Let's break down the problem step-by-step to understand why there is an interval over which the graph of the profit function decreases.
### Understanding the Profit Function
The given profit function is:
[tex]\[ y = -k^2 + 100x - 180 \][/tex]
Here, [tex]\( y \)[/tex] represents the profit and [tex]\( x \)[/tex] represents the price at which soccer balls are sold.
### Characteristics of a Quadratic Equation
Firstly, this function is a quadratic equation in terms of [tex]\( x \)[/tex]. A quadratic function is typically in the format:
[tex]\[ y = ax^2 + bx + c \][/tex]
Where:
- [tex]\( a \)[/tex] is the coefficient of [tex]\( x^2 \)[/tex]
- [tex]\( b \)[/tex] is the coefficient of [tex]\( x \)[/tex]
- [tex]\( c \)[/tex] is the constant term
In our case:
- The coefficient [tex]\( a = -k^2 \)[/tex] is negative because it is multiplied by [tex]\( -k^2 \)[/tex]. This negative coefficient makes the parabola open downwards.
- The coefficient [tex]\( b = 100 \)[/tex]
- The constant [tex]\( c = -180 \)[/tex]
### Analyzing the Profit Function
Because the coefficient [tex]\( a \)[/tex] is negative, the parabola opens downwards. This means that the quadratic function has a maximum point (vertex) and the profit decreases on either side of this vertex.
### Vertex of the Parabola
The vertex of a quadratic function [tex]\( y = ax^2 + bx + c \)[/tex] can be found at:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Given our specific function:
[tex]\[ x = -\frac{100}{2(-k^2)} = \frac{100}{2k^2} = \frac{50}{k^2} \][/tex]
This [tex]\( x \)[/tex] value represents the price at which the profit is maximized. On either side of this vertex, the profit decreases.
### Interval over Which the Graph Decreases
On one side of the vertex, the profit decreases as [tex]\( x \)[/tex] increases beyond the optimal price. On the other side, the profit decreases as [tex]\( x \)[/tex] decreases below the optimal price. This is because the quadratic curve representing the profit function has a maximum point.
### Price Impact on Profit
When considering real-world implications:
- If the soccer balls are too expensive, fewer will be sold, reducing profit. This happens because high prices discourage customers from buying, leading to fewer sales and thus reduced profit.
It is similar if the price is too low: the store won't cover costs efficiently, decreasing total profit.
### Conclusion
In summary, the profit function decreases after reaching its maximum because it is modeled by a quadratic equation with a negative coefficient for the [tex]\( x^2 \)[/tex] term, resulting in a downward-opening parabola. This makes sense because excessively high prices result in fewer sales, ultimately reducing profits.
So the answer is:
If the soccer balls are too expensive, fewer will be sold, reducing profit.
