Complete parts​ (a)-(c).
x
1
2
3
4
5
6

y
negative 2−2
negative 1−1
00
44
66
1414
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Part 1
​a) Find a cubic function that models the data in the table. Report the model with three decimal places.
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Part 2
​b) Find a linear function that models the data. Report the model with three decimal places.
yequals=enter your response here
Part 3
​c) Visually determine which model is the better fit for the data.

Linear

Cubic

The plot shows the data points (in red) along with the cubic (blue) and linear (green) models. Visually, the cubic model seems to fit the data points more closely, especially capturing the curvature in the data, whereas the linear model is a straight line and does not capture the changes in slope as effectively.

**Conclusion**: The cubic model is the better fit for the data compared to the linear model.

Do you have any questions or need further details?

### Related Questions
1. How can you calculate the residuals for both models to quantify the fit?
2. What is the significance of each coefficient in the cubic equation?
3. How would the models change if another data point was added?
4. How does the sum of squared errors differ between the two models?
5. Can you derive the cubic and linear equations by hand using matrix operations?
6. What is the role of the constant term in both the cubic and linear equations?
7. How do the graphs of higher-degree polynomials generally behave compared to linear functions?
8. What are some real-world examples where a cubic model would be more appropriate than a linear model?

**Tip**: Always check the residuals and other goodness-of-fit measures, such as R-squared, to compare different models quantitatively.

Explore how cubic and linear functions are used to model data points in a mathematical problem. Understand the process of selecting the best-fit model visually and quantitatively. Suitable for Grades 9-12 students.

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