To solve the problem of graphing the function $f(x) = \frac{-8}{-x - 2}$, we will analyze the function step-by-step by identifying the asymptotes, key points, and sketching the graph.

Step 1: Rewrite the Function for Clarity

The given function is:

$f(x) = \frac{-8}{-x - 2}$

Simplify the denominator:

$f(x) = \frac{-8}{-(x + 2)} = \frac{8}{x + 2}$

Thus, the function becomes:

$f(x) = \frac{8}{x + 2}$

Step 2: Identify the Vertical Asymptote

A vertical asymptote occurs when the denominator equals zero because division by zero is undefined. Set $x + 2 = 0$:

$x = -2$

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

Step 3: Identify the Horizontal Asymptote

To determine the horizontal asymptote for a rational function:

• Compare the degrees of the numerator and denominator.

Here, the numerator $8$ is degree $0$ (constant), and the denominator $x + 2$ is degree $1$ (linear).

When the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is:

$y = 0$

Step 4: Plot Key Points

To sketch the graph, pick values of $x$ on either side of the vertical asymptote $x = -2$.

For $x = -3$:

$f(-3) = \frac{8}{-3 + 2} = \frac{8}{-1} = -8$ Point: $(-3, -8)$

For $x = -1$:

$f(-1) = \frac{8}{-1 + 2} = \frac{8}{1} = 8$ Point: $(-1, 8)$

Step 5: Sketch the Graph

1. Vertical asymptote at $x = -2$.
2. Horizontal asymptote at $y = 0$.
3. Plot the points $(-3, -8)$ and $(-1, 8)$ as reference points.

Step 6: Graph Behavior

• As $x \to -2^-$ (from the left), $f(x) \to -\infty$.
• As $x \to -2^+$ (from the right), $f(x) \to \infty$.
• As $x \to \pm \infty$, $f(x) \to 0$ (horizontal asymptote).

Final Answer

1. Vertical asymptote: $x = -2$
2. Horizontal asymptote: $y = 0$
3. Plot points $(-3, -8)$ and $(-1, 8)$.

Would you like more explanation or further graphing help? Here are some related questions:

1. How do you determine vertical asymptotes for rational functions?
2. How do horizontal asymptotes depend on the degree of polynomials?
3. What happens when a rational function has no horizontal asymptote?
4. How do you identify points to graph rational functions accurately?
5. What is the significance of asymptotes in real-world applications?

Tip: Always simplify the function first to make analysis easier!