To graph the hyperbola given by the equation:

$\frac{x^2}{16} - \frac{y^2}{9} = 1$

we need to follow these steps:

1. Identify the center, vertices, and asymptotes:

   • This hyperbola is centered at the origin $(0,0)$ because there are no $h$ and $k$ values in the equation.
   • The equation is in the standard form $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, where $a^2 = 16$ and $b^2 = 9$. Thus, $a = 4$ and $b = 3$.

2. Vertices:

   • For hyperbolas of the form $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, the vertices are at $(\pm a, 0)$.
   • Therefore, the vertices are $(\pm 4, 0)$, which are $(4, 0)$ and $(-4, 0)$.

3. Asymptotes:

   • The equations of the asymptotes for hyperbolas in this form are $y = \pm \frac{b}{a} x$.
   • Substituting $a = 4$ and $b = 3$, the asymptote equations are $y = \pm \frac{3}{4} x$.

4. Graphing the Hyperbola:

   • Draw the center at the origin $(0,0)$.
   • Plot the vertices at $(4,0)$ and $(-4,0)$.
   • Draw the asymptotes $y = \frac{3}{4}x$ and $y = -\frac{3}{4}x$. These lines pass through the origin and have slopes of $\pm \frac{3}{4}$.
   • Sketch the hyperbola opening left and right, approaching but never touching the asymptotes.

5. Domain and Range:

   • For the hyperbola $\frac{x^2}{16} - \frac{y^2}{9} = 1$:
     • The domain (set of all possible $x$-values) is $x \in (-\infty, -4] \cup [4, \infty)$.
     • The range (set of all possible $y$-values) is $y \in (-\infty, \infty)$.

Here's the graph of the hyperbola:

1. Draw a set of coordinate axes.
2. Mark the center at $(0,0)$.
3. Plot the vertices at $(4,0)$ and $(-4,0)$.
4. Draw the asymptotes: $y = \frac{3}{4}x$ and $y = -\frac{3}{4}x$.
5. Sketch the branches of the hyperbola, opening to the left and right, approaching the asymptotes.

Would you like more details or have any questions?

Here are some related questions:

1. How do you derive the standard form of a hyperbola from its general equation?
2. What are the differences between the equations of horizontal and vertical hyperbolas?
3. How do you determine the foci of a hyperbola?
4. How do you find the lengths of the transverse and conjugate axes of a hyperbola?
5. How do you determine the eccentricity of a hyperbola?
6. How do the asymptotes help in sketching the graph of a hyperbola?
7. How can you verify if a given equation represents a hyperbola?
8. What are the properties of hyperbolas that differentiate them from other conic sections?

Tip: When graphing a hyperbola, always start by identifying the center, vertices, and asymptotes. These key elements guide the accurate sketching of the hyperbola's shape.