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The problem asks to fit a parabola \( y = a + bx + cx^2 \) to the given data and estimate \( y \) when \( x = 10 \). The data provided is as follows:

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

We need to perform a **quadratic regression** to find the best-fit coefficients \( a \), \( b \), and \( c \) for the model \( y = a + bx + cx^2 \). Then, we will use these coefficients to estimate the value of \( y \) at \( x = 10 \).

I will calculate these coefficients using the least-squares method and estimate \( y \) for \( x = 10 \). Let me calculate that.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like me to guide you through the quadratic regression steps manually, or do you want to try this calculation on your own?

Here are five related questions to further explore this problem:

1. How is quadratic regression different from linear regression?
2. What method can you use to solve the coefficients in a quadratic equation manually?
3. How can you check if a quadratic model fits the data well?
4. What are some real-life applications of fitting quadratic models to data?
5. How would the result change if we fit a different type of function (e.g., cubic) to the data?

**Tip:** To manually calculate the regression, you can create a system of equations based on the sum of squares of residuals, or you can use tools like Excel, Python, or a graphing calculator.

Fit a parabola y = a + bx + cx^2 to the following data and estimate y when x = 10. The data provided is: x = [1, 2, 3, 4, 5, 6, 7, 8, 9], y = [2, 6, 7, 8, 10, 11, 11, 10, 9]

Perform a quadratic regression to fit the equation y = a + bx + cx^2 to the data points x = [1, 2, 3, 4, 5, 6, 7, 8, 9], y = [2, 6, 7, 8, 10, 11, 11, 10, 9]. Learn how to find the best-fit coefficients using the least squares method and estimate y when x = 10.

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