To determine the value of $b$ in the quadratic regression equation $y = ax^2 + bx + c$ that models the profit, we follow these steps:

1. Input the data points from the table: $(x, y)$, where $x$ is the number of jackets sold, and $y$ is the profit.

   • Points: $(100, 7700), (120, 9144), (98, 7554), (106, 8136), (122, 9287), (105, 8067)$.

2. Use a quadratic regression method to calculate the coefficients $a$, $b$, and $c$. This can be done using a graphing calculator, software like Excel, or mathematical tools that support regression.

Let me compute the quadratic regression for this dataset. numpy as np

Data points (x: jackets sold, y: profit)

x = np.array([100, 120, 98, 106, 122, 105]) y = np.array([7700, 9144, 7554, 8136, 9287, 8067])

Perform quadratic regression using numpy's polyfit (degree 2 for quadratic)

coefficients = np.polyfit(x, y, 2) # Returns [a, b, c]

Extract the coefficient 'b' and round to two decimal places

b_coefficient = round(coefficients[1], 2) coefficients, b_coefficientThe quadratic regression equation is:

$y = -0.0475x^2 + 82.66x - 89.77$

The value of $b$ is 82.66 (rounded to two decimal places).

Would you like a detailed explanation of how the quadratic regression works or help with any related questions?

Here are five related questions to deepen your understanding:

1. How are the coefficients $a$, $b$, and $c$ interpreted in a quadratic regression equation?
2. What steps can be taken to perform this calculation manually without software?
3. Can you find the vertex of the parabola $y = -0.0475x^2 + 82.66x - 89.77$ and interpret its meaning in this context?
4. How does quadratic regression differ from linear regression, and when should it be used?
5. How do outliers in the data affect the results of quadratic regression?

Tip: For accuracy, always verify your regression calculations with a graph to ensure the model fits the data well.