To determine which statement is true about a parabola, let's analyze each statement based on the characteristics of parabolas:
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The y-intercept and the x-intercepts share the same point.
- This is generally false. The y-intercept is the point where the parabola crosses the y-axis (x=0), while x-intercepts are where it crosses the x-axis (y=0). They are not typically the same point unless the vertex is on both axes, which is rare.
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The vertex and the x-intercepts share the same point.
- This is also false. The vertex is the highest or lowest point of the parabola, depending on its direction (upward or downward). The x-intercepts are points where the parabola meets the x-axis. Only in specific cases (like the parabola touching the x-axis at the vertex) would they coincide, but generally, this statement is not true.
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There is only one x-intercept.
- This can be true for certain parabolas (specifically, those that touch the x-axis at one point, known as a "double root"). However, it is not universally true for all parabolas, since most parabolas can have two x-intercepts or none.
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The y-intercept and the vertex share the same point.
- This is generally false. While there are specific cases where the vertex could be on the y-axis, usually they are different points, as the vertex is defined by the parabola's maximum or minimum value, not by crossing the y-axis.
In summary, most statements provided are not true in general cases of parabolas. However, the third statement ("There is only one x-intercept") has a condition under which it can be true. Thus, while it is not universally true, it is the only statement that can hold in specific cases.
Final answer: There is only one x-intercept.
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