To find the quadratic regression curve $y(x) = ax^2 + bx + c$ for the given points (1, 4), (3, 6), (4, 5), and (5, 3), we can use a quadratic regression method, typically solved using a matrix or statistical tool. Let's proceed step-by-step:

The quadratic regression equation is:

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

We need to solve for the coefficients $a$, $b$, and $c$ that minimize the error between the quadratic model and the given points.

1. Set up the system of equations using the points:

For each point $(x, y)$, substitute into the quadratic equation $y = ax^2 + bx + c$.

• For $(1, 4)$: $4 = a(1)^2 + b(1) + c$ → $a + b + c = 4$
• For $(3, 6)$: $6 = a(3)^2 + b(3) + c$ → $9a + 3b + c = 6$
• For $(4, 5)$: $5 = a(4)^2 + b(4) + c$ → $16a + 4b + c = 5$
• For $(5, 3)$: $3 = a(5)^2 + b(5) + c$ → $25a + 5b + c = 3$

2. Solve the system of equations:

We now solve the following system of equations:

1. $a + b + c = 4$
2. $9a + 3b + c = 6$
3. $16a + 4b + c = 5$
4. $25a + 5b + c = 3$

I'll calculate the values of $a$, $b$, and $c$.

It seems I overestimated the number of points required for a quadratic regression (3 points would suffice). However, let's use a more appropriate method to perform quadratic regression for these points, and I'll determine the correct coefficients using a different approach.

Let me try a different calculation method.The quadratic regression equation that best fits the given points is:

$y(x) = -0.6136x^2 + 3.4227x + 1.2000$

Would you like further details or have any questions?

Here are five related questions to expand on this concept:

1. How is quadratic regression different from linear regression?
2. Can you use this method to predict values for $x$ beyond the given points?
3. How can technology like graphing calculators or software be used to check regression results?
4. How do you interpret the coefficients in a quadratic regression equation?
5. What are the conditions under which a quadratic regression model is appropriate?

Tip: You can use quadratic regression to model data that forms a parabolic shape, indicating the presence of a maximum or minimum.