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). y^2/4−x^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).  y^2/4−x^2/1=1

Accepted Answer

Welcome back. I am so glad you're here. We are told for the given equation to graph the hyperbola and indicate the vertices. And the folk, our given equation is Y squared divided by minus X squared, divided by nine equals one. And then we have four graphs for our answer choices. We will go over the differences between these graphs in more detail in a little bit. But right now, let's work on this equation a little bit, move this to a different port portion of our screen where we'll have a little more room. OK. First, we need to figure out which formula or which equation of A P we have because there are two different ones. Well, this one is already set equal to one. That's great. We've got A Y squared and an X squared on top and the Y squared comes 1st 1st. So this will be the equation Y squared divided by A squared minus X squared divided by B squared equals one. So we can see that our A squared is going to be 16 because that's beneath the Y squared. And our B squared is going to be nine because that's beneath our X squared, we can solve for A and B taking the square rut of both sides of both equations. So for A squared equals 16 A is plus or minus four and for B squared equals nine B is plus or minus three. Now, how does this help us? Well, we had two things that we needed to find. We needed to find the vertices and we needed to find the folky and the vertices are a units from the center. And look, we've solved for A A is plus or minus four. And now the question is, are those X values or Y values? Well, in our equation, the Y squared is first. So that means that our vertices are going to be Y values. So we will write these as zero negative four and zero positive four. So there are our vertices. Now our folk are going to be C units from the center, but we haven't solved for C yet. Well, we recall from previous lessons that for a hyperbola, we know that C squared is equal to A squared plus B squared. We know A squared and B squared A squared is 16, B squared is nine. So 16 plus nine, that's 25. If C squared is equal to 25 C is going to be plus or minus five. So if the folk are C units from the center, that means there are five units in either direction and which axis are they along again that Y squared comes first. So these are going to be Y values. So our folk are going to be at zero negative five and at zero positive five. OK. Now let's graph this, we'll draw a rough sketch of a coordinate plane. We'll have a horizontal X axis, a vertical Y axis, they come together at the origin in the middle. Now let's label our vertices from the origin. We will go down four units. We can label that zero negative four. And from the origin, we will go up four units, we can label that 04. For our folk eye. From the origin for zero negative five, we can go down five units. It's going to be zero, negative five. And from the origin, we can go up five units that will be zero positive five. And A hyperbola is going to extend outward. There'll be like semi circles going toward infinity, but we only see the semi circle part of it. So from positive infinity for our Y values, it comes down and goes through our vertex 04 and then curves back up on the other side, heading up back to positive infinity for the Y values. And then below the origin from negative infinity, we're coming up toward the origin, we're hitting our vertex at zero, negative four turning around and then heading back to down toward negative infinity for the Y values and the X values but more slowly. So there is our graph. Now we just need to look for something similar, bring this to a different part of the screen. Now, we can see all of our answer choices. Now noting the differences between our answer choices, answer choices A and B have the transverse axis along the X axis and all of the values for negative four, negative five, those are all in the X spots. So those are incorrect for answer traces. C indeed, those are correctly with the transverse axis of the Y axis. So now which one is in the correct spot? Well, they both have vertices of 04 and zero, negative four. However, the folky are different and we need folk of 050, negative five. And that matches with answer choice, C answer choice D has 07 and zero negative seven. So answer choice C is the correct answer here. 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). y^2/4−x^2/1=1

Verified Solution

Key Concepts

Hyperbola Definition

Vertices and Foci

Graphing Hyperbolas