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Understanding the slope is crucial when dealing with linear functions, such as the demand function for Blu-ray sales. It's essentially the "steepness" of the line graph or how much one variable changes concerning a change in another. To calculate the slope, we need two data points. In our scenario, these data points are prices and units sold: (250, 12) and (200, 15).

The formula for calculating the slope, "m," is \[ m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \]
Here,

• \(y_1 = 12\) and \(y_2 = 15\) represent the number of units sold.
• \(x_1 = 250\) and \(x_2 = 200\) are the prices.

Plugging these into the formula gives us:\[ m = \frac{{15 - 12}}{{200 - 250}} = \frac{3}{-50} = -\frac{1}{16} \]
Therefore, the slope of our demand function is \(-\frac{1}{16}\), indicating that for every $1 decrease in price, 1/16 more units are sold.

The y-intercept is a crucial component in formulating the demand equation. It represents the point where the line crosses the y-axis, meaning the theoretical demand when the price is zero. To find the y-intercept, or "b," in the linear equation \(y = mx + b\), we need the slope "m" and one data point.

From our slope calculation, \(m = -\frac{1}{16}\), and choosing the point \((250, 12)\), we substitute these into the equation:

• \(12 = \left(-\frac{1}{16}\right) \cdot 250 + b\)
• Solve for \(b\).

This gives:

• \(12 = -15.625 + b\)
• \(b = 12 + 15.625 = 27.625\)

The y-intercept, therefore, is \(27.625\). In the context of the demand function, it suggests that if Blu-ray players were free, an estimated 27.625 units would be demanded per day.

Graphing the demand function helps visualize how price impacts sales. The graph represents our determined linear equation \(D(p) = -\frac{1}{16}p + 27.625\). To create this graph, you use both the slope and the y-intercept.

Begin by plotting the y-intercept on the graph, the point \((0, 27.625)\). This is where the line will begin on the y-axis. Then use the slope to determine the direction and steepness of your line:

• Start at \((0, 27.625)\).
• Move down 1 unit and right 16 units (following the slope \(-\frac{1}{16}\)).

From these two points, draw a straight line, extending it across the graph. The resulting line shows the negative relationship between price and demand: as price decreases, demand increases. This visual representation helps in quickly understanding how pricing impacts sales.

The domain of the demand function defines the range of potential prices. It determines the input values (price) for which the function (demand) is valid. In real-world scenarios, particularly those involving sales, we must consider realistic values to represent prices.

In this case, the domain is driven by the condition that prices must be non-negative, as negative prices don't make sense. Thus, the domain of our Blu-ray player demand function becomes:

• All non-negative real numbers, or mathematically, \(p \geq 0\).

This domain ensures that our demand equation aligns with practical pricing strategies. By definition, it restricts the function to sensible pricing values usable by a business to predict sales.