https://file.aiquickdraw.com/mathbot/file/172568831253234etofzv.jpg

The graph shown in the image represents a parabola, which is the graph of the function \( f(x) \). To determine the graph of its derivative \( f'(x) \), let's analyze the shape of the parabola:

1. **Shape of the function \( f(x) \)**: The graph is a downward-opening parabola, which suggests that the function \( f(x) \) is a quadratic function of the form \( f(x) = ax^2 + bx + c \) where \( a < 0 \) because it opens downward.

2. **Behavior of the derivative \( f'(x) \)**:
   - The slope of the tangent line to \( f(x) \) is zero at the vertex of the parabola. In this case, the vertex occurs at \( x = 0 \). Therefore, \( f'(x) = 0 \) at \( x = 0 \).
   - The slope is positive to the left of the vertex (the function is increasing) and negative to the right of the vertex (the function is decreasing). This indicates that the derivative should start positive, cross zero at \( x = 0 \), and then become negative.

3. **Nature of \( f'(x) \)**: Since the derivative of a quadratic function is a linear function, the graph of \( f'(x) \) will be a straight line that decreases from positive to negative as \( x \) increases.

### Key Features of the Derivative Graph:
- \( f'(0) = 0 \) (since the vertex is at \( x = 0 \)).
- The derivative is positive for \( x < 0 \) and negative for \( x > 0 \).
- The graph of \( f'(x) \) will be a straight line with a negative slope.

Do you want me to help you verify the correct graph of the derivative or explain this concept in more detail?

Here are 5 related questions:
1. What is the relationship between the concavity of a function and the sign of its second derivative?
2. How do you find the derivative of a quadratic function using the power rule?
3. What does the sign of the derivative tell us about the increasing or decreasing behavior of a function?
4. How do critical points relate to the derivative of a function?
5. Can you explain the role of inflection points and how they are related to the second derivative?

**Tip**: Always check where the slope of the function is zero—this helps to locate critical points where the derivative is zero.

Select the correct graph of the derivative of the function f(x) given below (a downward parabola).

Learn how to analyze the graph of a quadratic function and determine its derivative using visual graph analysis. Explore key concepts such as increasing and decreasing intervals and the behavior of the derivative. Suitable for students in Grades 10-12.

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