To determine which quadratic inequality represents the graph, we need to analyze the given options by considering the characteristics of the parabola described by the equations.
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Identify the vertex form of the parabola. The general vertex form of a quadratic is \(y = a(x - h)^2 + k\), where \((h, k)\) is the vertex of the parabola.
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Analyze the coefficient (a):
- If \(a > 0\), the parabola opens upwards.
- If \(a < 0\), the parabola opens downwards.
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Determine the inequality symbols:
- For "less than" (\(<\)), the region below the parabola is shaded.
- For "greater than" (\(>\)), the region above the parabola is shaded.
- For "less than or equal to" (≤), the region below the parabola, including the parabola, is shaded.
- For "greater than or equal to" (≥), the region above the parabola, including the parabola, is shaded.
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Given options:
- \(y < 2(x + 1)^2 - 4\): Opens upward (since \(2 > 0\)), shading below the parabola.
- \(y > 2(x + 1)^2 - 4\): Opens upward, shading above the parabola.
- \(y ≤ 2(x + 1)^2 - 4\): Opens upward, shading below the parabola, including the boundary.
- \(y < -2(x + 1)^2 - 4\): Opens downward (since \(-2 < 0\)), shading below this downward parabola.
To make a definitive choice, it is essential to observe the direction of opening (upward or downward) and which area (above or below) is shaded in the given graph.
If the parabola opens upward and the region below is shaded, the correct choices would be either \(y < 2(x + 1)^2 - 4\) or \(y ≤ 2(x + 1)^2 - 4\).
If the region above the parabola is shaded, then the correct choices would be either \(y > 2(x + 1)^2 - 4\).
If you can clarify whether the parabola opens upward or downward and whether the region shaded is above or below, I can confirm which inequality accurately represents the graph.