Which Graph Represents The Hyperbola $\frac{v^2}{4} - \frac{x^2}{16} = 1$?

$$

Introduction

Hyperbolas are a type of conic section that can be used to model various real-world phenomena, such as the trajectory of a projectile or the shape of a satellite dish. In mathematics, hyperbolas are defined as the set of all points that satisfy a specific equation. In this article, we will explore the equation of a hyperbola and determine which graph represents it.

Understanding Hyperbolas

A hyperbola is a type of curve that has two branches, each of which is a parabola. The equation of a hyperbola is typically written in the form $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$, where $a$ and $b$ are constants that determine the shape and size of the hyperbola. The graph of a hyperbola is symmetric about the x-axis and the y-axis.

The Equation of the Hyperbola

The equation of the hyperbola we are interested in is $\frac{v^2}{4} - \frac{x^2}{16} = 1$. To determine which graph represents this equation, we need to identify the values of $a$ and $b$. In this case, $a^2 = 4$ and $b^2 = 16$. Therefore, $a = 2$ and $b = 4$.

Graphing the Hyperbola

To graph the hyperbola, we need to find the vertices and the asymptotes. The vertices are the points where the branches of the hyperbola intersect the x-axis. The asymptotes are the lines that the branches of the hyperbola approach as they extend towards infinity.

Finding the Vertices

The vertices of the hyperbola are the points where the branches intersect the x-axis. To find the vertices, we need to set $y = 0$ in the equation of the hyperbola and solve for $x$. This gives us:

$$\frac{0^2}{4} - \frac{x^2}{16} = 1$$

Simplifying this equation, we get:

$$-\frac{x^2}{16} = 1$$

Multiplying both sides by $-16$, we get:

$$x^2 = -16$$

Taking the square root of both sides, we get:

$$x = \pm 4$$

Therefore, the vertices of the hyperbola are the points $(4, 0)$ and $(-4, 0)$.

Finding the Asymptotes

The asymptotes of the hyperbola are the lines that the branches approach as they extend towards infinity. To find the asymptotes, we need to set $y = 0$ in the equation of the hyperbola and solve for $x$. This gives us:

$$\frac{0^2}{4} - \frac{x^2}{16} = 1$$

Simplifying this equation, we get:

$$-\frac{x^2}{16} = 1$$

Multiplying both sides by $-16$, we get:

$$x^2 = -16$$

Taking the square root of both sides, we get:

$$x = \pm 4$$

Therefore, the asymptotes of the hyperbola are the lines $y = \pm \frac{2}{4} (x - 4)$ and $y = \pm \frac{2}{4} (x + 4)$.

Graphing the Hyperbola

Now that we have found the vertices and the asymptotes, we can graph the hyperbola. The graph of the hyperbola is a pair of branches that intersect at the vertices and approach the asymptotes as they extend towards infinity.

Conclusion

In this article, we have explored the equation of a hyperbola and determined which graph represents it. We have found the vertices and the asymptotes of the hyperbola and graphed it. The graph of the hyperbola is a pair of branches that intersect at the vertices and approach the asymptotes as they extend towards infinity.

Which Graph Represents the Hyperbola?

Based on the equation of the hyperbola, we can determine which graph represents it. The graph that represents the hyperbola is the one that has the vertices at $(4, 0)$ and $(-4, 0)$ and the asymptotes $y = \pm \frac{2}{4} (x - 4)$ and $y = \pm \frac{2}{4} (x + 4)$.

Graphs of Hyperbolas

Here are some graphs of hyperbolas for comparison:

Graph 1

This graph represents a hyperbola with the equation $\frac{y^2}{4} - \frac{x^2}{16} = 1$. The vertices of this hyperbola are the points $(4, 0)$ and $(-4, 0)$, and the asymptotes are the lines $y = \pm \frac{2}{4} (x - 4)$ and $y = \pm \frac{2}{4} (x + 4)$.

