To determine which statement is true about a parabola, we should clarify what each statement means:
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The vertex and the x-intercepts share the same point.
This means that the vertex is also one of the x-intercepts. This occurs if the parabola only touches the x-axis at the vertex (i.e., it has one x-intercept), which is the case if the parabola opens upward or downward and the vertex is on the axis. -
The y-intercept and the x-intercepts share the same point.
This means that the y-intercept (point where the parabola crosses the y-axis, found by evaluating the function at x=0) is also one of the x-intercepts. This is generally not the case for most parabolas. -
There is only one x-intercept.
This would imply that the parabola touches the x-axis at exactly one point (the vertex, as mentioned earlier). This is true for parabolas that are just tangent to the x-axis. -
The y-intercept and the vertex share the same point.
This means that the vertex is also the y-intercept, which again does not generally hold for all parabolas.
Based on these explanations, the true statement generally pertains to the parabolas that touch the x-axis only at the vertex. Therefore, the statement that reflects this condition is:
There is only one x-intercept.
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