To graph the function $f(x) = \frac{x - 2}{x + 1}$, we need to analyze its horizontal and vertical asymptotes and plot an additional point. Here’s the step-by-step process:

1. Identify the Vertical Asymptote

The vertical asymptote occurs where the denominator is zero (since division by zero is undefined).

For $f(x) = \frac{x - 2}{x + 1}$:

• Set $x + 1 = 0$, which gives $x = -1$.

So, the vertical asymptote is at $x = -1$.

2. Identify the Horizontal Asymptote

For rational functions, the horizontal asymptote is determined by comparing the degrees of the polynomial in the numerator and the polynomial in the denominator.

In this case, both the numerator and denominator are degree 1 polynomials. When the degrees are the same, the horizontal asymptote is the ratio of the leading coefficients.

Since the leading coefficients of $x$ in both the numerator and denominator are 1:

• The horizontal asymptote is $y = \frac{1}{1} = 1$.

So, the horizontal asymptote is at $y = 1$.

3. Plot an Additional Point

To better sketch the graph, let’s find an additional point on the curve.

Choose $x = 0$ (a simple point to calculate): $f(0) = \frac{0 - 2}{0 + 1} = \frac{-2}{1} = -2$ So, the point $(0, -2)$ is on the graph.

4. Sketching the Graph

• Draw the vertical asymptote as a dashed line at $x = -1$.
• Draw the horizontal asymptote as a dashed line at $y = 1$.
• Plot the point $(0, -2)$.
• The function will approach the asymptotes but never cross them.

Would you like a visual sketch or further details on interpreting asymptotes?

Here are some follow-up questions:

1. What happens to the graph of $f(x)$ as $x$ approaches positive or negative infinity?
2. How would the graph change if we altered the numerator to $x + 2$?
3. How do vertical and horizontal asymptotes help in understanding the behavior of rational functions?
4. What other points would be useful to plot for a more accurate graph of $f(x)$?
5. Can a rational function have both a horizontal and an oblique asymptote?

Tip: Always check for simplification opportunities in rational functions before analyzing asymptotes and plotting points.