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The given problem asks which graph could represent the quadratic function \( y = -ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are positive constants.

### Analysis of the quadratic function:
The general form of the equation is \( y = -ax^2 + bx + c \), which is a **parabola** that opens **downward** because the coefficient of \( x^2 \) (which is \( -a \)) is negative.

### Key characteristics of the graph:
- The parabola opens **downward** (because of the negative \( a \)).
- Since \( b \) is positive, the parabola will have a positive slope at the vertex and will open downward after that.
- The constant \( c \) shifts the graph **upward**, so the parabola doesn't start at the origin.

### Graph matching:
- We need to look for a graph that opens **downward** and has the vertex above the x-axis, which indicates the positive \( c \) shift.

### Examining the options:
- **Graph A:** This graph opens downward and has its vertex above the x-axis, which fits the description.
- **Graph B, C, D:** These either open upward or do not match the description.

### Conclusion:
The correct graph is **A** because it opens downward and has its vertex above the x-axis.

Do you have any questions or would you like further details?

Here are 5 related questions for you:
1. What happens to the graph if \( a \) becomes negative but \( b \) and \( c \) remain positive?
2. How would the graph change if \( c \) were negative instead of positive?
3. What does the value of \( b \) control in a quadratic equation?
4. What are the x-intercepts of a general quadratic function?
5. How does the axis of symmetry relate to the vertex of a quadratic function?

**Tip:** The vertex form of a quadratic equation is a useful way to quickly identify the vertex and direction of a parabola!

Which of the following could be the graph of y = -ax^2 + bx + c given that a, b, and c are positive constants?

Learn how to identify the correct graph for the quadratic function y = -ax^2 + bx + c, where a, b, and c are positive constants. Understand how the parabola opens downward and the role of the constants in shifting the graph. Suitable for high school students.

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