To answer why there is an interval over which the graph of the quadratic equation [tex]\( y = -6x^2 + 100x - 180 \)[/tex] decreases, let's delve into the properties of quadratic functions and the context of the problem.
1. Understanding the Quadratic Equation:
- The given equation, [tex]\( y = -6x^2 + 100x - 180 \)[/tex], describes a parabola.
- The coefficient of [tex]\( x^2 \)[/tex] (which is -6) is negative, indicating that the parabola opens downward.
2. Finding the Vertex of the Parabola:
- The vertex of a parabola represented by [tex]\( y = ax^2 + bx + c \)[/tex] occurs at [tex]\( x = -\frac{b}{2a} \)[/tex].
- For our equation, [tex]\( a = -6 \)[/tex], [tex]\( b = 100 \)[/tex], and [tex]\( c = -180 \)[/tex].
- Calculating the x-coordinate of the vertex:
[tex]\[
x = -\frac{100}{2 \cdot (-6)} = -\frac{100}{-12} = \frac{100}{12} \approx 8.33
\][/tex]
- Therefore, the x-coordinate of the vertex is approximately 8.33 dollars.
3. Interval Over Which the Graph Decreases:
- A downward-facing parabola increases until it reaches its vertex, and then it starts to decrease.
- Since it's downward-facing, the graph decreases for all [tex]\( x \)[/tex] values greater than the vertex.
- This means for [tex]\( x > 8.33 \)[/tex], the profit [tex]\( y \)[/tex] decreases.
4. Contextual Explanation:
- If the soccer balls are too expensive, fewer will be sold, reducing profit.
- Initially, increasing the price of soccer balls might increase profit because selling each ball at a higher price will outweigh the potential reduction in the number sold.
- However, beyond a certain price point (vertex), the high price will significantly reduce the number of soccer balls sold, causing total profit to decrease.
- Thus, for [tex]\( x\)[/tex] (cost per soccer ball) greater than approximately 8.33 dollars, the store will sell fewer soccer balls resulting in decreased profit.
Therefore, the reason the graph decreases for higher values of [tex]\( x \)[/tex] (cost per soccer ball) is because if soccer balls are too expensive, fewer will be sold, thereby reducing the store's profit. The interval over which the graph decreases is for [tex]\( x\)[/tex] values greater than approximately 8.33 dollars.
