Revenue The revenue R (in dollars) generated by the sale of...

Revenue The revenue $R$ (in dollars) generated by the sale of $x 32$ -inch televisions is modeled by $R=375 x-\frac{3}{2} x^{2}$ . Use a graphing utility to graph the function and approximate the sales that will maximize revenue.

Key Concepts

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically expressed in the form ax² + bx + c. They graph as parabolas, and their shape (opening upward or downward) is determined by the leading coefficient. This concept is central to modeling scenarios where relationships have a single peak or trough.

Vertex of a Parabolic Function

The vertex of a parabola represents the maximum or minimum point of a quadratic function, depending on the sign of the coefficient of the squared term. For functions modeling revenue, the vertex gives the sales level at which revenue is maximized. It can be found using the formula -b/(2a), which is a fundamental tool in optimization.

Optimization

Optimization in the context of mathematical functions involves finding the maximum or minimum values that a function can achieve. When applied to revenue models, optimization helps determine the best number of units to sell for maximizing revenue, using methods such as identifying the vertex in quadratic functions.

Graphical Analysis

Graphical analysis involves using a graphing utility or other visual tools to plot a function and identify key features like intercepts, vertex, and areas of increase or decrease. It is an effective way to approximate and verify solutions for optimization problems, such as locating the revenue maximum point for a given sales model.

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