For $$ \frac{(y+4)^{2}}{25}-\frac{(x+6)^{2}}{64}=1 $$, $$ \begin{array}{ll} ext { (a) Identify the center. } & ext { (b) Identify the vertices. } \ ext { (c) Identify the foci. } & ext { (d) Write equations for the asymptotes. } \ ext { (e) Graph the hyperbola. }\end{array} $$ Give exact answers. Express numbers in simplest form using integers, fractions, or radicals.

Question

For $$ \frac{(y+4)^{2}}{25}-\frac{(x+6)^{2}}{64}=1 $$, $$ \begin{array}{ll}	ext { (a) Identify the center. } & 	ext { (b) Identify the vertices. } \ 	ext { (c) Identify the foci. } & 	ext { (d) Write equations for the asymptotes. } \ 	ext { (e) Graph the hyperbola. }\end{array} $$ Give exact answers. Express numbers in simplest form using integers, fractions, or radicals.

UpStudy AI · Accepted Answer

It seems that I am unable to graph the hyperbola directly. However, I can guide you on how to graph it based on the information provided.

### Steps to Graph the Hyperbola:

1. **Plot the Center:**
   - Mark the point $$(-6, -4)$$ on the coordinate plane.

2. **Plot the Vertices:**
   - Mark the vertices at $$(-6, 1)$$ and $$(-6, -9)$$.

3. **Plot the Foci:**
   - Calculate the approximate values of the foci:
     - $$ -4 + \sqrt{89} \approx -4 + 9.43 \approx 5.43 $$ (Focus 1)
     - $$ -4 - \sqrt{89} \approx -4 - 9.43 \approx -13.43 $$ (Focus 2)
   - Mark the foci at $$(-6, 5.43)$$ and $$(-6, -13.43)$$.

4. **Draw the Asymptotes:**
   - Use the equations of the asymptotes:
     - For $$y = \frac{5}{8}(x + 6) - 4$$:
       - Find points by substituting values for $$x$$ (e.g., $$x = -6$$ gives $$y = -4$$).
     - For $$y = -\frac{5}{8}(x + 6) - 4$$:
       - Similarly, find points by substituting values for $$x$$.
   - Draw dashed lines through these points to represent the asymptotes.

5. **Sketch the Hyperbola:**
   - The hyperbola opens upwards and downwards, approaching the asymptotes but never touching them. Draw the two branches of the hyperbola, ensuring they pass through the vertices and are symmetric about the center.

By following these steps, you will have a complete graph of the hyperbola. If you have graphing software or graph paper, it will help visualize the hyperbola more accurately.
- **Center:** $$(-6, -4)$$ - **Vertices:** $$(-6, 1)$$ and $$(-6, -9)$$ - **Foci:** $$\left(-6, -4 + \sqrt{89}\right)$$ and $$\left(-6, -4 - \sqrt{89}\right)$$ - **Asymptotes:** - $$y = \frac{5}{8}(x + 6) - 4$$ - $$y = -\frac{5}{8}(x + 6) - 4$$ To graph the hyperbola, plot the center, vertices, and foci, then draw the asymptotes and sketch the hyperbola branches accordingly.

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