 ###### Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

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# The Hyperbola - Concept

Carl Horowitz ###### Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

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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.