you could say it's the absolute value of B minus K this would actually always work it'll always give you kind of the positive distance so if we knew what what the absolute value of B minus K is if we knew this distance then we just put it in half the directrix is going to be that distance half the distance above and then the focus is going to be half the distance below so let's see if we can figure this out and we can figure this out because we see in this I guess you could say this equation you can see where they where B minus K is involved 1 over 2 times B minus K needs to be equal to negative 1/3 so let's solve for B minus K so we get we get 1 over 2 times B minus K is going to be equal to negative 1/3 once again this corresponds to that it's going to be equal to negative 1/3 we could take the reciprocal of both sides we get 2 times B minus K is equal to is equal to 3 is equal to 3 now we can divide both sides we can divide both sides by - and so we're going to get we're going to get B B minus K is equal to is equal to was that three-halves three-halves B minus K is equal to let me make sure I happen to be a negative 3 so this has to be negative three-halves and so if you took the absolute value of B minus K you're going to get you're going to get positive three-halves or if you took K minus B you're going to get positive three-halves so just like that using this part look just just actually matching the negative one-third to this part of this equation we're able to solve for the absolute value of B minus K which is going to be the distance between the y axis in the y direction between the focus and the directrix so this distance right over here is three-halves so what is half that distance so the reason why i care about half that distance is because then i can calculate where the focus is because is going to be half that distance below the vertex and I can say well that and whatever that distance is it's going to be that distance also above the directrix so half that distance so 1/2 times three-halves is equal to three-fourths so just like that we're able to figure out the directrix it's going to be 3/4 above this so I could say the directrix so let me see I'm running out of space the directrix is going to be y is equal to the y coordinate of the focus so 23 I'm sorry the y coordinate of the vertex would be careful my little language it's going to be equal to the y-coordinate of the vertex plus plus 3/4 plus 3/4 so plus 3/4 which is equal to 26 over 4 which is equal to what is that that's equal to 6 and 1/2 so this right over here actually I got pretty close when true it is actually going to be the directrix Y is equal to 6 and 1/2 and the focus well we know the x coordinate of the focus a is going to be equal to 1 and B is going to be 3/4 less than the y coordinate of the directrix so 23 over 4 - 3/4 it's going to be twenty three over four twenty three over four - three fourths which is 20 over 4 which is just equal to it is just equal to five and we are done that's the focus one comma five directrix is y is equal to six and a half