Problem 37 - Intermediate Algebra

Question

Use a graphing calculator to graph each equation. See Using Your Calculator: Graphing Hyperbolas. $$\frac{x^{2}}{9}-\frac{y^{2}}{4}=1$$

Step-by-Step Solution

Answer

Graph the equation \(y = \pm \sqrt{4(\frac{x^2}{9} - 1)}\) using a graphing calculator.

Step 1: Recognize the Equation Format

The given equation \(\frac{x^2}{9}-\frac{y^2}{4}=1\) is in the standard form of a hyperbola, \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). In this case, \(a^2 = 9\) and \(b^2 = 4\). This tells us the hyperbola is centered at the origin and opens left-right because the \(x^2\) term is positive.

Step 2: Determine Key Features

From the standard form, we calculate \(a = 3\) (since \(a^2 = 9\)) and \(b = 2\) (since \(b^2 = 4\)). These values help us identify the vertices of the hyperbola as \((\pm 3, 0)\). The slopes of the asymptotes are \(\pm \frac{b}{a} = \pm \frac{2}{3}\).

Step 3: Set Up Your Calculator

Input the equation into your graphing calculator. Convert the hyperbola equation into its two part forms by solving for \(y\): \(y = \pm \sqrt{4(\frac{x^2}{9} - 1)}\). This will allow the calculator to graph the two separate branches of the hyperbola.

Step 4: Graph the Hyperbola

Use your graphing calculator to enter the two equations derived from splitting \(y = \pm \sqrt{4(\frac{x^2}{9} - 1)}\). Set an appropriate window that includes the vertices \((3,0)\) and \((-3,0)\) to ensure the full hyperbola is visible.

Key Concepts

Hyperbola Equation

Hyperbola equations are mathematical expressions that describe hyperbolas. These are types of conic sections, formed by the intersection of a plane and a double napped cone. Unlike circles and ellipses, hyperbolas have two distinct branches. The general form of a hyperbola equation with a horizontal transverse axis is given by:

• \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)

In this equation:- The terms are subtracted, not added, indicating a hyperbola rather than an ellipse.- \(a^2\) and \(b^2\) represent squared values that determine the distances from the center to the vertices along the major axis and the distances to the endpoints of the conjugate axis, respectively. The transverse axis of this hyperbola is horizontal because the \(x^2\) term is positive, and the resulting shape opens to the sides. It is important to identify these characteristics when analyzing the hyperbola equation to accurately plot the graph.

Using Graphing Calculator

Graphing calculators are invaluable tools for visualizing hyperbolas, especially when their equations are complex. To effectively use a graphing calculator for plotting hyperbolas, follow these simplified steps:

• Convert the hyperbola equation into its two separate branches for explicit notation, since one equation translates into two curves. For the given equation: \(y = \pm \sqrt{4(\frac{x^2}{9} - 1)}\).
• Enter both equations into the calculator. Remember to input them separately to ensure each half of the hyperbola is captured. This means inputting \( y = \sqrt{4(\frac{x^2}{9} - 1)} \) and \( y = -\sqrt{4(\frac{x^2}{9} - 1)} \).
• Adjust your view window to include the relevant features of the hyperbola. Set your window using the vertices or intercepts as references.

By accurately setting up your calculator, you can then press graph to view both branches of the hyperbola, ensuring it aligns correctly with the expected mathematical behavior.

Standard Form of a Hyperbola

The standard form of a hyperbola helps to clearly understand its structure and properties. In order to graph a hyperbola or extract information about its key features, recognize its standard form:

• \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)
• \( \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \)

The distinction between these forms lies in the position of the positive term, which informs the orientation of the hyperbola:- If the \(x^2\) term is first and positive, the hyperbola opens left and right.- If the \(y^2\) term is first and positive, the hyperbola opens upward and downward.In the example equation \(\frac{x^2}{9} - \frac{y^2}{4}=1\), \(a^2 = 9\) leads to \(a = 3\) and \(b^2 = 4\) gives \(b = 2\), defining essential information such as the distance from the center to vertices and rotation of the hyperbola. A correct understanding of the standard form ensures accurate graphing and analysis of hyperbolas. By mastering these forms, students can confidently tackle any hyperbola-related problems.