91Ó°ÊÓ

In a hyperbola, the vertices are crucial points that lie on the hyperbola itself, representing the closest points to each other on either branch. Given the standard form of a hyperbola with a horizontal transverse axis, the equation can be written as:\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\]Here, the vertices are located at \((\pm a, 0)\). For our specific hyperbola, the value of \(a\) is found by rewriting the equation and identifying the denominator under \(x^2\). From the steps provided, we know \(a = \frac{1}{3}\). Thus, the vertices are located at the points \(\left(\pm \frac{1}{3}, 0\right)\).
This means the vertices are slightly set apart on the horizontal axis, indicating the primary direction of the hyperbola's branches opening out.

The foci of a hyperbola are two fixed points located along the transverse axis, further apart than the vertices. These points are used to define the shape of the hyperbola.
In the context of a hyperbola in the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the distance to the foci from the center of the hyperbola can be calculated with the formula:\[c^2 = a^2 + b^2\]Given \(a = \frac{1}{3}\) and \(b = \frac{1}{4}\), we calculate \(c\) as:\[c = \sqrt{\left(\frac{1}{3}\right)^2 + \left(\frac{1}{4}\right)^2} = \sqrt{\frac{25}{144}} = \frac{5}{12}\]This tells us that the foci are located at \(\left(\pm \frac{5}{12}, 0\right)\). The foci are always further from the center than the vertices, which in this case are closer to the origin, mirroring the stronger curves on the graph.

Asymptotes are lines that the hyperbola approaches but never actually meets. These lines help guide the shape and direction of the hyperbola's branches.
For a hyperbola with a standard form equation \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the equations for the asymptotes are determined by:\[y = \pm \frac{b}{a}x\]In our example, with \(a = \frac{1}{3}\) and \(b = \frac{1}{4}\), the slopes of the asymptotes are calculated to be \(\frac{b}{a} = \frac{1/4}{1/3} = \frac{3}{4}\).
Thus, the asymptotes are described by the lines:

• \(y = \frac{3}{4}x\)
• \(y = -\frac{3}{4}x\)

These lines intersect at the center of the hyperbola (origin) and provide a visual guide for the hyperbola's branches as they extend infinitely without touching the asymptotes.

The equation of a hyperbola is fundamental in defining its shape and properties. A hyperbola can be expressed in a couple of standard forms, but our specific scenario involves the form where the transverse axis is horizontal:\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\]In this equation:

• \(a^2\) is associated with the horizontal axis (\(x\)-axis).
• \(b^2\) is associated with the vertical axis (\(y\)-axis).

Rewriting our initial hyperbola equation \(9x^2 - 16y^2 = 1\) into this form, we get:\[\frac{x^2}{\left(\frac{1}{3}\right)^2} - \frac{y^2}{\left(\frac{1}{4}\right)^2} = 1\]This rewritten equation clearly shows the relationship between \(x\) and \(y\), helping to understand which axis the hyperbola spreads across, showing a symmetric pattern with respect to the center. It is this equation that provides the foundation for plotting and analyzing the hyperbola.