The task requires finding a quadratic regression curve that fits the given points: $(0, 0)$, $(-3, 3)$, $(2, 2)$, and $(-5, 5)$. The equation of a quadratic function is generally:

$y(x) = ax^2 + bx + c$

To find the coefficients $a$, $b$, and $c$, I will use quadratic regression. Let me calculate it.The quadratic regression equation that fits the given points $(0, 0)$, $(-3, 3)$, $(2, 2)$, and $(-5, 5)$ is:

$y(x) = 0.2x^2 + 0.0828x + 0.7241$

Would you like further details or have any questions?

Here are 5 related questions for practice:

1. How would the regression change if another point was added to the dataset?
2. How does the least squares method find the best-fitting quadratic curve?
3. Can quadratic regression be applied to non-parabolic datasets effectively?
4. How do outliers affect the accuracy of quadratic regression?
5. What are the differences between linear and quadratic regression?

Tip: Always ensure that your dataset is appropriate for quadratic modeling before applying quadratic regression. If the data isn't parabolic, consider alternative models.