To determine which statement about the parabola is true, let's analyze each option provided:
-
The y-intercept and the vertex share the same point.
- This is generally not true. The y-intercept occurs when \( x = 0 \), and the vertex has its own specific coordinates based on the equation of the parabola.
-
The vertex and the x-intercepts share the same point.
- This is not generally true for parabolas. The vertex can be different from the x-intercepts unless the parabola is tangent to the x-axis at the vertex, which is a specific case.
-
There is only one x-intercept.
- This can be true for parabolas that are tangent to the x-axis, meaning they touch the axis at exactly one point. However, many parabolas have two x-intercepts or none (depending on their orientation and position).
-
The y-intercept and the x-intercepts share the same point.
- This is generally not true. The y-intercept occurs when \( x = 0 \), and the x-intercepts can occur at different x-values.
Based on general properties of parabolas, the only statement that can potentially be true in specific cases is:
"There is only one x-intercept."
However, without additional context specifying the parabola's specific characteristics, the most generally applicable statements (for all types of parabolas) would be that the first set of statements are not true. Thus, under normal considerations:
None of the statements are universally true for all parabolas without specific conditions.
If you are looking for a specific case, please clarify the conditions of the parabola for a more tailored analysis.