To determine which statement is true about the parabola, we need to understand the key concepts of parabolas and their intercepts and vertex.
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X-Intercepts: A parabola can have 0, 1, or 2 x-intercepts depending on its orientation and position relative to the x-axis. So the statement "There is only one x-intercept" can be true in specific cases (like a parabola that just touches the x-axis) but is not universally true for all parabolas.
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Y-Intercept and Vertex Sharing a Point: The y-intercept of a parabola is found by setting \( x = 0 \) in its equation, and the vertex is the maximum or minimum point of the parabola. Generally, these two points do not share the same coordinates.
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Vertex and X-Intercepts Sharing a Point: The vertex of the parabola can coincide with an x-intercept only in specific cases, such as when the parabola opens upwards and has a minimum point at the x-axis. However, this is not a general rule.
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Y-Intercept and X-Intercepts Sharing a Point: It is very rare for a y-intercept (which occurs at \( x = 0 \)) to also be an x-intercept unless the parabola crosses the x-axis exactly at that point (which is very situational and not likely for most parabolas).
Based on the analysis above, none of the statements provided can be deemed universally true about all parabolas. However, if you were to evaluate specific cases (like the vertex of a parabola located at the y-intercept), then a specific statement might hold true.
If you have a particular parabola in mind, feel free to provide more details for a more accurate assessment!