Find the Foci and Asymptotes Step by Step

all right so in this example we're going to find the foci as well as the asymptotes and just to have a little fun I'll probably find most of everything a lot of times when I get going I just like to keep on solving and maybe even do a graph I don't know we'll see how much time we have so let's go and take a look at this problem um or at least to be able to understand what exactly we're dealing with because this is a hyperbola not some weird Parabola all right so the main thing that I want you to understand is when you're trying to identify the foci as well as the asymptotes is you need to know the standard form and the orientation of the hyperbola that you are going to be dealing with okay so here is the standard form of a hyperbola when you have an X as what we call your leading term now it's always a squared minus B squared all right but whenever your a squared is under the X right because sometimes we swap the X and the Y's in the numerator that depends on the orientation so this is what's really important when your a is under the X what that means is you have a horizontal transverse axis I do not know what this hyperbola looks like yet but what I do know is I have a horizontal transverse axis so therefore my hyperbola is going to look something like that right now again remember we're going to have a center we're going to have Foci what we're looking for I'm sorry those are vertices here's your Foci right over here and then you're going to have some nice asymptotes going on here all right so we need to understand if we have a horizontal transverse axis from our Center which is h k our Foci are going to be going left and right so let me just go and let's just label everything so therefore we're going to figure out everything anyways foreign okay so um we have our Center HK right that's labeled right there our vertices are going to be h plus or minus a comma K right because again we're going horizontal right so we say well what is our a well that's the next step we're going to want to do anyways right here we have this right so we can identify the center and that's kind of easy Center is remember it's x minus H right x minus H so x minus 7 that means my Center here is 7 comma five right don't do negative seven negative five No it's h and k is right there 7 and 5 is right there that's the formula x minus h x minus K so 7 5 is going to be your Center now my a is just going to be if a squared is equal to 16 then my a is equal to a 4 my B is equal to a 5. and they say well what about your c c is part of my Foci how do you find your C well the nice thing about hyperbolas is everybody kind of remembers this a squared plus b squared equals c squared like a the Pythagorean theorem and like yeah it basically is like so I always just remember that way hyperbola Pythagorean theorem when you look at how it kind of works you can see that relationship but for right now I just want to be like all right so to find c squared that is going to be c squared is equal to a16 plus 825 so therefore that's going to be a 30 that's going to be a 41. so c squared equals 41. so C equals the square root of 41. now what is the square root of 41 if we were to graph this you know it's approximately 6 point something because 6 squared is 36 and 7 squared is 49. so it's going to be somewhere in protein there but I'll just leave this as the square root square root of 41. okay so we have my center right and if I want to be able to find let's just find the vertices because that's easy so vertices is going to be 7 plus 4 and minus 4 right so 7 plus or minus 4 comma 5. and again we can just kind of simplify this for our teacher right that'd be 11 comma 5 and that would be a 3 comma 5. then if we want to be able to find the foci we're going to do the same thing now this one we're not going to simplify for that teacher because I have a square root of 41. I can't really add or subtract the square root of 41 you know without being approximating with the 7. so I'm just going to leave that as a 7 plus or minus 6 square root of 1 come on comma 5. all right and then last but not least ladies and gentlemen is going to be our asymptotes so the cool thing is once you know the equation of the asymptotes right now again it is different if it's a horizontal transverse axis or a vertical transverse axis the only difference is the A and B are swapped but for a horizontal transverse axis this is our formula so again we already found a we already found B we already found H we already found K it's literally just a plug and a chug so we go ahead and take my B five don't do 25 so it's a 5 over my a which is down to 16. it's a 4 comma and x minus my H so that's x minus 7. and then here's the weird one though it's plus k right so a lot of times students will get this confused a lot of times we write actually the formula as y minus K so if you're like oh it's a negative five but again that's not a part of the formula if K is actually 5 so it's going to be a plus A5 all right now again I like to just kind of graph this so I'm sorry I'm going to take a little bit extra take here so I want two three four five six seven one two one two three four five I don't know why I'm going down there we go has a horizontal transverse axis and then 5 6 7 8 9 10 11. and then what was that three so one two three there you go so you have something like that something like that nobody really knows graphing this you could simplify this to put this in a slope intercept form but either way guys you're going to have two low lines that's going to look like this and that is how you graph the parabola but the main thing is there is your Foci and there is your asymptotes hopefully this video was helpful for you and if it was you're gonna love the next video however here