Problem 39 Graph each hyperbola with center... [FREE SOLUTION]

Chapter 12: Problem 39

Graph each hyperbola with center shifted away from the origin. $$ \frac{y^{2}}{36}-\frac{(x-2)^{2}}{49}=1 $$

Short Answer

Expert verified

Identify the center at $ (2, 0) $. Plot vertices at $(2, 6)$ and $(2, -6)$. Draw asymptotes using equations $y = \frac{6}{7}(x-2)$ and $y = -\frac{6}{7}(x-2)$. Sketch the hyperbola opening vertically.

Step by step solution

- Identify the Standard Form

Recognize that the given equation $\frac{y^{2}}{36}-\frac{(x-2)^{2}}{49}=1$ matches the standard form of a vertically oriented hyperbola $ \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 $ where $(h, k)$ is the center.

- Determine the Center

Identify $h$ and $k$. In this case, $k = 0$, and the center is $(2, 0)$.

- Identify $a$ and $b$

Compare the given equation to the standard form to find $a^2$ and $b^2$. Here, $a^2 = 36$ so $a = 6$, and $b^2 = 49$ so $b = 7$.

- Plot the Center

Mark the center of the hyperbola at $(2, 0)$ on the coordinate plane.

- Draw the Transverse and Conjugate Axes

From the center $(2, 0)$, move $a = 6$ units along the y-axis (up and down from the center) to get the vertices at $(2, 6)$ and $(2, -6)$. Move $b = 7$ units along the x-axis (left and right from the center) to plot the ends of the conjugate axis at $(-5, 0)$ and $(9, 0)$.

- Sketch the Asymptotes

The equations of the asymptotes can be derived from the slope given by $ \frac{a}{b} $ and $ \frac{b}{a} $. Thus, the equations are $ y = \frac{6}{7}(x-2)$ and $ y = -\frac{6}{7}(x-2)$. Draw these lines through the center.

- Draw the Hyperbola

Sketch the hyperbola by drawing branches opening vertically, approaching the asymptotes but never touching them.

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Key Concepts

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

hyperbola

A hyperbola is a type of conic section formed when a plane intersects both nappes (the upper and lower halves) of a double cone. Hyperbolas look like two separate curves that open in opposite directions. Unlike circles and ellipses, hyperbolas have two branches. Each branch approaches a set of asymptotes but never meets them. Understanding hyperbolas is crucial in advanced algebra and geometry.

standard form of a hyperbola

The standard form of a hyperbola's equation depends on its orientation. There are two primary forms:

Vertical Hyperbola: $\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$
Horizontal Hyperbola: $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$

In both forms, $(h, k)$ is the center of the hyperbola. The values $a^2$ and $b^2$ determine the distances from the center.

center of a hyperbola

The center of a hyperbola (ewline$(h, k)$ ewline) is the point equidistant from the vertices and the foci. Identifying the center is the first step in graphing a hyperbola. In the given equation ewline $\frac{y^{2}}{36}-\frac{(x-2)^{2}}{49}=1$, ewline it is clear that the center is at $(2, 0)$. This shift from the origin must be accurately marked on your graph.

transverse axis

The transverse axis of a hyperbola is the line that passes through its vertices. For hyperbolas in the standard form ewline $\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$, ewline the transverse axis is vertical, as the hyperbola opens upwards and downwards. For the given equation, the transverse axis lies along the y-axis and extends 6 units from the center. This gives us the vertices at $(2, 6)$ and $(2, -6)$.

asymptotes of a hyperbola

The asymptotes of a hyperbola are the lines that the branches of the hyperbola approach but never touch. They provide a framework for sketching the hyperbola. For a hyperbola with the equation ewline $\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$, the asymptotes have slopes of $ \frac{a}{b} $ and $ -\frac{a}{b} $. In the given equation, the asymptotes are ewline $ y = \frac{6}{7}(x-2) $ and ewline $ y = -\frac{6}{7}(x-2) $. Drawing these lines helps in sketching the hyperbola accurately.

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91影视

Graph each hyperbola with center shifted away from the origin. $$ \frac{y^{2}}{36}-\frac{(x-2)^{2}}{49}=1 $$

Short Answer

Step by step solution

- Identify the Standard Form

- Determine the Center

- Identify \(a\) and \(b\)

- Plot the Center

- Draw the Transverse and Conjugate Axes

- Sketch the Asymptotes

- Draw the Hyperbola

Key Concepts

hyperbola

standard form of a hyperbola

center of a hyperbola

transverse axis

asymptotes of a hyperbola

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