Economics tells us that as a company raises its prices, it will sell fewer products. As a result, we can use a quadratic model to describe a company's profits,y , as a function of the price per item,x . The table below shows a certain company's price for one of its products and the annual profit from that product.

Price of item (dollars)
Profit (thousands of dollars)
10	132.0
15	169.5
20	188.0
25	187.5
30	168.0
35	129.5
40	72.0

Use the quadratic regression feature of your calculator to find the parabola of best fit for the data. Round coefficients to 3 decimal places.

The quadratic equation that best fits the data, based on the quadratic regression, is:

\[
y = -0.380x^2 + 17.000x - 0.000
\]

where $ y $ represents the profit (in thousands of dollars) and $ x $ is the price of the item (in dollars).

Do you want more details or have any questions?

Here are 5 related questions to explore:

1. How would the company's profits change if the price were increased to $45?
2. What is the price that maximizes the company's profits based on this model?
3. How would the company's profits change if the price were decreased to $5?
4. Can this model predict profits for prices outside the given range? What are the risks of extrapolating?
5. What factors, aside from price, could affect the accuracy of this quadratic model in predicting profits?

**Tip:** Quadratic regression is useful when there is a turning point in the data, which often models maximum or minimum values, like profit maximization in business cases.

Economics tells us that as a company raises its prices, it will sell fewer products. As a result, we can use a quadratic model to describe a company's profits, y, as a function of the price per item, x. The table below shows a certain company's price for one of its products and the annual profit from that product.

Price of item (dollars)
Profit (thousands of dollars)
10  132.0
15  169.5
20  188.0
25  187.5
30  168.0
35  129.5
40  72.0

Use the quadratic regression feature of your calculator to find the parabola of best fit for the data. Round coefficients to 3 decimal places.

This problem explores how to use quadratic regression to model a company's profit as a function of the price per item, based on data. Learn how to calculate the best-fitting quadratic equation using a calculator. Data points include prices ranging from $10 to $40, with profits in thousands of dollars.

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