Problem 27 - Algebra and Trigonometry

Question

Use a graphing device to graph the hyperbola. $$ x^{2}-2 y^{2}=8 $$

Step-by-Step Solution

Answer

The hyperbola opens along the x-axis with vertices at \( (\pm \sqrt{8}, 0) \).

Step 1: Recognize the Equation Form

The given equation is a hyperbola in the form \( x^2 - 2y^2 = 8 \). A standard form for hyperbolas oriented along the x-axis is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). First, we need to rewrite the equation in this standard form.

Step 2: Convert to Standard Form

To convert the given equation to the standard form, divide all terms by 8: \[ \frac{x^2}{8} - \frac{2y^2}{8} = 1 \] Simplify to get: \[ \frac{x^2}{8} - \frac{y^2}{4} = 1 \] This represents a hyperbola centered at the origin \((0,0)\) opening along the x-axis.

Step 3: Identify Components of the Hyperbola

From the standard form \( \frac{x^2}{8} - \frac{y^2}{4} = 1 \), identify: \( a^2 = 8 \) and \( b^2 = 4 \). Therefore, \( a = \sqrt{8} \approx 2.83 \) and \( b = 2 \). The slopes of the asymptotes are given by \( \pm \frac{b}{a} = \pm \frac{2}{\sqrt{8}} = \pm \frac{1}{\sqrt{2}} \approx \pm 0.707 \).

Step 4: Graph the Hyperbola

Plot the vertices at \( (\pm a, 0) = (\pm \sqrt{8}, 0) \) approximately \( (\pm 2.83, 0) \). Draw the asymptotes through the origin with slopes \( \pm 0.707 \). The hyperbola approaches these asymptotes without touching them.

Step 5: Verify with a Graphing Device

Using a graphing device, input the function or the implicit plot command to visualize the hyperbola. Check that \( x^2 - 2y^2 = 8 \) appears as expected with the correct orientation, center, and asymptotes.

Key Concepts

Standard Form of Hyperbola

Understanding the standard form of a hyperbola is key to graphing and analyzing these unique curves. The standard form for a hyperbola centered at the origin \((0, 0)\) is either \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) or \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\). The choice depends on the orientation of the hyperbola:

• The first form is used when the hyperbola opens along the x-axis.
• The second form applies when it opens along the y-axis.

Here, \(a\) and \(b\) represent the distances from the center to the vertices and co-vertices, respectively. These values are derived from \(a^2\) and \(b^2\) in the equation. This form simplifies the process of identifying key features, such as vertices and asymptotes, necessary for graphing.

Converting Hyperbola Equations

Converting a hyperbola equation into its standard form requires a bit of algebraic manipulation, often involving division to match the pattern of the standard form. In our example, the initial equation given was \(x^2 - 2y^2 = 8\). To convert this, each term must be divided by 8, transforming the equation into \(\frac{x^2}{8} - \frac{y^2}{4} = 1\).
This straightforward division method allows the equation to follow the form\(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), where \(a^2 = 8\) and\(b^2 = 4\). This conversion reveals important characteristics of the hyperbola, such as its orientation and center.

Asymptotes of Hyperbolas

The asymptotes of a hyperbola are crucial as they guide the shape and orientation of the curve but never intersect it. They serve as imaginary boundary lines that the branches of the hyperbola approach as they extend outward. Once the equation \(\frac{x^2}{8} - \frac{y^2}{4} = 1\) is in standard form, calculating the slopes of these lines is straightforward. For a hyperbola centered at the origin opening along the x-axis, the slopes of the asymptotes are given by \(\pm \frac{b}{a}\). In this case, \(b = 2\) and \(a = \sqrt{8}\), resulting in slopes \(\pm \frac{1}{\sqrt{2}} \approx \pm 0.707\).
These slopes help sketch the asymptotes accurately, contributing to a correctly rendered graph of the hyperbola.

Graphing with a Graphing Device

Modern graphing devices or software offer a streamlined way to visualize hyperbolas, ensuring accuracy and clarity. Whether using a graphing calculator or computer software, inputting the hyperbola's equation correctly is essential. For the hyperbola \(x^2 - 2y^2 = 8\), ensure it is input in the standard form \(\frac{x^2}{8} - \frac{y^2}{4} = 1\) if necessary. This allows the tool to appropriately interpret its components, like vertices and asymptotes.
The device will typically enable you to see the hyperbola drawn with its characteristic branches extending towards but never reaching the asymptotes. Using tools, students can check their hand-drawn graphs for alignment with calculated features, enhancing both understanding and confidence in graphing abilities.