https://file.aiquickdraw.com/mathbot/file/1725132118333f3i2zxk5.jpg

Which inequality below is represented by the following graph:



A. y≥−2x2+6x+8
B. y<2x2−8x+8
C. y>2x2−8x−8
D. y≤2x2−8x−8

To determine which inequality is represented by the graph, let's analyze the features of the graph first.

### Analysis of the Graph:
1. **Parabolic Shape**: The graph shows a parabola that opens upwards. The shaded region is above the parabola, which suggests an inequality involving "greater than or equal to" (≥) or "greater than" (>).

2. **Vertex of the Parabola**: From the graph, the vertex of the parabola appears to be around \( (2, 0) \).

3. **Shading**: Since the shaded area is outside (above) the parabola, we are looking for an inequality with "greater than" (y > or y ≥).

### Determining the Correct Inequality:
Now, let's evaluate each option:

A. **y ≥ -2x² + 6x + 8**:  
   - This option suggests a parabola that opens downward (negative coefficient for \(x^2\)).  
   - Since the parabola in the graph opens upwards, this option is **incorrect**.

B. **y < 2x² - 8x + 8**:  
   - This option suggests shading below a parabola that opens upwards (positive coefficient for \(x^2\)).  
   - The shading in the graph is above the parabola, so this option is **incorrect**.

C. **y > 2x² - 8x - 8**:  
   - This option suggests shading above a parabola that opens upwards (positive coefficient for \(x^2\)).  
   - The graph shows shading above the parabola, so this option is **correct**.

D. **y ≤ 2x² - 8x - 8**:  
   - This option suggests shading below a parabola that opens upwards.  
   - The graph shows shading above the parabola, so this option is **incorrect**.

### Conclusion:
The correct answer is **C. y > 2x² - 8x - 8**.

Would you like more details or have any questions?

Here are 5 related questions that might interest you:
1. How do you determine the vertex of a parabola from its equation?
2. What are the differences between the inequalities involving parabolas (>, ≥, <, ≤)?
3. How does the coefficient of \(x^2\) affect the direction in which a parabola opens?
4. How can we find the axis of symmetry for a given parabolic equation?
5. How would the graph change if we altered the constants in the quadratic expression?

**Tip:** When analyzing graphs of inequalities, always pay attention to the direction of shading to determine the correct inequality.

Discover how to identify the correct inequality from a graph analysis. Analyze the parabolic shape, vertex, and shading direction to determine which inequality best fits the graph. Suitable for high school students studying quadratic equations and inequalities.

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