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Current time:0:00Total duration:5:15

Vertices & direction of a hyperbola

Video transcript

which of the following graphs can represent the hyperbola y squared over 9 minus x squared over 4 is equal to 1 and we have our 4 choices here choices a and C open up to the top and the bottom or up and down choices B and D you can see D here open to the left and the right and you can see within the ones that open up to the left to the right or up and or the up/down ones they have different vertices so I encourage you to pause the video and see if you can figure out which of the following graphs represent the equation of a hyperbola or the graph of this equation right over here alright so there's a bunch of ways to think about it one thing you might see what's the center of this hyperbola and since in our equation we just have a simple Y squared or a simple x squared we know that the center is going to be at 0 0 if the center was anywhere else if the center if the center was at the point H comma K then this equation would be Y squared minus the y coordinate of the center sorry not Y squared it would be Y minus K so Y minus the y coordinate of the center squared over 9 minus X minus the x coordinate of the center squared over 4 is equal to 1 and in this case well this is just the case where K and H or H and K are both equal to 0 so we just get you could view this as Y minus 0 squared and X minus 0 squared so the center in this case is going to be at 0 0 and you see that for all of them now the next question you might ask is well is this going to be opening up and down or is going to be opening left and right and the key thing to realize is is you need to look at whichever term and when just written in standard form like this when you have Y minus K squared over something squared minus X minus H squared over something squared is equal to 1 or it could be the other way around the X square the X term might be positive and then the Y term would be negative if we're dealing with a hyperbola so the key is to just look at whichever term is positive that will tell you which direction the hyperbola opens in since the Y term here is the one that is positive it tells us that this hyperbola is going to open up up and down now you could just memorize that but I that's never too satisfying I always want to know why does that why does that work and the key thing to realize is if the y term is positive then you could set the other term equal to zero and the way that you would set the other term equal to zero in this case is by making your x equal to the x-coordinate of your Center and that's zero so if X is if you X is equal to the x-coordinate of your Center and this term becomes zero you can actually solve this equation you can solve y squared over 9 equals one so if X is equal to zero so X is equal to the x-coordinate of its center then this term goes away and you would get Y squared over 9 is equal to 1 or Y squared is equal to 9 or Y is equal to plus or minus 3 so you know that the coordinates the x-coordinate of the center plus or minus 3 that there on the hyperbola that there and so you know that's going to open upwards and downwards so you go to the center the x-coordinate the center plus 3 and minus 3 are on the hyperbola notice over here plus 3 and minus 3 are not on this hyperbola in fact if plus 3 and minus 3 were on this hyperbola you wouldn't be able to open up to the right and the left and so that's why whichever term is positive that is the direction the that you open up or down with or if it's so if the X term was positive we would be opening to the left and the right for the exact same reason and you can see if if we did the other way around if we had the Y equaling the y coordinate of the center so this term if the y term was zeroed out you would end up with negative x squared over 4 is equal to 1 which is the same thing as x squared over 4 is equal to negative 1 which is equal to x squared is equal to negative 4 I just multiplied both sides by negative 1 there and then I multiplied both sides by 4 and this has no solution and so that's why we know that we're not going to intercept the we're not going to intercept the line we're never going to have a situation where y is equal to the y-coordinate of the center so that's so y is never y is never going to be equal to zero in this case in cases b and d y there are points where y equals zero so the thing to realize is whichever term is positive that and whatever variable that is so if it's for the y variable that's the direction that we're going to open up in and when i figured out what the actual vertices are we saw that the point 0 plus or minus 3 are on the graph so a looks like a really good candidate if we look at the other choice that opens up and down it doesn't have 0 plus or minus 3 on the graph it has 0 plus or minus 2 so we can feel pretty good that about choice a