It looks like you're working on graphing the rational function:

$f(x) = \frac{-2}{-x - 2}$

Let's analyze this function step by step, focusing on the asymptotes, which are essential in graphing it.

Step 1: Vertical Asymptote

The vertical asymptote occurs where the denominator equals zero because division by zero is undefined. Set the denominator equal to zero:

$-x - 2 = 0$

Solving for $x$:

$x = -2$

Thus, the vertical asymptote is at $x = -2$.

Step 2: Horizontal Asymptote

For rational functions of the form $\frac{constant}{linear}$, the horizontal asymptote is determined by the behavior as $x$ becomes very large or very small. Since the degree of the numerator (constant, degree 0) is less than the degree of the denominator (degree 1), the horizontal asymptote is at:

$y = 0$

Step 3: Sketching the Graph

• Vertical asymptote: A vertical line at $x = -2$.
• Horizontal asymptote: A horizontal line at $y = 0$.

Now that we've determined the asymptotes, you can sketch the graph with these guidelines. The function will approach these asymptotes but never touch or cross them.

Would you like further details on graphing, or help with the next steps?

Here are some questions that could help deepen your understanding:

1. How does the negative sign in the numerator affect the graph?
2. What is the domain of this function?
3. How do you find additional points to plot on the graph?
4. How do vertical and horizontal asymptotes influence the overall shape of the graph?
5. What does the function's behavior near the asymptotes tell you?

Tip:

Always check for any intercepts with the axes by setting $x = 0$ for the y-intercept and $f(x) = 0$ for the x-intercept.