In Exercises 1–4, find the vertices and locate the foci of each hyperbola with the given equation. Then match each equation to one of the graphs that are shown and labeled (a)–(d). x^2/4−y^2/1=1

Question

In Exercises 1–4, find the vertices and locate the foci of each hyperbola with the given equation. Then match each equation to one of the graphs that are shown and labeled (a)–(d).  x^2/4−y^2/1=1

Accepted Answer

Welcome back. I am so glad you're here. And we are asked for the given equation to graph the hyperbola and indicate the vertices and the folk. So our given equation is X squared divided by 16 minus Y squared, divided by nine equals one. And our answer choices are four graphs. We will examine the differences between those graphs in a little bit. But for now, let's move this to a different part of the screen where we have a little bit more room to work. And what we want to do first is determine which equation for a hyperbola we are dealing with. So we've got our X squared first in the first fraction here. So the equation here and because this is centered about the origin is going to be X squared divided by A squared minus Y squared, divided by B squared equals one. And we can see from this that A squared is equal to 16 because 16 is underneath the X squared and B squared is equal to nine because nine is underneath the Y squared. And from these two equations we can solve for A and B, we take the square of both sides A is going to be equal to the squared of 16, which is plus or minus four and B is equal to the square of nine, which is plus or minus three. Well, why are these important? Well, because we were asked to find two things before graphing the vertices and the folky, the vertices are a units from the origin. And in this case, they are going to be four units from the origin. But along which axis, well, in this equation, the X squared comes first. So they're going to be along the X axis. So we would write these vertices as the X values. So they're going to be negative 40 and 40 are our vertices. And how about the folk? The folk are C units from the origin but we haven't solved for C yet. We found A and B. Well, we recall from previous lessons that for hyperbola C squared is equal to A squared plus B squared, we know a squared that's 16, we know B square, that's nine, 16 plus nine is 25. If C squared equals 25 then C is plus or minus five. So the folk are five units from the origin. But along which axis again, the transverse axis here is along the X axis because that comes first. So the five and the negative five are the X values. We have folk at negative 50 and at 50. And now we can graph this. So we draw a rough sketch of a coordinate plane. We have a vertical y axis, a horizontal X axis. They come together at the origin in the middle and to graph our vertices first from the origin, we have the point negative 40. So we will go down four units. No, excuse me, negative 40 is not down four units. That's to the left from the origin four units. And here is negative 40. And from the origin 40 is to the right four units label that 40 and then our are at negative 50. So from the origin, we're going to the left five units negative 50. And from the origin to the right five units at 50. And now hyperbola, those are going to look like they are, it's almost like a semi circle but heading away from the origin. And so we're coming from negative infinity for our X values. And then we are going to come intersect with our vertex at negative 40, turn around and head back toward negative infinity for the X values. And then on the right side of the origin coming from positive infinity for the X values turn hit our vertex at 40, then turn around and head back to positive infinity for the X values. And there is our graph. Now we just need to find the matching one. So we'll go up to a different part of our screen where we can see the answer choices and we examine our answer choices, answer choices A and B have the transverse axis along the X axis. As we do answer traces C and D have the transverse axis along the y axis, which is not what we have. So we can eliminate answer choices C and D. Now, what's the difference between A and B? Well, they both have vertices at negative 40 and 40. That's great. But the folk, I are at different places for answer choice A, they're at negative 50 and 50. And for answer choice B they are at negative 70 and 70. Ours are at negative 50 and 50. So answer choice A works well done. We'll catch you on the next one.

In Exercises 1–4, find the vertices and locate the foci of each hyperbola with the given equation. Then match each equation to one of the graphs that are shown and labeled (a)–(d). x^2/4−y^2/1=1

Verified Solution

Key Concepts

Hyperbola Definition

Vertices and Foci

Graphing Hyperbolas