To determine the equation of the rational function represented by the graph, let us analyze the key features of the graph and compare them with the given options:

1. Vertical Asymptote: The graph has a vertical asymptote at $x = 3$. This means the denominator of the rational function must have a factor of $x - 3$.

2. End Behavior: The graph's branches extend towards negative infinity on one side and positive infinity on the other, indicating that the function has a negative leading coefficient.

3. Slant Asymptote: The graph suggests a linear slant asymptote, confirming the numerator is linear.

Based on these observations:

• Options (a) and (b) have $x - 3$ in the denominator, consistent with the vertical asymptote.
• Option (a) has a negative leading coefficient, matching the behavior of the graph.
• Option (b) has a positive leading coefficient, which does not match the graph's behavior.

Thus, the correct answer is (a) $y = \frac{-2x + 1}{x - 3}$.

Would you like further clarification or detailed analysis?

Here are five related questions to deepen your understanding:

1. How do vertical asymptotes relate to the factors of the denominator?
2. What is the significance of the leading coefficient's sign in rational functions?
3. How can you find the equation of a rational function given a graph with multiple asymptotes?
4. What is the role of slant asymptotes in determining the numerator's degree relative to the denominator's?
5. How can graphing software or calculators help verify rational function equations?

Tip: Always check the behavior near vertical asymptotes and infinity to match the graph with the correct rational function equation.