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You have provided a problem where we are asked to fit a parabola of the form \( y = a + bx + cx^2 \) to a given set of data points, with \( x \) as the independent variable. The data table is:

\[
\begin{array}{|c|c|c|c|c|c|c|c|c|c|}
\hline
x & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
y & 2 & 6 & 7 & 8 & 10 & 11 & 11 & 10 & 9 \\
\hline
\end{array}
\]

### To solve this:

We need to use the method of least squares to find the best-fitting values of \( a \), \( b \), and \( c \) for the quadratic model \( y = a + bx + cx^2 \).

This involves solving the system of normal equations derived from minimizing the sum of squared residuals. We will set up the following system:

\[
\begin{aligned}
\sum y &= a \cdot n + b \sum x + c \sum x^2, \\
\sum xy &= a \sum x + b \sum x^2 + c \sum x^3, \\
\sum x^2y &= a \sum x^2 + b \sum x^3 + c \sum x^4,
\end{aligned}
\]
where \( n \) is the number of data points (in this case, \( n = 9 \)).

Let me calculate the necessary sums from the data table and solve for \( a \), \( b \), and \( c \).The best-fitting quadratic equation for the given data points is:

\[
y = -0.93 + 3.52x - 0.27x^2
\]

Here, the coefficients are:
- \( a = -0.93 \) (approximately),
- \( b = 3.52 \) (approximately),
- \( c = -0.27 \) (approximately).

Would you like to go into more detail on any part of this process?

Here are some related questions you could explore:
1. How can we check the goodness of fit for this model?
2. What is the significance of the coefficients \( a \), \( b \), and \( c \) in real-world data fitting?
3. How would this process change for a cubic or higher-degree polynomial fit?
4. Can we visualize the fitted parabola along with the given data points?
5. How do we handle outliers in least squares regression?

**Tip**: When fitting a curve to data, always verify the model by calculating the residuals (differences between observed and predicted values) to check for systematic errors.

Fit a parabola of the form y = a + bx + cx^2 to the following data by taking 'x' as the independent variable: x: 1, 2, 3, 4, 5, 6, 7, 8, 9; y: 2, 6, 7, 8, 10, 11, 11, 10, 9.

Learn how to fit a parabola to the data points (x: 1-9, y: 2-9) using the least squares method. This problem involves quadratic curve fitting and solving for coefficients a, b, and c. Suitable for advanced high school or college students.

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