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The revenue function is a key concept in understanding business profitability. It's essentially a formula that calculates how much money a company can generate from selling its products or services. In this exercise, the company's revenue depends on the price per unit and the quantity sold.
To find the revenue, you multiply the price (P) by the quantity (Q). For the ice cream company, the initial price is set at \(4.00, and the quantity demanded is 4000 units. However, the scenario changes as price adjustments are made. For each \)0.25 decrease in price, the demand increases by 200 units.
Thus, the price can be expressed as:
\[ P = 4 - 0.25x \]
where \( x \) represents the number of $0.25 price decreases. Meanwhile, the quantity is:
\[ Q = 4000 + 200x \]
Together, these expressions are multiplied to form the revenue function:
\[ R(x) = (4 - 0.25x)(4000 + 200x) \]
Understanding this function is crucial as it combines both variables that impact the revenue directly.

A quadratic equation is a polynomial equation of the second degree. In the context of this revenue problem, the revenue function is represented as a quadratic equation. Understanding the properties of quadratic equations helps in determining the maximum or minimum values of the function.
When the revenue function is expanded, it takes on the standard quadratic form:
\[ R(x) = -50x^2 - 200x + 16000 \]
The general structure of a quadratic equation is \( ax^2 + bx + c \), where "a", "b", and "c" are constants. Here, \( a = -50 \), \( b = -200 \), and \( c = 16000 \).
The significance of the quadratic equation in this problem is that it depicts a parabola. Because the leading coefficient \( a \) is negative, this parabola opens downwards. Hence, the vertex of the parabola signifies the maximum point, providing the price and quantity at which revenue is maximized.

The vertex of a parabola plays a critical role in optimization problems, such as maximizing revenue. For a downward-opening parabola, the vertex represents the highest point. It is crucial to find this point to determine the maximum revenue.
The formula for finding the vertex of a quadratic equation \( ax^2 + bx + c \) is:
\[ x = -\frac{b}{2a} \]
This provides the \( x \)-value at which the maximum or minimum occurs. In this calculation, substituting \( a = -50 \) and \( b = -200 \) into the vertex formula gives:
\[ x = -\frac{-200}{2(-50)} = 2 \]
This value of \( x \), when plugged back into the expressions for price and quantity, determines the optimal price and quantity that achieve maximum revenue. For our problem, setting \( x = 2 \) gives a price of $3.50 and a demand of 4400 units.
Identifying the vertex is essential for making informed decisions about pricing strategies to maximize revenue.