The function given is $f(x) = \frac{x - 2}{x + 1}$, and the task is to graph it by placing horizontal and vertical asymptotes and an additional point on the graph.

Step 1: Identify Vertical Asymptote

The vertical asymptote occurs when the denominator of the function is zero, i.e., when $x + 1 = 0$. This gives $x = -1$, so the vertical asymptote is at $x = -1$.

Step 2: Identify Horizontal Asymptote

For rational functions, the horizontal asymptote depends on the degrees of the numerator and denominator. Here, both the numerator and denominator are linear (degree 1). The horizontal asymptote is determined by the ratio of the leading coefficients, which are both 1. Hence, the horizontal asymptote is at $y = 1$.

Step 3: Additional Point

We can choose a value for $x$ to plot an additional point. Let’s take $x = 0$: $f(0) = \frac{0 - 2}{0 + 1} = -2$ So, the point $(0, -2)$ is on the graph.

Graphing Steps:

• Draw a vertical asymptote at $x = -1$.
• Draw a horizontal asymptote at $y = 1$.
• Plot the point $(0, -2)$.
• The graph will approach the asymptotes as $x$ goes to infinity or negative infinity.

Would you like a detailed graph or further explanation?

Here are 5 related questions to expand your understanding:

1. How do you find vertical asymptotes for more complex rational functions?
2. What changes in the graph when the numerator has a higher degree than the denominator?
3. How would you determine the slant asymptote if it existed?
4. What happens to the graph if we add a constant to the numerator or denominator?
5. How would the graph look if we reversed the signs in the function?

Tip: Always check for holes in the graph when the numerator and denominator share common factors.