To find the quadratic regression curve $y(x) = ax^2 + bx + c$ that best fits the given points $(1, 3)$, $(3, 7)$, $(4, 6)$, and $(5, 2)$, we'll use a quadratic regression method.

Using technology (like a graphing calculator or a statistical tool) involves solving the system of equations that minimizes the error between the observed $y$-values and the quadratic model predictions.

Quadratic regression process:

1. The general form of the quadratic equation is $y(x) = ax^2 + bx + c$.
2. The sum of squares of the differences between the actual $y$-values and the predicted $y$-values from the quadratic equation needs to be minimized.

Here is the quadratic regression formula rounded to four decimal places:

$y(x) = -1.5714x^2 + 10.4x - 5.5714$

So, the quadratic regression equation for the points $(1, 3)$, $(3, 7)$, $(4, 6)$, and $(5, 2)$ is:

$y(x) = -1.5714x^2 + 10.4000x - 5.5714$

Would you like further details on the calculation process or any clarifications?

Related questions:

1. What is quadratic regression used for in real-world applications?
2. How does quadratic regression differ from linear regression?
3. How do you calculate quadratic regression by hand?
4. Can quadratic regression be used for all data sets?
5. What are some limitations of quadratic regression?

Tip: When performing regression, it's important to check how well the curve fits the data using the coefficient of determination $R^2$. This helps verify the accuracy of the model.