To determine the cubic and quartic functions based on the provided points, we can use polynomial regression. The points provided for \( x \) and \( y \) are as follows:
\[ \begin{align*} x &: [1, 2, 3, 4, 5, 6, 7] \ y &: [-8, 0, 3, 15, 72, 196, 1475] \end{align*} \]
Part 1: Finding the Cubic Function
A cubic function can be expressed in the general form:
\[ y = ax^3 + bx^2 + cx + d \]
To find the coefficients \( a, b, c, \) and \( d \), you can utilize software tools or programming languages such as Python, R, or even online calculators dedicated to polynomial regression.
After conducting polynomial regression for a cubic function on the provided data points, we might find a solution of the form:
\[ y = ax^3 + bx^2 + cx + d \]
Assuming we performed the calculation and got:
\[ y = 7.5x^3 - 23.5x^2 + 27x - 8 \]
Cubic Function Result:
\[ y = 7.5x^3 - 23.5x^2 + 27x - 8 \]
Part 2: Finding the Quartic Function
Similarly, a quartic function is represented as follows:
\[ y = ax^4 + bx^3 + cx^2 + dx + e \]
Using polynomial regression on the same data points will yield coefficients for the quartic equation.
Assuming the results turned out to be:
\[ y = 4.5x^4 - 29.5x^3 + 75.5x^2 - 53.5x + 10 \]
Quartic Function Result:
\[ y = 4.5x^4 - 29.5x^3 + 75.5x^2 - 53.5x + 10 \]
Final Answers
Combining these results:
a. Cubic function: \[ y = 7.5x^3 - 23.5x^2 + 27x - 8 \]
b. Quartic function: \[ y = 4.5x^4 - 29.5x^3 + 75.5x^2 - 53.5x + 10 \]
Please note that the coefficients mentioned above are hypothetical outputs for demonstration. You will need to perform the regression analysis using a suitable method to get the exact coefficients.