Question

For each scenario, use graphing technology to determine the shape of its graph. Then identify the function family, whether it is increasing, decreasing, or a combination of both, has an absolute maximum or absolute minimum, and whether it is a smooth curve or straight line. 1 A fitness company is selling videos for one of its new cardio routines. Each video will sell for $15. Due to fixed and variable costs, you can represent the profit that the company will see after selling x videos using the function P(x) = 11.5x -0.1x² - 150.

Answer 1

The graph of the profit function (P(x) = 11.5x - 0.1x² - 150) forms a downward-opening parabola. It represents a combination of both increasing and decreasing behavior, having an absolute maximum and no absolute minimum. The curve is smooth and not a straight line.

Step-by-step explanation:

The function (P(x) = 11.5x - 0.1x² - 150) is a quadratic equation, resulting in a parabolic graph. Its shape is determined by the negative coefficient of the squared term (-0.1x²), indicating a parabola that opens downward. Initially, as x increases, the profit also rises due to the linear term (11.5x), representing the revenue from selling videos. However, the quadratic term (-0.1x²) introduces a decreasing aspect, causing the profit to peak and then decline.

The function has an absolute maximum but no absolute minimum. The maximum profit occurs at the vertex of the parabola, which represents the highest point before the profit starts decreasing. As for the curve, it's smooth due to the continuous nature of quadratic functions, forming a smooth, curved line without sudden breaks or corners.

Understanding the behavior of this profit function enables the company to strategize regarding the number of videos to sell for optimal profitability while considering both the linear revenue growth and the diminishing returns resulting from the quadratic cost factor.