Problem 29 - Algebra and Trigonometry

Question

Use a graphing device to graph the hyperbola. $$ \frac{y^{2}}{2}-\frac{x^{2}}{6}=1 $$

Step-by-Step Solution

Answer

Graph a hyperbola with vertices at (0, ±√2), asymptotes y = ±(√3/3)x, opening vertically.

Step 1: Identify the Hyperbola Equation Standard Form

The given equation is \( \frac{y^{2}}{2} - \frac{x^{2}}{6} = 1 \). This matches the standard form of a hyperbola centered at the origin: \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \), with \( a^2 = 2 \) and \( b^2 = 6 \). Here, the y-term comes first, which tells us the hyperbola opens vertically.

Step 2: Calculate the Asymptotes

For a hyperbola in the form \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \), the asymptotes are given by the equations \( y = \pm \frac{a}{b}x \). Calculate these as \( y = \pm \sqrt{\frac{2}{6}}x = \pm \frac{1}{\sqrt{3}}x = \pm \frac{\sqrt{3}}{3}x \).

Step 3: Determine the Vertices

The vertices of the hyperbola are located at \( (0, \pm a) \). Since \( a = \sqrt{2} \), the vertices are \( (0, \sqrt{2}) \) and \( (0, -\sqrt{2}) \).

Step 4: Plot the Hyperbola

Using a graphing device, plot the hyperbola. Draw the asymptotes as straight lines through the origin with slopes \( \pm \frac{\sqrt{3}}{3} \). Mark the vertices at \( (0, \sqrt{2}) \) and \( (0, -\sqrt{2}) \). The arms of the hyperbola will curve away from each other along these guidelines, opening vertically.

Step 5: Verify the Graph Characteristics

Ensure that the arms of the hyperbola approach but do not touch the asymptotes. The vertices should be correctly plotted at \( (0, \sqrt{2}) \) and \( (0, -\sqrt{2}) \), and the hyperbola should open upwards and downwards.

Key Concepts

Asymptotes

Asymptotes are imaginary lines that a curve approaches as it heads towards infinity. They are extremely useful in graphing hyperbolas, as they provide a framework that a hyperbola follows closely but never touches. For the given hyperbola equation \[ \frac{y^{2}}{2} - \frac{x^{2}}{6} = 1 \], the asymptotes are calculated from the equation:

• \( y = \pm \frac{a}{b}x \), where \( a^2 = 2 \) and \( b^2 = 6 \).

Plugging the values, we find:\[ y = \pm \frac{1}{\sqrt{3}}x = \pm \frac{\sqrt{3}}{3}x \].These lines cross through the origin of the graph and are crucial because they define the direction in which the arms of the hyperbola will expand. As the graph stretches towards the asymptotes, the curves approach but never meet these lines.

Vertices

Vertices are key points associated with hyperbolas and are vital in the graphing of these conic sections. They are the points where the hyperbola makes its closest approach to the center. For a hyperbola centered at the origin with a standard form of equation:\[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \], the vertices are located vertically at \( (0, \pm a) \).In our given problem, \( a^2 = 2 \) giving us:

• \( a = \sqrt{2} \)
• The vertices are \( (0, \sqrt{2}) \) and \( (0, -\sqrt{2}) \).

These points are essential for sketching out the hyperbola, as they represent the closest proximity of the hyperbolic arms and serve as guides for graphing the figure accurately.

Graphing Hyperbolas

Graphing a hyperbola involves understanding its distinctive shape and the role of its asymptotes and vertices. Hyperbolas, part of the conic sections, have a unique form characterized by two separate curves or "arms" that mirror each other.To graph our hyperbola, we follow these steps:

• Draw the asymptotes first, which are straight lines through the origin with slopes \( \pm \frac{\sqrt{3}}{3} \).
• Mark the vertices \( (0, \sqrt{2}) \) and \( (0, -\sqrt{2}) \).
• Sketch the hyperbola opening up and down around these guidelines.

When plotting, it's important to note:

• The vertices are used to set the width of the hyperbola's arms.
• The asymptotes act as a directional guide.
• Ensure that the arms curve away without touching the asymptotes.

Using a graphing device can make this process precise and quick.

Conic Sections

Conic sections are curves obtained by intersecting a cone with a plane at various angles. This group includes circles, ellipses, parabolas, and hyperbolas.Hyperbolas, like in our given equation, have their unique attributes:

• They have two distinct branches.
• Their shape is driven by their asymptotes.
• Formally expressed by equations similar to: \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \)

The position of the hyperbola changes based on which squared term is positive:

• If the \( y^2 \) term is positive, the hyperbola opens vertically, as seen in our problem.
• Conversely, if the \( x^2 \) term is positive, the arms open horizontally.

Fully understanding conic sections, specifically hyperbolas, involves recognizing their important features like asymptotes and vertices. This comprehensive knowledge is crucial for accurately sketching and interpreting these intriguing curves.