We need to perform a quadratic regression to express the profit $P(x)$ as a function of the number of items sold $x$. A quadratic equation has the general form:

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

Step 1: Perform Quadratic Regression

Using the data points provided:

• $x = [1, 2, 3, 6, 9, 14]$
• $P(x) = [78.4, 92, 95.8, 48.4, -87.2, -509.2]$

We'll calculate the quadratic regression coefficients $a$, $b$, and $c$.

Step 2: Estimate Profit for Selling 8 Items

Once we obtain the quadratic equation $P(x)$, we'll substitute $x = 8$ to estimate the profit from selling 8 items.

Step 3: Estimate the Number of Items for a Profit of $13

We'll solve the quadratic equation for $P(x) = 13$ to find the number of items that need to be sold to make a profit of $13.

Let me calculate these values for you.The quadratic regression equation for the profit as a function of the number of items sold is:

$P(x) = -4.9x^2 + 28.3x + 55.0$

1. Profit from selling 8 items:

Substituting $x = 8$ into the equation:

$P(8) = -32.2 \, \text{dollars}$

Thus, the estimated profit from selling 8 items is $-32.20$ dollars.

2. Number of items to be sold for a profit of $13:

By solving the equation $P(x) = 13$, we find two possible solutions:

$x \approx 7 \quad \text{or} \quad x \approx -1.22$

Since the number of items cannot be negative, the estimated number of items needed to achieve a profit of $13 is approximately 7 items.

Let me know if you need further details, or if you have any questions!

Here are some related questions:

1. What is the meaning of negative profit in this context?
2. How would the shape of the graph change for larger values of $x$?
3. What are some limitations of using quadratic regression for this dataset?
4. Can the quadratic model be used to predict profits for extremely high values of $x$?
5. How would a linear regression compare to a quadratic regression for this data?

Tip: Always check the model's predictions for reasonable values within the range of your data.