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.