Question

1. (20 points) daily revenues, in millions of dollars, for a corporation are given in the table below. use a system of equations to fit a quadratic function (f(x) = ax^2 + bx + c) to the data, where (x) is the number of days after day 4. round your coefficients to the nearest hundredth.

\

$$\begin{tabular}{|c|c|} \\hline day & revenue \\\\ \\hline 4 & 192 \\\\ 6 & 284 \\\\ 8 & 364 \\\\ \\hline \\end{tabular}$$

(f(x) =)

Explanation:

Step1: Define the independent variable

Let \(x\) be the number of days after Day 4.

$$x = \text{Day} - 4$$

Step2: Determine the data points

Calculate \(x\) and \(f(x)\) for each given day.

$$
LATEXBLOCK0
$$

Step3: Find the constant term

Substitute \(x = 0\) into the quadratic function.

$$f(0) = a(0)^2 + b(0) + c = 192 \implies c = 192$$

Step4: Set up system of equations

Substitute \(x = 2\) and \(x = 4\) with \(c = 192\).

$$
LATEXBLOCK1
$$

Step5: Solve for the coefficients

Solve the system of linear equations for \(a\) and \(b\).

$$
LATEXBLOCK2
$$

Step6: Write the final function

Substitute the coefficients back into the quadratic form.

$$f(x) = -1.5x^2 + 49x + 192$$

Answer:

\(f(x) = -1.5x^2 + 52x + 192\)