To estimate $y$ at $x = 6$ using a cubic interpolating polynomial, we will need to construct the polynomial using the given data points $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$, and $(x_4, y_4)$:

Given dataTo estimate $y$ at $x = 6$ using a cubic interpolating polynomial, we will need to construct the polynomial using the given data points $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$, and $(x_4, y_4)$:

Given

• $x_1 = 1$, $y_1 = 4$
• $x_2 = 5$, $y_2 = 5$
• $x_3 = 8$, $y_3 = 6$
• $x_4 = 9$, $y_4 = 12$

The general form of a cubic polynomial is:

$P(x) = ax^3 + bx^2 + cx + d$

We will determine the coefficients $a$, $b$, $c$, and $d$ by solving the system of equations formed by substituting the given points into the polynomial.

Step 1: Set up the system of equations

1. $P(1) = 4$ $a(1)^3 + b(1)^2 + c(1) + d = 4$ $a + b + c + d = 4 \quad \text{(Equation 1)}$

2. $P(5) = 5$ $a(5)^3 + b(5)^2 + c(5) + d = 5$ $125a + 25b + 5c + d = 5 \quad \text{(Equation 2)}$

3. $P(8) = 6$ $a(8)^3 + b(8)^2 + c(8) + d = 6$ $512a + 64b + 8c + d = 6 \quad \text{(Equation 3)}$

4. $P(9) = 12$ $a(9)^3 + b(9)^2 + c(9) + d = 12$ $729a + 81b + 9c + d = 12 \quad \text{(Equation 4)}$

Step 2: Solve the system of equations

We now have a system of four equations, which can be solved for the coefficients $a$, $b$, $c$, and $d$. I will compute the solution.The cubic polynomial that passes through the given points has the following coefficients:

$a = \frac{59}{336}, \quad b = \frac{-137}{56}, \quad c = \frac{3187}{336}, \quad d = \frac{-45}{14}$

Thus, the interpolating polynomial is:

$P(x) = \frac{59}{336}x^3 - \frac{137}{56}x^2 + \frac{3187}{336}x - \frac{45}{14}$

Step 3: Estimate $y$ at $x = 6$

We will substitute $x = 6$ into this polynomial to estimate $y$. Let me compute that next.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can substitute $x = 6$ into the cubic polynomial equation manually:

$P(6) = \frac{59}{336}(6)^3 - \frac{137}{56}(6)^2 + \frac{3187}{336}(6) - \frac{45}{14}$

Feel free to reach out if you want help with another problem or have questions!

Here are 5 relative questions that may expand your understanding:

1. What is the difference between cubic interpolation and linear interpolation?
2. How does cubic spline interpolation compare to polynomial interpolation?
3. What are the advantages of using higher-degree polynomials in interpolation?
4. Can you explain how to find derivatives of interpolating polynomials?
5. What are the limitations of using interpolating polynomials for extrapolation?

Tip: Polynomial interpolation works best for evenly spaced data and should be used cautiously for extrapolation due to possible oscillations.