he table below shows the profit, �(�), in dollars, from selling � items.

**�**

1

2

3

6

9

14

**�(�)**

78.4

92

95.8

48.4

-87.2

-509.2

Using your calculator to do a quadratic regression, express the profit as a function of the number of items sold.
�(�)=  
*Round all numbers to 1 decimal place.*



Using your quadratic regression, estimate the profit from selling 8 items?  Select an answer items dollars 
*Round to 2 decimal places.*

Using your quadratic regression, estimate the number of items that need to be sold to make a profit of $13 ?  Select an answer items dollars 

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.

The table below shows the profit, P(x), in dollars, from selling x items. x: [1, 2, 3, 6, 9, 14], P(x): [78.4, 92, 95.8, 48.4, -87.2, -509.2]. Using your calculator to do a quadratic regression, express the profit as a function of the number of items sold. P(x) = ? Round all numbers to 1 decimal place. Using your quadratic regression, estimate the profit from selling 8 items? Round to 2 decimal places. Using your quadratic regression, estimate the number of items that need to be sold to make a profit of $13?

Learn how to express the profit from selling items as a quadratic function using regression. The problem involves calculating profit based on the number of items sold, with data points including [1, 2, 3, 6, 9, 14] for items sold and corresponding profits. Includes estimating profit for selling 8 items and determining the number of items needed to reach a $13 profit.

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