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

1. **Identify and draw the vertical asymptote**: The vertical asymptote occurs where the denominator is zero. So, we solve:
   \[
   -x + 3 = 0 \implies x = 3
   \]
   Therefore, the vertical asymptote is \( x = 3 \).

2. **Identify and draw the horizontal asymptote**: For large values of \( x \), the function \( f(x) \) can be approximated by the ratio of the leading coefficients of the numerator and the denominator. Thus, we have:
   \[
   f(x) \approx \frac{2x}{-x} = -2
   \]
   Therefore, the horizontal asymptote is \( y = -2 \).

3. **Find and plot the x-intercept**: The x-intercept occurs where the numerator is zero. So, we solve:
   \[
   2x - 3 = 0 \implies x = \frac{3}{2}
   \]
   Therefore, the x-intercept is at \( \left( \frac{3}{2}, 0 \right) \).

4. **Find and plot the y-intercept**: The y-intercept occurs where \( x = 0 \). So, we calculate:
   \[
   f(0) = \frac{2(0) - 3}{-0 + 3} = \frac{-3}{3} = -1
   \]
   Therefore, the y-intercept is at \( (0, -1) \).

5. **Plot additional points**: To better sketch the graph, we can calculate a few more points by plugging in values of \( x \) into the function.

Now let's plot these points and draw the asymptotes on the graph:

1. **Vertical Asymptote**: \( x = 3 \)
2. **Horizontal Asymptote**: \( y = -2 \)
3. **x-intercept**: \( \left( \frac{3}{2}, 0 \right) \)
4. **y-intercept**: \( (0, -1) \)

Here is the graph:

1. Draw a dashed vertical line at \( x = 3 \).
2. Draw a dashed horizontal line at \( y = -2 \).
3. Plot the points \( \left( \frac{3}{2}, 0 \right) \) and \( (0, -1) \).
4. Draw the curve of the rational function, which approaches the asymptotes but never touches them.

If you need a detailed graph or further assistance, let me know.

Would you like more details or have any questions?

Here are 5 numbered questions you can ask:
1. How do you find the asymptotes of a rational function?
2. What are the steps to plot a rational function?
3. How do you determine the x-intercept of a rational function?
4. How do you determine the y-intercept of a rational function?
5. What are the characteristics of rational function graphs?

**Tip:** When graphing a rational function, always start with identifying and plotting the asymptotes, as they provide the framework for how the function behaves.

Learn how to graph the rational function f(x) = (2x - 3) / (-x + 3) with detailed steps including finding asymptotes, intercepts, and plotting points. Understand the behavior of rational functions in this comprehensive guide.

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