quadratic equation graph by finding the price, net price, earning

The graph above represents the quadratic function \(E(x) = -2x^2 + 10x + 5\), which models earnings based on a given variable \(x\) (such as the number of units sold or a price adjustment). 

### Key Observations:
1. **Vertex**: The peak of the parabola represents the maximum earnings. This occurs at the vertex of the parabola, which can be calculated using \(x = -\frac{b}{2a}\). Here, it is around \(x = 2.5\).
2. **Earnings Decrease**: As \(x\) increases beyond a certain point, the earnings start to decrease due to the negative coefficient of \(x^2\), indicating diminishing returns or increasing costs.
3. **Zero Earnings**: The points where the graph intersects the x-axis are the break-even points, where earnings are zero.

Would you like to dive deeper into any of these points or have any specific questions? 

### Follow-up Questions:
1. How do you find the vertex of a quadratic function?
2. What does the coefficient of \(x^2\) indicate in a financial context?
3. How can the break-even points be calculated for this function?
4. What real-life scenarios can this quadratic model represent?
5. How do you modify the quadratic function for different business situations?

**Tip:** Quadratic equations are often used in optimization problems, such as maximizing profit or minimizing cost. Understanding the graph's vertex and intercepts is crucial for these analyses.

Understand how to graph quadratic equations to analyze earnings based on price optimization. Explore key concepts such as finding the vertex for maximum earnings, the significance of the quadratic coefficient, and determining break-even points. Ideal for Grades 10-12.

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