Question

graph the hyperbola.
\\(\frac{y^2}{25} - \frac{x^2}{9} = 1\\)

Explanation:

Step1: Identify the standard form

The standard form of a hyperbola is \(\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}} = 1\). For the given equation \(\frac{y^{2}}{25}-\frac{x^{2}}{9}=1\), we have \(a^{2}=25\), so \(a = 5\), and \(b^{2}=9\), so \(b = 3\).

Step2: Find the vertices

The vertices of a hyperbola of the form \(\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1\) are at \((0,\pm a)\). Substituting \(a = 5\), the vertices are \((0,5)\) and \((0, - 5)\).

Step3: Find the asymptotes

The equations of the asymptotes for a hyperbola \(\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1\) are \(y=\pm\frac{a}{b}x\). Substituting \(a = 5\) and \(b = 3\), the asymptotes are \(y=\frac{5}{3}x\) and \(y=-\frac{5}{3}x\).

To graph the hyperbola:

1. Plot the vertices \((0,5)\) and \((0,-5)\).
2. Draw the asymptotes \(y = \frac{5}{3}x\) (passing through the origin with a slope of \(\frac{5}{3}\)) and \(y=-\frac{5}{3}x\) (passing through the origin with a slope of \(-\frac{5}{3}\)).
3. Sketch the hyperbola approaching the asymptotes.

Answer:

Graph the hyperbola with vertices \((0,5)\) and \((0,-5)\) and asymptotes \(y=\frac{5}{3}x\) and \(y =-\frac{5}{3}x\) as described above.