To fit a quadratic model of the form $y(x) = ax^2 + b$ to the points $(-1, -2)$, $(1, -1)$, and $(2, 1)$ using the least squares method, we aim to minimize the sum of squared residuals.

1. Set up the system of equations

We want to find $a$ and $b$ such that the difference between the predicted $y$ values (from $y = ax^2 + b$) and the actual $y$ values is minimized. The residual for each data point $(x_i, y_i)$ is:

$r_i = y_i - (ax_i^2 + b)$

The goal is to minimize the sum of squared residuals:

$S(a, b) = \sum_{i=1}^{3} r_i^2 = \sum_{i=1}^{3} (y_i - (ax_i^2 + b))^2$

Let's write out the system of equations explicitly using the three points $(-1, -2)$, $(1, -1)$, and $(2, 1)$:

• For $(-1, -2)$: $-2 = a(-1)^2 + b = a + b$
• For $(1, -1)$: $-1 = a(1)^2 + b = a + b$
• For $(2, 1)$: $1 = a(2)^2 + b = 4a + b$

This gives us the following system of equations:

a + b &= -2 \\ a + b &= -1 \\ 4a + b &= 1 \end{aligned}$$ ### 2. Solve the system of equations We can solve these equations step-by-step. The first two equations can be simplified as they both have $$a + b$$: $$a + b = -2 \quad \text{and} \quad a + b = -1$$ This is a contradiction. The system is inconsistent, and therefore, the exact quadratic model $$y(x) = ax^2 + b$$ cannot perfectly fit the three data points. ### 3. Conclusion: We need a model with more parameters To better fit the points, we would need to introduce another term (e.g., a linear term or a cubic term). However, based on your request, there seems to be a misunderstanding in the problem setup. Would you like to proceed with a different model, or should we continue with a better-suited form for your question?