Find the center, transverse axis, vertices, foci, and asymptotes. Graph the equation. \( \frac{x^{2}}{64}-\frac{y^{2}}{9}=1 \)

The given equation is \[ \frac{x^{2}}{64} - \frac{y^{2}}{9} = 1 \] This is the standard form of a hyperbola centered at the origin \((0, 0)\) with a horizontal transverse axis. The general form of a hyperbola is \[ \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \] where \(a\) and \(b\) are the distances related to the vertices and foci. ### Step 1: Identify \(a\) and \(b\) From the equation, we can identify: - \(a^2 = 64\) \(\Rightarrow a = \sqrt{64} = 8\) - \(b^2 = 9\) \(\Rightarrow b = \sqrt{9} = 3\) ### Step 2: Find the center The center of the hyperbola is at the origin: \[ \text{Center} = (0, 0) \] ### Step 3: Find the vertices The vertices of a hyperbola with a horizontal transverse axis are located at \((\pm a, 0)\): \[ \text{Vertices} = (8, 0) \text{ and } (-8, 0) \] ### Step 4: Find the foci The foci are located at \((\pm c, 0)\), where \(c\) is calculated using the formula: \[ c = \sqrt{a^2 + b^2} \] Calculating \(c\): \[ c = \sqrt{64 + 9} = \sqrt{73} \] Thus, the foci are: \[ \text{Foci} = \left(\sqrt{73}, 0\right) \text{ and } \left(-\sqrt{73}, 0\right) \] ### Step 5: Find the asymptotes The equations of the asymptotes for a hyperbola centered at the origin with a horizontal transverse axis are given by: \[ y = \pm \frac{b}{a} x \] Calculating \(\frac{b}{a}\): \[ \frac{b}{a} = \frac{3}{8} \] Thus, the equations of the asymptotes are: \[ y = \frac{3}{8} x \quad \text{and} \quad y = -\frac{3}{8} x \] ### Summary of Results - **Center**: \((0, 0)\) - **Transverse Axis**: Horizontal - **Vertices**: \((8, 0)\) and \((-8, 0)\) - **Foci**: \(\left(\sqrt{73}, 0\right)\) and \(\left(-\sqrt{73}, 0\right)\) - **Asymptotes**: \(y = \frac{3}{8} x\) and \(y = -\frac{3}{8} x\) ### Step 6: Graph the hyperbola To graph the hyperbola, plot the center, vertices, and foci. Draw the asymptotes as dashed lines. The hyperbola opens to the left and right from the vertices. 1. Plot the center at \((0, 0)\). 2. Plot the vertices at \((8, 0)\) and \((-8, 0)\). 3. Plot the foci at approximately \((8.54, 0)\) and \((-8.54, 0)\). 4. Draw the asymptotes through the center with slopes of \(\frac{3}{8}\) and \(-\frac{3}{8}\). 5. Sketch the hyperbola opening left and right, approaching the asymptotes. This completes the analysis and graphing of the hyperbola given by the equation \(\frac{x^{2}}{64} - \frac{y^{2}}{9} = 1\).