https://file.aiquickdraw.com/mathbot/file/1729518885583eqnrzjs3.jpg

To solve this, we need to analyze the polynomial equation \( y = 3x^3 + 5x^2 - x + 3 \) and match its key features (end behavior, turning points, and zeros) with one of the provided graphs.

### Step-by-step approach:

1. **Degree and End Behavior**:
   - The equation is cubic (\(x^3\)), so its graph will have end behaviors characteristic of odd-degree polynomials.
   - As \(x \to +\infty\), the graph tends towards \(+\infty\) (since the leading coefficient, 3, is positive).
   - As \(x \to -\infty\), the graph tends towards \(-\infty\).
   - This behavior matches graphs **a** and **b**, where the right side of the graph goes up, and the left side goes down.

2. **Turning Points**:
   - A cubic function can have up to 2 turning points (local maxima and minima).
   - Graphs **a** and **b** show two turning points, while **c** and **d** have only one.

3. **Zeros (Roots)**:
   - To find the x-intercepts (zeros), we need to approximate or factor the polynomial. However, visually, we can check:
     - In graph **a**, the graph crosses the x-axis around \(x = -3\), \(x = 1\), and \(x = 2\), which suggests three real roots.
     - In graph **b**, there appear to be only two roots.
   
   Since we expect the polynomial to have three real roots, graph **a** seems more fitting.

### Conclusion:
Based on the end behavior, turning points, and the number of zeros, the graph that best represents the equation \( y = 3x^3 + 5x^2 - x + 3 \) is **graph a**.

Would you like further details or have any questions? Here are some related questions you might find helpful:

1. How can we algebraically find the roots of a cubic polynomial?
2. What does the derivative tell us about turning points of the graph?
3. How can we verify end behavior using limits?
4. What are the implications of the sign of the leading coefficient for the graph's orientation?
5. How can graph transformations affect the shape of a polynomial curve?

**Tip**: When analyzing polynomials, always focus on the degree (for end behavior) and the leading coefficient to predict the graph's overall shape.

Use end behaviours, turning points, and zeros to determine which graph represents the polynomial equation y = 3x^3 + 5x^2 - x + 3.

Learn how to match a polynomial graph to the equation y = 3x^3 + 5x^2 - x + 3 by analyzing end behaviors, turning points, and zeros. This problem introduces key concepts in graph analysis and cubic functions. Suitable for students in Grades 10-12.

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