To analyze the given rational function $f(x) = \frac{-3}{x + 2}$, let's identify the horizontal and vertical asymptotes.

1. Horizontal Asymptote:

The horizontal asymptote for a rational function $f(x) = \frac{a}{x + b}$ (where $a$ and $b$ are constants and $x$ appears only in the denominator) is $y = 0$. This is because as $x$ approaches infinity or negative infinity, $f(x)$ approaches zero.

For this function: $y = 0$ So, you should draw a horizontal line along the $x$-axis in the left graph and write $y = 0$ as the horizontal asymptote.

2. Vertical Asymptote:

The vertical asymptote occurs where the denominator equals zero, as this makes the function undefined.

Set the denominator equal to zero: $x + 2 = 0 \Rightarrow x = -2$ Thus, $x = -2$ is the vertical asymptote. Draw a vertical line at $x = -2$ in the right graph and write $x = -2$ as the vertical asymptote.

Would you like more details on asymptotes or have any questions?

Follow-up Questions:

1. What are asymptotes, and how do they affect the shape of a graph?
2. How can we determine if a function has an oblique asymptote?
3. How do horizontal and vertical asymptotes influence the limits of a function?
4. How do we find asymptotes for more complex rational functions?
5. What are some real-world examples where asymptotes appear in graphs?

Tip:

For rational functions, remember that horizontal asymptotes depend on the degrees of the numerator and denominator, while vertical asymptotes are found by setting the denominator equal to zero.