Here is the graph of the hyperbola given by the equation:

$\frac{(x + 2)^2}{16} - \frac{(y - 3)^2}{25} = 1$

Summary of Key Elements:

• Center: $(-2, 3)$
• Vertices: $(-6, 3)$ and $(2, 3)$
• Asymptotes: The dashed red lines represent the asymptotes with slopes $\pm \frac{5}{4}$.

If you notice any gaps in the branches, it's because the hyperbola doesn't exist for all values of $x$; only when $\frac{(x + 2)^2}{16} > 1$ does the square root give real values.

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Would you like to explore more on hyperbolas or another concept?

Here are five related questions to consider:

1. What is the effect of changing $a$ and $b$ on the shape of the hyperbola?
2. How do you derive the asymptote equations for hyperbolas?
3. How do vertical and horizontal hyperbolas differ?
4. Can you determine the foci of the given hyperbola?
5. How do you transform this hyperbola to the origin-centered form?

Tip: For a hyperbola, the further the asymptotes diverge, the narrower the branches will appear.