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The image shows a graph of a quadratic function \( f(x) = ax^2 + bx + c \) and asks to determine the values of \( a \), \( b \), and \( c \). From the graph, we observe a parabola that opens downwards and has the following key points:

- Vertex at \( (2, 4) \)
- The x-intercepts at \( x = 0 \) and \( x = 4 \)

### Step-by-Step Solution

1. **Determine the equation format:**  
   The vertex form of a quadratic function is:
   \[
   f(x) = a(x - h)^2 + k
   \]
   where \((h, k)\) is the vertex. Here, the vertex is \( (2, 4) \), so:
   \[
   f(x) = a(x - 2)^2 + 4
   \]

2. **Use x-intercepts to find \( a \):**  
   The x-intercepts are at \( x = 0 \) and \( x = 4 \), so:
   \[
   f(0) = 0 \quad \text{and} \quad f(4) = 0
   \]

   Substitute \( x = 0 \) into the vertex form:
   \[
   0 = a(0 - 2)^2 + 4 \implies 0 = a(4) + 4 \implies -4 = 4a \implies a = -1
   \]

3. **Find the standard form coefficients:**

   Now that we have \( a = -1 \), substitute \( a \) back into the vertex form:
   \[
   f(x) = -1(x - 2)^2 + 4
   \]

   Expand to convert it to standard form:
   \[
   f(x) = -1(x^2 - 4x + 4) + 4
   \]
   \[
   f(x) = -x^2 + 4x - 4 + 4
   \]
   \[
   f(x) = -x^2 + 4x
   \]

   Therefore, the coefficients are:
   \[
   a = -1, \; b = 4, \; c = 0
   \]

### Conclusion

- \( a = -1 \)
- \( b = 4 \)
- \( c = 0 \)

Would you like more details, or have any questions?

Here are 5 related questions to consider:

1. How do you find the vertex of a parabola from its equation?
2. What is the axis of symmetry for the quadratic function given?
3. How does changing the value of \( a \) affect the graph of a quadratic function?
4. What is the significance of the discriminant in quadratic equations?
5. Can you convert any quadratic function into vertex form?

**Tip:** Practice plotting different quadratic functions to understand how each coefficient affects the shape and position of the parabola.

Given the graph of the quadratic function f(x) = ax^2 + bx + c, determine the values of a, b, and c.

Explore how to determine the coefficients a, b, and c of the quadratic function f(x) = ax^2 + bx + c using the graph, focusing on the vertex form and x-intercepts. Suitable for students in Grades 9-11.

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