91Ó°ÊÓ

To effectively graph a hyperbola, understanding its standard form is crucial. The standard form of a hyperbola can be integral in rearranging and interpreting the equation we are working with. A hyperbola's equation can be expressed in two standard forms:

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

For the given exercise, the equation \(3y^{2} - 2x^{2} = 15\) needs to be rearranged into a standard form. By dividing each term by 15, we arrive at the standard form \(\frac{y^{2}}{5} - \frac{x^{2}}{7.5} = 1\). Here, the hyperbola is oriented vertically because the \(y^{2}\) term is positive and comes first. This orientation decision is crucial because it influences how we plot the graph and position the hyperbola along the axes.

Vertices and co-vertices are fundamental elements when graphing hyperbolas. They serve as critical points that help indicate the shape and orientation of the hyperbola:

• Vertices: These points are on the hyperbola itself and represent the maximum distance from the center along the hyperbola's main axis. For our exercise, we determine the vertices based on \(a\), where \(a = \sqrt{5}\). Since it is a vertical hyperbola, these vertices are positioned at (0, ±\(\sqrt{5}\)).
• Co-vertices: Unlike vertices, co-vertices do not lie on the hyperbola. They assist in forming the rectangle used to sketch the asymptotes. For this hyperbola, \(b = \sqrt{7.5}\), and the co-vertices are at (±\(\sqrt{7.5}\), 0).

The center of the hyperbola is at the origin (0,0), which simplifies our calculations and plotting. By marking vertices and co-vertices, we can easily visualize the hyperbola's envelope and help guide its drawing on the graph.

Graphing utilities are powerful tools for visually presenting mathematical equations, making them especially handy for graphing hyperbolas. Here's how they can be utilized effectively:

• Input the Equation: Enter the rearranged standard form of the hyperbola into the graphing tool. For our example, this is \(\frac{y^{2}}{5} - \frac{x^{2}}{7.5} = 1\).
• Plot the Key Points: Use the graphing utility to plot the center, vertices, and co-vertices based on our earlier calculations.
• Sketch the Asymptotes: Graphing utilities often have features to draw asymptotes, which pass diagonally through corners of the rectangle formed by vertices and co-vertices, guiding the curvature of the hyperbola.
• Visualize the Hyperbola: The utility will assist you in drawing the branches of the hyperbola that pass through the vertices. It helps to confirm the hyperbola's direction and curvature visually.

Using graphing utilities enhances precision and saves time, allowing you to focus more on understanding the hyperbola's behavior rather than the manual plotting mechanics.