Find the cubic function that is the best fit for the data in the table below.
b. Find the quartic function that is the best fit for the data in the table.

x
1
2
3
4
5
6
7
y
-4
0
10
30
5
190
1500

To find the cubic and quartic functions that best fit the given data, we can use polynomial regression. Let's start by fitting a cubic function \( y = ax^3 + bx^2 + cx + d \) and a quartic function \( y = ax^4 + bx^3 + cx^2 + dx + e \) to the data points provided. The goal is to determine the coefficients \(a\), \(b\), \(c\), and \(d\) (and \(e\) for the quartic function) that minimize the difference between the actual data points and the values predicted by the polynomial functions.

The data points are:
- (1, -4)
- (2, 0)
- (3, 10)
- (4, 30)
- (5, 5)
- (6, 190)
- (7, 1500)

Let's calculate the coefficients for both cubic and quartic fits.The best-fit cubic and quartic functions for the given data points are:

### Cubic Function:
\[ y = 36.64x^3 - 352.58x^2 + 980.06x - 724.86 \]

### Quartic Function:
\[ y = 12.70x^4 - 166.57x^3 + 745.13x^2 - 1298.82x + 712.14 \]

These functions represent the polynomials that best fit the provided data points.

Would you like more details or have any questions about the fitting process?

Here are some further explorations you might find interesting:
1. How do these polynomial functions compare visually to the data points?
2. What is the residual sum of squares (RSS) for each polynomial fit?
3. How would the best-fit function change with more data points?
4. Can these models be used for reliable extrapolation beyond the given data range?
5. What are the limitations of using polynomial regression for fitting data?
6. How sensitive are the polynomial coefficients to slight changes in the data points?
7. What is the meaning of overfitting, and how does it relate to the degree of the polynomial?
8. How can we evaluate the goodness-of-fit for these models?

**Tip:** Higher-degree polynomials can fit more complex data but may also lead to overfitting, where the model describes random noise rather than the underlying relationship.

Learn how to find the best fit cubic and quartic functions for a given set of data points using polynomial regression. Detailed solutions and coefficients provided.

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