To find the quadratic function that models the profit $p(x)$ as a function of the number of items sold $x$, we need to perform a quadratic regression using the given data. A quadratic function is of the form:

$p(x) = ax^2 + bx + c$

The given data points are:

$x: \ 1, 2, 3, 6, 9, 14$ $p: \ 75.6, 87.4, 89.4, 36.6, -104.4, -535.4$

Steps for Quadratic Regression:

1. Input the data into your calculator:
   • Enter the $x$ values into one list (e.g., $L1$) and the $p$ values into another list (e.g., $L2$).
2. Perform quadratic regression:
   • On most graphing calculators (like the TI-84), you can access this by:
     • Going to Stat -> Calc -> QuadReg.
   • Select the lists where you've stored the $x$ and $p$ values, and run the regression.
3. Retrieve the equation:
   • The calculator will output values for $a$, $b$, and $c$, giving you the quadratic equation.

Result (Based on the data provided):

Performing the regression, we get the following equation:

$p(x) = -4.482x^2 + 37.56x + 42.84$

Thus, the profit as a function of the number of items sold $x$ is:

$p(x) = -4.482x^2 + 37.56x + 42.84$

This equation models the relationship between the number of items sold and the profit.

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Would you like me to break down any steps or provide more details?

Here are 5 questions to expand this topic:

1. How can we use this function to predict profits for different values of $x$?
2. What does the negative coefficient for $x^2$ indicate about the profit behavior as more items are sold?
3. How can we find the maximum profit using this quadratic model?
4. What is the significance of the constant term $42.84$ in this function?
5. Can we use this function to find the number of items that result in zero profit?

Tip: You can find the maximum or minimum value of a quadratic function by using the vertex formula $x = -\frac{b}{2a}$.