To graph the rational function $f(x) = \frac{3x + 3}{x + 2}$, we will follow these steps:

1. Identify the vertical asymptote: This occurs where the denominator is zero. For $x + 2 = 0$, $x = -2$.

2. Identify the horizontal asymptote: For a rational function $\frac{P(x)}{Q(x)}$ where the degree of $P(x)$ and $Q(x)$ are the same, the horizontal asymptote is $\frac{\text{leading coefficient of } P(x)}{\text{leading coefficient of } Q(x)}$. Here, it is $y = \frac{3}{1} = 3$.

3. Find the x-intercept: Set the numerator equal to zero and solve for $x$. For $3x + 3 = 0$, $x = -1$.

4. Find the y-intercept: Set $x = 0$. For $f(0) = \frac{3(0) + 3}{0 + 2} = \frac{3}{2}$.

5. Sketch the graph:

   • Plot the asymptotes.
   • Plot the intercepts.
   • Determine the behavior around the asymptotes by choosing points near $x = -2$ and analyzing the sign of $f(x)$.

Let's plot the graph based on these findings.

Graphing Steps:

1. Vertical Asymptote: $x = -2$
2. Horizontal Asymptote: $y = 3$
3. x-intercept: $x = -1$
4. y-intercept: $y = \frac{3}{2}$

Here's a rough sketch of the graph:

1. Draw the vertical asymptote at $x = -2$.
2. Draw the horizontal asymptote at $y = 3$.
3. Plot the intercepts: $(-1, 0)$ and $(0, \frac{3}{2})$.
4. Behavior around asymptotes: For values of $x$ approaching -2 from the left, $f(x)$ will tend to $-\infty$, and from the right, $f(x)$ will tend to $\infty$.

For a precise and neat graph, it is advisable to use graphing software or graphing calculator tools. However, this sketch will provide a good starting point for understanding the behavior of the function.

Would you like more detailed steps on how to plot it precisely, or do you have any other questions?

Here are some related questions you might find useful:

1. What is the behavior of the function as $x \to \pm \infty$?
2. How do you find oblique asymptotes for rational functions?
3. Can you determine the domain of the function?
4. How would you find the critical points of the function?
5. What is the significance of horizontal asymptotes in the graph of a function?
6. How does the degree of the numerator and denominator affect the asymptotes?
7. What happens to the graph of $f(x)$ if you change the coefficients of $x$ and the constant term?
8. Can you explain the difference between holes and vertical asymptotes in rational functions?

Tip: Always analyze the end behavior and asymptotes first when graphing rational functions to understand their overall shape and direction.