To determine which statement is true about the parabola based on the provided information, we need to analyze each option. You're saying that the graph shows a curve going through the point \( y = 5 \).
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A. The y-intercept and the x-intercepts share the same point.
- This would mean that the parabola crosses the y-axis at the same point where it crosses the x-axis. This is unlikely unless the parabola touches the x-axis at the y-intercept, which is a very specific case.
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B. The y-intercept and the vertex share the same point.
- This is also unlikely unless the parabola opens upwards or downwards and is symmetric in such a way that the vertex is at the y-intercept.
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C. The vertex and the x-intercepts share the same point.
- This would mean that the parabola touches the x-axis at the vertex (which is a special case of a double root where the vertex is the minimum or maximum point).
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D. There is only one x-intercept.
- This scenario describes a parabola that is tangent to the x-axis at one point, which would mean the vertex is at the x-axis (making it the x-intercept).
Given that the graph goes through \( y = 5 \), it's possible that the parabola can have one x-intercept if it opens downward and its vertex is at or above 5.
Without additional details on the shape or specific location of the x-intercept and whether the parabola opens upwards or downwards, the safest assumption based on typical parabolic behavior is:
D. There is only one x-intercept.
This is true if the vertex of the parabola is directly at the x-axis (indicating a perfect square trinomial) or it may not easily cross the x-axis at more than one point, depending on its shape. However, more context about the parabola is needed for a definitive answer. If the parabola opens upward or downward and is positioned in such a way that it only touches the x-axis, then option D is indeed the most plausible as true.
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