91Ó°ÊÓ

Every business aims to achieve maximum profit, and understanding quadratic functions can help with this. In the quadratic function provided, \( P = -15 + 10n - 0.2n^2 \), \( P \) stands for profit in thousands of dollars, and \( n \) is the number of thousands of widgets sold. This function shows that profit increases with each widget sold but at a decreasing rate due to the negative \( n^2 \) term. It also implies a maximum profit will be reached before the profit starts declining. This point is called the vertex of the parabola.
To find the number of widgets sold for maximum profit, use the formula for the vertex of a parabola \( n = -\frac{b}{2a} \), where \( a = -0.2 \) and \( b = 10 \). Solving yields \( n = \frac{10}{0.4} = 25 \). However, since the formula applies up to 15,000 widgets, the company achieves maximum profit beyond this range, and practical evaluation must be within limitations.

Concavity tells us how the graph of a function curves. In this context, because the quadratic coefficient \( a \) is negative in the profit function, the graph is concave down. When the graph is concave down, it signifies the profit increases as more widgets are sold but at a decreasing rate.
The second derivative \( P''(n) = -0.4 \) confirms concave down, as \( P''(n) < 0 \). This reflects diminishing returns — every additional widget sold contributes less to the profit than the previous one. Understanding concavity helps businesses recognize when further production may yield less financial benefit and informs decisions on the optimal quantity of widgets to produce.

The break-even point is the sales level where the revenue equals costs, resulting in zero profit. In other words, it's the point where the profit function crosses the horizontal axis. For the widget producer, solving the function \( 0 = -15 + 10n - 0.2n^2 \) using the quadratic formula helps find these critical points.

• The quadratic formula: \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = -0.2 \), \( b = 10 \), and \( c = -15 \).
• Solving gives two values: approximately \( 1.605 \) and \( 9.395 \).

These translate to around 1,605 and 9,395 widgets required to break even. Selling widgets below these points results in a loss, while selling beyond the highest break-even point results in a profit. These indicators are crucial for financial decision-making and planning.

Graphing functions provides a visual representation, making it easier to understand relationships between variables. For the quadratic profit function \( P = -15 + 10n - 0.2n^2 \), the graph is a downward-opening parabola.
Graphs help:

• Visualize maximum and minimum values quickly, aiding in identifying the vertex.
• Observe the concavity to understand the profit behavior as sales volume changes.
• Identify intercepts, such as the break-even points, more intuitively.

Plotting values of \( n \) from 0 to 15 on the x-axis (thousand widgets) and corresponding \( P \) on the y-axis (thousand dollars) will help chart the path to optimal production levels and economic insights for strategic decisions.
Each point on the graph reflects a different sales level and its financial outcome, unlocking a deeper understanding of sales strategies.