The Hyperbola - Concept
A hyperbola is a type of conic section that is formed by intersecting a cone with a plane, resulting in two parabolic shaped pieces that open either up and down or right and left. Similar to a parabola, the hyperbola pieces have vertices and are asymptotic. The hyperbola is the least common of the conic sections.
We're going to look at the equation for a hyperbola and a hyperbola equation looks very very similar to that of an ellipse instead of having a plus sign now we're dealing with a minus sign in between our two variables. Okay so what I want to do is just sort of look at a couple key things that we know how to figure out and then we'll talk more about the graph.
Okay so to find the x intercept we let y equals 0, when we let y equals 0 this term disappears and we're left with x squared over 16 is equal to 1 so what we need is x squared over 16 they'll be the same value so x has to be plus or minus 4 okay. To find our y intercept we let x equal 0 so that makes this term disappear and we're left with negative y squared over 4 equals 1. Trick here is that we have a negative y squared, negative y squared is always going to be positive so what we end up with is a negative positive number equal to 1. That can't happen so we actually do not have any y intercepts okay?
The last thing is the domain is what that's what I want to look at and to find the domain what I'll think about is we know that this term y squared over 4 has to be positive okay? It can't be, it can be 0 and it can be up so what that tells us is we know that x squared over 16 has to be greater than or equal to 1 because if we are subtracting a positive number from this term giving us 1 that means this has to be greater than 1 already okay? Solving this out what that tells us x squared is greater than or equal to 1 oops sorry x squared is equal to 16 take the square root of both sides what we end up with is x squared is greater than or equal to 4 or x is less than or equal to negative 4. The numeric value has to be bigger than 4 either positive or negative, so graphically what that tells us is we have a point here and a point here and that everything on this graph is going to be out there's not going to be anything in the middle okay? So actually we'll talk a little more about what this graph looks like but it's going to end up looking something like two sideways parabolas facing away from each other okay? And we'll talk more about how we actually know this okay.
Let's go look at some general equations for hyperbolas. What I have up here is two different hyperbolas one vertical one horizontal, the vertical one is on the bottom because it's going up and down and a horizontal one going side to side okay? There's a lot of things going on here but what I want to do is first and foremost just draw parallels to what we know about ellipses okay? So really whatever the variable is that's first it's going to dictate the way that the parabola is facing. This is the variable that's going to have intercepts so therefore we're going to be going whatever direction that dictates x first up and down sorry x first side to side, y first up and down okay? So if we had an equation for an ellipse, what we would end up doing is looking at the term underneath the x and that would become part of the x radius as I call it compared the x and the y so see which is the major or minor axis but basically they're going to become the radii for the particular section.
We do the exact same thing for hyperbolas okay? So you look at the term underneath the x and that is going to go out x units, look at the term underneath the y you're going to go up y units, so what you can do is you can create sort of 4 key points 4 where if it was an ellipse that ellipse would go through except instead of connecting them as a circle what you really want to do is create a box so where those two points would meet in each quadrant draw the point and then draw this little dotted box. There's different technical terms for it, some books will call it the fundamental rectangle I just call it the silly box because what the silly box really does is gives us the tools to find our asymptotes okay? So if you connect the corners of your box the opposite corners, those are going to end up being the asymptotes for this curve so you have one from one corner to the other and then you do the opposite way as well okay? So what else do we know about this? We know that our intercept is going to be that same value the square root of whatever is underneath the x squared or y squared the leading term and then just connect the points going towards your asyymptote making sure to get near your asymptote but not cross it okay? So I just talked about the horizontal one, the vertical one is exactly the same except instead of dealing with the x squared first we're dealing with the y squared first and that's going to tell us we're going up and down.
Okay so other little things that we know, some language the transverse is the term you'll hear from time to time and basically what that is referring to is the distance between the two points that we have plotted on our curve okay so sort of the equivalent of an axis in an ellipse except it varies depending on which way it is facing so your transverse for this side to side hyperbola is going to be this, where your transverse for your up and down is going to be this it doesn't really depend on which is bigger or smaller like it does in ellipse.
The other axis is going to be called the conjugate okay? so the conjugate for a side to side hyperbola is going to be from top to bottom and the kinds you get for a up and down hyperbola is going to be from side to side okay? Couple other pieces of information, the ends of the transverse the actual points on your graph are called your vertices the opposing points the ends of the conjugate are called your co-vertices okay? A lot of information hope you're following along and the last little piece of information is the focus, foci okay and those are going to lie inside the curves so depending on which ways it faces it depends on which axis it's on and the relationship between your terms is in this case a squared plus b squared is equals to c squared okay? Easy way to remember this all is that the equation for hyperbola is has a negative sign between but the relationship between your axes and your foci is addition so you have one plus one minus. Ellipse is just the opposite to find the equation you're adding inside the equation and then you're subtracting to find your relationship between your foci okay so it's really a lot of parallels between the equation for a hyperbola and the equation for an ellipse so hopefully if you get comfortable with one of them you can then take that knowledge and just with a little bit of tweak apply it to the other.