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At what integer values) of x do Function C and Function D have the same value of y？

To determine the integer values of \( x \) where Functions C and D have the same \( y \)-value, we need to analyze the graphs of these functions.

1. **Function C**: This appears to be a parabola opening upwards. Without specific coordinates, it resembles the standard form of a quadratic function \( y = ax^2 + bx + c \).

2. **Function D**: The function \( f(x) = x^2 \) is provided in the image. This is a standard quadratic function with the vertex at the origin \((0, 0)\) and opens upwards.

### Steps to Solve:
1. **Identify the Equation of Function C**: By examining the graph closely, identify key points on the parabola (especially the vertex and intercepts if possible).
  
2. **Compare Function C and Function D**: For the integer values of \( x \), calculate \( y \) for both Function C and \( f(x) = x^2 \). The \( x \)-values for which \( y_C = y_D \) will give the required integer solutions.

#### Based on the visual inspection of the graph:
- If Function C closely matches \( y = x^2 \), the solutions may occur where the two graphs intersect.

However, without precise coordinates, let's consider key points where the graphs of \( y = x^2 \) (Function D) intersect the parabola of Function C. These intersection points are likely to be symmetric around the vertex.

Would you like me to derive the equation of Function C from the graph, or would you prefer to analyze it with more detailed coordinates?

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### Related Questions:
1. What is the vertex form of a quadratic function, and how can it be used to describe Function C?
2. How does the symmetry of a parabola help in finding points of intersection with another quadratic function?
3. Can we always expect two distinct intersection points when comparing two parabolic functions? Why or why not?
4. How do the coefficients in a quadratic equation affect the shape and position of the parabola on the graph?
5. How would the solutions change if Function D were shifted or reflected?

### Tip:
When comparing quadratic functions, always consider the discriminant to check if and where the functions intersect.

Discover how to find integer solutions for Functions C and D where their y-values are equal. Learn about quadratic equations, graph intersections, and symmetry of parabolas. Suitable for high school students studying algebra and graphing quadratic functions.

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