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In the context of a hyperbola, the vertices are key points that define its shape. For the hyperbola given by the equation \( \frac{y^2}{36} - \frac{x^2}{16} = 1 \), we know it's a vertical hyperbola centered at the origin (0,0). The standard form of a vertical hyperbola \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \) is used to find the location of the vertices.The value \(a\) is crucial in determining the position of the vertices. Here, \(a^2 = 36\), which means \(a = 6\). This tells us the vertices are located at

• (0, a) which is (0, 6), and
• (0, -a) which is (0, -6).

These points signify the positions on the y-axis where the hyperbola intercepts, forming the top and bottom of its vertical shape.Understanding vertices helps in sketching the basic framework of the hyperbola, giving you a clear visualization of its overall structure.

The foci of a hyperbola are points located along the hyperbola's axis of symmetry, further apart than the vertices. They play a key role in defining the hyperbola's properties and location.To find the foci of our hyperbola, we use the relationship:\[c = \sqrt{a^2 + b^2}\]Where \(a^2 = 36\) and \(b^2 = 16\), giving us:\(c = \sqrt{36 + 16} = \sqrt{52} = 2\sqrt{13}\).For a vertical hyperbola like \( \frac{y^2}{36} - \frac{x^2}{16} = 1 \), the calculated foci are at:

• (0, c) which is (0, 2\sqrt{13}), and
• (0, -c) which is (0, -2\sqrt{13}).

These points demonstrate the real physical 'stretch' of the hyperbola, extending further than the vertices do. The foci direct the characteristic nature and wideness of the hyperbola, as every point on the hyperbola is such that the difference of the distances to the foci is constant.

Asymptotes are lines that a hyperbola approaches but never actually touches. They act as a guide to the hyperbola's shape, showing the direction and inclination of its branches.For a vertical hyperbola like \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \), the asymptotes are given by the equations:\[ y = \pm \frac{a}{b} x \]Substituting \(a = 6\) and \(b = 4\), we find:\[ y = \pm \frac{6}{4} x = \pm \frac{3}{2} x \]These are the asymptote lines \(y = \frac{3}{2} x\) and \(y = -\frac{3}{2} x\) that crisscross through the origin.

• They provide a vital guide for sketching the hyperbola, as its branches will get increasingly closer to these lines without crossing them.
• Asymptotes highlight that as you move away from the center, the branches progressively "open up" within these boundaries.

Understanding the asymptotes helps to graph the hyperbola accurately and understand its expansive nature.