To determine which quadratic inequality accurately represents the graph, we need to consider the features of the quadratic function given in the options.
The general form of a quadratic function is: \[ y = a(x - h)^2 + k \] where \( (h, k) \) is the vertex of the parabola and \( a \) determines the direction (upwards for \( a > 0 \), downwards for \( a < 0 \)) and the width of the parabola.
Let's analyze the options provided:
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\( y \leq 2(x + 1)^2 - 4 \):
- This represents a parabola that opens upwards (since \( a = 2 > 0 \)) with the vertex at \( (-1, -4) \). The graph would be shaded below or on the parabola due to the "≤" symbol.
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\( y < 2(x + 1)^2 - 4 \):
- Like the previous option, this parabola also opens upwards with vertex at \( (-1, -4) \), but the graph would be shaded below the parabola.
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\( y > 2(x + 1)^2 - 4 \):
- This also has the same vertex and direction as the previous parabolas, but it would shade above the parabola.
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\( y < -2(x + 1)^2 - 4 \):
- Here, the parabola opens downwards (since \( a = -2 < 0 \)) with the vertex at \( (-1, -4) \), and the graph would be shaded below the parabola.
To select the appropriate inequality, note these key points from the graph:
- Identify the direction of the parabola (upward or downward).
- Determine if the shading is above or below the parabola.
Without seeing the actual graph, in this context, if the parabola opens upwards and you want the region below it, then the correct options are the first and second inequalities. If you want points above the parabola, then option three is applicable.
Give the graph's details, if the inequality should shade below an upward-opening parabola, pick \( y \leq 2(x+1)^2 - 4 \) for inclusion of the boundary. If not strict, choose \( y < 2(x+1)^2 - 4 \).
In contrast, if the parabola opens downwards and you seek the area below it, select option four. Since you didn’t specify the orientation of the parabola or the regions required, that context is crucial for a definitive choice.
1 answer