Graph the hyperbola. Locate the foci and find the equations of th...

Question

Graph the hyperbola. Locate the foci and find the equations of the asymptotes. 4y^2 - x^2 = 16

Accepted Answer

Hello, today we're gonna be using the given equation to identify the graph of the hyperbola. So what we are given is nine, Y squared minus four, X squared is equal to 36. Now, before we continue, we need to go ahead and set the equation equal to one. And in order to do that, we're going to divide the entirety of the equation by 36 doing this is going to give us Y squared over four minus X squared over nine is going to equal to one. And this is the standard form of the equation of the hyperbola that's given to us as Y squared over A squared minus X squared over B squared is equal to one. So this is a hyperbola that's located at the origin. And since the Y square term is given to us first, this is a hyperbola that opens up and down along the Y axis and whose is located on the Y axis as well. So before we continue, let's go ahead and identify the SM total box of the hyperbola. And we can do that by identifying our A and B values. Currently, we have A squared is equal to four and B squared is equal to nine. If we take the square root of both sides of the A equation, we get A is equal to plus or minus two. And if we take the square root of both sides of the B equation, we get B is equal to plus or minus three. So what this means is we have this hyperbola on an axis. So let's go ahead and draw the axis first and again, the transverse axis is the Y axis. So a represents the length of the box along the transverse axis. And since it's given to us as plus or minus two, this is going to be a box that's above and below the origin two units for a total height of two. So let's go ahead and just quickly graph that. And for the B variable, we are told that B is equal to plus or minus three. Now, B represents the length of the box along the X axis and that's going to be three units to the right and left. So this as in total box is going to have a total length of six. So currently our ass in total box is a box whose length is six and whose height is four. Now, what we want to go ahead and do is identify the equations for the diagonal asy totes. The equations of the diagonal asy totes is given to us as Y is equal to plus or minus A over B times X. And we know what the values of A and B are A is equal to two and B is equal to three. So if we plug this in to the equation of the ascent totes, we get two equations for both of our diagnose asymptotes Y is going to equal to 2/3 X and Y is going to equal to negative 2/3 X. So that's the equations for the ascent totes. Now let's go ahead and identify the vertices of the hyperbola. The vertices are going to be located on the transverse axis which is the Y axis. And we know that the length of the total box on the transverse axis is two units above and below. So we're going to be adding two and subtracting two to the Y value of the origin. So for the vertices, we are given zero comma zero minus two and zero comma zero plus two. And this is going to give us the locations of the vertices which is going to be zero, comma negative two and zero comma two. So just to quickly put down our vertex, we're going to have this here and here at plus or zero com plus or minus two. Now we need to identify the location of the. Now, since the Y axis is the main axis, the equation of the size is given to us as C is equal to the square root of A squared plus B squared. A squared is equal to four and B squared is equal to nine. So we plug this in and that's going to give us the square root of nine plus four and that is going to equal to the square root of 13. Now, the square root of 13 cannot be simplified. So we're going to leave it as plus or minus the square root of 13. And we're going to add this to the Y value of the origin since the gonna be located on the Y axis. So that's going to give us the values or the coordinates zero, comma zero plus the square root of 13 and zero, comma zero minus the square root of 13. And simplifying this is going to give us the locations of the, which are going to be at zero comma square root of 13 and zero comma negative square root of 13, which is gonna be roughly located here on the axis. So now that we have all the information, we can go ahead and draw the hyperbole with respect to the Asy totes. And that's gonna roughly look like this. And now we want to go ahead and select the graph that accurately represents all this information. And that answer is going to be a now just to verify a says that the Asy totes are Y is equal to 2/3 X and Y is equal to negative 2/3 X which we've identified here. The foci are located at zero comma square root 13 and zero comma negative square root 13, which we've identified down below. And the vertices are located at zero comma two and zero comma negative two, which we've identified here. So all the information is accurately represented on this graph. Therefore, the answer to this problem is going to be a. So I hope this video helps you in understanding how to identify the graph of a hyperbola. And I'll go ahead and see you all in the next video.

Graph the hyperbola. Locate the foci and find the equations of the asymptotes. 4y^2 - x^2 = 16

Key Concepts

Hyperbola Definition

Foci of a Hyperbola

Asymptotes of a Hyperbola

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