To determine which function represents profit, \( P(x) \), as a function of \( x \), the price per music class, let's analyze each of the given functions:
### Function Analysis:
1. \( P(x) = -(x-6)^2 + 116 \):
- This function is a quadratic equation in the form of \( -(x-6)^2 + 116 \).
- It represents a downward-opening parabola (due to the negative sign in front of the squared term).
- The vertex of this parabola is at \( (6, 116) \), which indicates that the maximum profit is 116 when \( x = 6 \).
2. \( P(x) = (x+6)^2 + 116 \):
- This function is also a quadratic equation, but in the form of \( (x+6)^2 + 116 \).
- It represents an upward-opening parabola (due to the positive sign in front of the squared term).
- The vertex of this parabola is at \( (-6, 116) \). Since the parabola opens upwards, it means there is a minimum profit at \( x = -6 \), which does not align with a realistic maximum profit scenario.
3. \( P(x) = -2(80 x) \):
- This function is linear and simplified to \( -160x \).
- It does not form a parabola and does not indicate a realistic profit function as it decreases indefinitely as \( x \) increases.
4. \( P(x) = -80\left(2^2\right) \):
- This function is a constant value, \( -320 \), and does not depend on \( x \).
- It does not depict any relationship between the profit and the price per class which is required for a profit function.
### Conclusion:
Based on the analysis of each function, the function \( P(x) = -(x-6)^2 + 116 \) is the one that realistically represents profit as a function of \( x \). This is because it models a scenario where there is a maximum profit value at a specific price point (vertex), which is consistent with how profits typically behave with respect to varying prices.
Therefore, the correct function is:
[tex]\[ P(x) = -(x-6)^2 + 116 \][/tex]