Graph 2

This graph represents a hyperbola with the equation $\frac{y^2}{16} - \frac{x^2}{4} = 1$. The vertices of this hyperbola are the points $(0, 4)$ and $(0, -4)$, and the asymptotes are the lines $y = \pm \frac{4}{16} (x - 0)$ and $y = \pm \frac{4}{16} (x + 0)$.

Graph 3

This graph represents a hyperbola with the equation $\frac{y^2}{16} - \frac{x^2}{16} = 1$. The vertices of this hyperbola are the points $(0, 4)$ and $(0, -4)$, and the asymptotes are the lines $y = \pm \frac{4}{16} (x - 0)$ and $y = \pm \frac{4}{16} (x + 0)$.

Conclusion

In conclusion, the graph that represents the hyperbola $\frac{v^2}{4} - \frac{x^2}{16} = 1$ is the one that has the vertices at $(4, 0)$ and $(-4, 0)$ and the asymptotes $y = \pm \frac{2}{4} (x - 4)$ and $y = \pm \frac{2}{4} (x + 4)$.

Introduction

Hyperbolas are a type of conic section that can be used to model various real-world phenomena, such as the trajectory of a projectile or the shape of a satellite dish. In mathematics, hyperbolas are defined as the set of all points that satisfy a specific equation. In this article, we will answer some frequently asked questions about hyperbolas.

Q: What is a hyperbola?

A: A hyperbola is a type of curve that has two branches, each of which is a parabola. The equation of a hyperbola is typically written in the form $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$, where $a$ and $b$ are constants that determine the shape and size of the hyperbola.

Q: What are the vertices of a hyperbola?

A: The vertices of a hyperbola are the points where the branches of the hyperbola intersect the x-axis. To find the vertices, we need to set $y = 0$ in the equation of the hyperbola and solve for $x$.

Q: What are the asymptotes of a hyperbola?

A: The asymptotes of a hyperbola are the lines that the branches of the hyperbola approach as they extend towards infinity. To find the asymptotes, we need to set $y = 0$ in the equation of the hyperbola and solve for $x$.

Q: How do I graph a hyperbola?

A: To graph a hyperbola, we need to find the vertices and the asymptotes. The vertices are the points where the branches of the hyperbola intersect the x-axis, and the asymptotes are the lines that the branches of the hyperbola approach as they extend towards infinity.

Q: What is the difference between a hyperbola and a parabola?

A: A hyperbola is a type of curve that has two branches, each of which is a parabola. A parabola is a type of curve that has only one branch. The equation of a parabola is typically written in the form $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants that determine the shape and size of the parabola.

Q: Can a hyperbola be a circle?

A: No, a hyperbola cannot be a circle. A hyperbola is a type of curve that has two branches, each of which is a parabola. A circle is a type of curve that is symmetric about its center and has a constant radius.

Q: Can a hyperbola be a line?

A: No, a hyperbola cannot be a line. A hyperbola is a type of curve that has two branches, each of which is a parabola. A line is a type of curve that is straight and has no curvature.

Q: How do I determine the equation of a hyperbola?

A: To determine the equation of a hyperbola, we need to know the values of $a$ and $b$. The equation of a hyperbola is typically written in the form $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$, where $a$ and $b$ are constants that determine the shape and size of the hyperbola.

Q: Can a hyperbola be a function?

A: Yes, a hyperbola can be a function. A function is a relation between a set of inputs and a set of possible outputs. A hyperbola is a type of curve that has two branches, each of which is a parabola. If the branches of the hyperbola intersect at a single point, then the hyperbola can be a function.

Q: Can a hyperbola be a relation?

A: Yes, a hyperbola can be a relation. A relation is a set of ordered pairs that satisfy a certain condition. A hyperbola is a type of curve that has two branches, each of which is a parabola. If the branches of the hyperbola intersect at multiple points, then the hyperbola can be a relation.

Conclusion

In conclusion, hyperbolas are a type of conic section that can be used to model various real-world phenomena. They are defined as the set of all points that satisfy a specific equation. In this article, we have answered some frequently asked questions about hyperbolas.