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The given curve is a downward-facing parabola, which means that the leading coefficient of the quadratic equation must be negative. This eliminates options **a** and **c**, as they both have positive coefficients for \( x^2 \).

Now let's focus on the remaining two options: **b** and **d**, both of which have a negative leading coefficient. 

- Looking at the graph, the parabola appears to intersect the y-axis at \( y = 1 \), which matches the y-intercept in **b**. 
- Option **d** has a y-intercept of \(-1\), which is not consistent with the graph.

Thus, the correct answer is **b. \( y = -2x^2 + 3x + 1 \)**.

Would you like more details, or have any questions?

Here are some related questions to explore:

1. How do you identify the direction of a parabola from its equation?
2. How do the coefficients of \(x^2\), \(x\), and the constant affect the shape of a parabola?
3. What are the steps to find the vertex of a quadratic equation?
4. How can you calculate the roots of a quadratic function?
5. How do transformations (like shifting) affect the graph of a parabola?

**Tip:** The sign of the coefficient of \(x^2\) determines whether a parabola opens upwards (positive) or downwards (negative).

Find the quadratic equation of the following curve?

Learn how to determine the quadratic equation from a given parabola graph. The problem focuses on understanding the leading coefficient, y-intercept, and the general form of quadratic functions. Suitable for students in Grades 9-12.

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