To graph the hyperbola given by the equation:

$\frac{x^2}{9} - \frac{y^2}{4} = 1,$

let’s break it into steps:

1. Recognize the form:

This is a horizontal hyperbola because the $x^2$ term is positive. The standard form for such a hyperbola is:

$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1,$

where:

• $a^2 = 9$, so $a = 3$,
• $b^2 = 4$, so $b = 2$.

2. Plot the center:

The center of the hyperbola is at $(0, 0)$.

3. Plot the vertices:

For a horizontal hyperbola, the vertices lie along the $x$-axis at a distance of $a = 3$ from the center:

• Vertices: $(3, 0)$ and $(-3, 0)$.

4. Find the slopes of the asymptotes:

The asymptotes for a horizontal hyperbola are given by: $y = \pm \frac{b}{a} x.$

Here: $\text{slope} = \pm \frac{2}{3}.$

5. Sketch the asymptotes:

Draw the lines passing through the origin with slopes $\frac{2}{3}$ and $-\frac{2}{3}$.

6. Draw the hyperbola:

• Sketch the hyperbola opening left and right, approaching the asymptotes but never touching them.
• Ensure the hyperbola passes through the vertices $(3, 0)$ and $(-3, 0)$.

Would you like me to graph this hyperbola for you?