Problem 40 Find the vertices and the asympt... [FREE SOLUTION]

Chapter 10: Problem 40

Find the vertices and the asymptotes of each hyperbola. $$ y^{2}-x^{2}=1 $$

Short Answer

Expert verified

The vertices of the hyperbola are at (0, 1) and (0, -1), and the asymptotes are given by the equations y = x and y = -x.

Step by step solution

Identify the center, a, and b

By comparing the given equation $y^{2} - x^{2} = 1$ with the standard equation of a hyperbola $\frac{(y-K)^{2}}{a^{2}} - \frac{(x-H)^{2}}{b^{2}} = 1$, we can see that the center (H, K) is at the origin (0, 0) because the terms for H and K are missing. This is also a case where a and b are both 1.

Determine the vertices

The vertices of the hyperbola are found by moving 'a' units vertically from the center since the y term is first in the equation. So, the vertices are at (0, 1) and (0, -1).

Find the asymptotes

For a hyperbola centered at (H, K) with equation $\frac{(y-K)^{2}}{a^{2}} - \frac{(x-H)^{2}}{b^{2}} = 1$, the asymptotes are $y = K \pm \frac{a}{b}(x - H)$. In this case, since the center is at the origin and a and b are both 1, the equations of the asymptotes are y = x and y = -x.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Asymptotes in a Hyperbola

Asymptotes in a hyperbola are straight lines that the curve approaches but never touches. They provide a framework that guides the shape of the hyperbola. Asymptotes give us an idea about the steepness and orientation of the hyperbola's branches.

They are significant in understanding the hyperbola's growth pattern.
For a hyperbola centered at (H, K), the general formula for asymptotes is: $ y = K \pm \frac{a}{b}(x - H) $
In the equation $ y^2 - x^2 = 1 $, the center is at the origin (0, 0). Thus, H and K are zero.
With a and b equal to 1, the asymptotes of this hyperbola are $ y = x $ and $ y = -x $.

The asymptotes act like invisible scaffolding, showing us the direction in which the branches of the hyperbola extend.

Locating the Vertices of a Hyperbola

Vertices are the points where the hyperbola changes direction. They are crucial in grasping the hyperbola's geometric structure and are directly tied to its center.

To determine vertices, identify the center of the hyperbola and measure 'a' units from it.
For the equation $ y^2 - x^2 = 1 $, there's no transformation affecting the center, meaning it stays at $ (0, 0) $.
Given that 'a' is 1 in $ y^2 - x^2 = 1 $, you move 1 unit from the center along the y-axis.
The vertices, therefore, are at the points $ (0, 1) $ and $ (0, -1) $.

These points are essential as they show where the branches of the hyperbola start to form.

The Standard Form of a Hyperbola

The standard form of a hyperbola is a crucial representation, allowing us to easily identify key characteristics. It sets the stage for locating the center, vertices, and asymptotes. Depending on the orientation, the standard form is slightly different.

When the y-term comes before the x-term in subtraction, the hyperbola opens vertically; hence the form is: $ \frac{(y-K)^2}{a^2} - \frac{(x-H)^2}{b^2} = 1 $
If it's the reverse, with the x-term first, the hyperbola opens horizontally.
Here, like in our problem $ y^2 - x^2 = 1 $, the y-term comes first, indicating a vertical opening.

Recognizing standard form quickly helps determine the nature and direction of the hyperbola.

Identifying the Center of a Hyperbola

The center of a hyperbola is its balancing point. When you understand the center's position, the rest of the hyperbola's attributes align easily around it.

In transforming a standard equation, the center $ (H, K) $ is indicated by its differences from the y and x terms.
For $ y^2 - x^2 = 1 $, there are no H or K values suggested, which means the center is at $ (0, 0) $.
If the equation had terms like $(y - 2)^2$ or $(x + 3)^2$, the center would shift to $ (2, -3) $.

The center simplifies the task of finding vertices and asymptotes and serves as a reference point for all transformations on the hyperbola.

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91影视

Find the vertices and the asymptotes of each hyperbola. $$ y^{2}-x^{2}=1 $$

Short Answer

Step by step solution

Identify the center, a, and b

Determine the vertices

Find the asymptotes

Key Concepts

Understanding Asymptotes in a Hyperbola

Locating the Vertices of a Hyperbola

The Standard Form of a Hyperbola

Identifying the Center of a Hyperbola

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