To determine which of the statements is true about a parabola, we need to analyze the characteristics of parabolas in relation to their intercepts and vertices:
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The y-intercept and the vertex share the same point.
This generally is not true; the vertex and y-intercept can be different points unless the vertex is exactly on the y-axis (for example, a vertical parabola with a vertex at (0, k) where k is the y-intercept). -
The y-intercept and the x-intercepts share the same point.
This is false except for very specific cases where the parabola passes through the y-axis at the same value as one of the x-intercepts (very unlikely). -
There is only one x-intercept.
This is true for parabolas that "touch" the x-axis at exactly one point, also known as having a "double root", which occurs when the vertex is on the x-axis. -
The vertex and the x-intercepts share the same point.
This is not generally true; the vertex can be above or below the x-axis, resulting in different points.
Given these analyses, the statement "There is only one x-intercept." could be true in the context of a specific parabola. Therefore, this would be the correct and true statement if we are considering the possibility of a parabola that only touches the x-axis at one point (the vertex).
Thus, the correct answer is: There is only one x-intercept.