To graph the function \( y = \frac{6x - 1}{3x} \), we need to follow these steps:

1. **Simplify the function**: Start by simplifying the given expression.
   \[
   y = \frac{6x - 1}{3x}
   \]
   We can split the fraction:
   \[
   y = \frac{6x}{3x} - \frac{1}{3x} = 2 - \frac{1}{3x}
   \]
   Thus, the simplified function is:
   \[
   y = 2 - \frac{1}{3x}
   \]

2. **Identify asymptotes and domain**:
   - **Vertical asymptote**: Occurs where the denominator is zero, i.e., \( x = 0 \).
   - **Horizontal asymptote**: As \( x \) approaches infinity or negative infinity, \( \frac{1}{3x} \) approaches zero. Thus, the horizontal asymptote is \( y = 2 \).
   - **Domain**: The function is defined for all \( x \neq 0 \).

3. **Plot key points**:
   - Calculate the function value at key points:
     - When \( x = 1 \):
       \[
       y = 2 - \frac{1}{3(1)} = 2 - \frac{1}{3} = \frac{5}{3}
       \]
     - When \( x = -1 \):
       \[
       y = 2 - \frac{1}{3(-1)} = 2 + \frac{1}{3} = \frac{7}{3}
       \]
     - When \( x = 3 \):
       \[
       y = 2 - \frac{1}{3(3)} = 2 - \frac{1}{9} = \frac{17}{9}
       \]
     - When \( x = -3 \):
       \[
       y = 2 - \frac{1}{3(-3)} = 2 + \frac{1}{9} = \frac{19}{9}
       \]

4. **Graph the function**:
   - Plot the asymptotes.
   - Plot the key points calculated.
   - Draw the curve approaching the asymptotes and passing through the key points.

Here's a rough sketch of the graph:

\[
\begin{array}{c|c}
\text{x} & \text{y} \\
\hline
1 & \frac{5}{3} \\
-1 & \frac{7}{3} \\
3 & \frac{17}{9} \\
-3 & \frac{19}{9} \\
\end{array}
\]

Graph:
```
    y
    |
3   |          *
    |          *                       *
2   |----------+-----------------------+---- x
    |                  *
    |
1   |                
    |                
    |
    |
    +------------------------------------
```

Would you like further details or have any questions?

Here are some additional questions to consider:
1. What is the behavior of the function as \( x \) approaches zero from the positive side?
2. What is the behavior of the function as \( x \) approaches zero from the negative side?
3. How does the graph change if the numerator is altered, for example, \( 6x - 1 \) to \( 6x + 1 \)?
4. What happens to the graph if the denominator changes, for example, \( 3x \) to \( x \)?
5. How would you find the intersection points with the x-axis and y-axis?

**Tip:** Always check for vertical asymptotes by setting the denominator equal to zero, as these are critical in understanding the behavior of the function.

Learn how to graph the rational function y = (6x - 1) / 3x by simplifying the expression, identifying vertical and horizontal asymptotes, and plotting key points. This solution covers key concepts in algebra and is suitable for students in Grades 10-12.

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