Let's analyze each of the statements provided about a parabola:
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The y-intercepts and the x-intercepts show the same point: This statement can be true in very specific cases (for example, when a parabola touches the y-axis at the same point it touches the x-axis). However, in general, this is not true for parabolas, as the y-intercept is the point where the graph crosses the y-axis (when \( x = 0 \)), and x-intercepts are points where the graph crosses the x-axis (when \( y = 0 \)).
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The y-intercept and the vertex share the same point: This statement is rarely true, as the vertex of a parabola is the point representing the minimum or maximum value of the function, while the y-intercept is where the parabola crosses the y-axis. They will generally be different points unless at a very specific configuration, but this is not common.
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There is only one x-intercept that vertex and the x-intercepts show the same point: This statement describes a scenario in which the parabola is tangent to the x-axis. This specific case happens when the discriminant (\( b^2 - 4ac \)) of the quadratic equation is zero, indicating a double root. This is true, in this case, the vertex is also the x-intercept, meaning they share the same point.
Conclusion: Among the given statements, the third one is true under the conditions of a parabola that has only one x-intercept (which coincides with the vertex).
Thus, the third statement is the true statement: "There is only one x-intercept that vertex and the x-intercepts show the same point."
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