Question
Determine which of the following statements is true about the parabola. (1 point) Responses There is only one x-intercept. There is only one x -intercept. The vertex and the x-intercepts share the same point. The vertex and the x -intercepts share the same point. The y-intercept and the vertex share the same point. The y -intercept and the vertex share the same point. The y-intercept and the x-intercepts share the same point. The y -intercept and the x -intercepts share the same point.
Answers
Answered by
GPT-4o mini
To determine which statement is true about a parabola, we need to analyze the properties of parabolas and their vertex, intercepts, and overall shape.
1. **There is only one x-intercept**: This occurs if the parabola is tangent to the x-axis (meaning it has a double root), which is a specific case but not a general property of all parabolas.
2. **The vertex and the x-intercepts share the same point**: This statement is true if the vertex is on the x-axis, which happens in the specific case where the parabola is a perfect square (has one x-intercept). Again, this is not generally true for all parabolas.
3. **The y-intercept and the vertex share the same point**: This is also a specific case where the vertex lies on the y-axis. However, not all parabolas have this property.
4. **The y-intercept and the x-intercepts share the same point**: This is generally false since the y-intercept is always at (0, f(0)) and the x-intercepts are the solutions to the equation \( y = 0 \), which usually do not coincide unless under very specific conditions.
Given these evaluations, the most consistently true statement applies to the condition where a parabola opens either upwards or downwards.
The **most general case** that includes cases with at least one x-intercept and possibly others is:
"There is only one x-intercept" is specific but can hold true in certain scenarios.
The correct and true statement from the provided options heavily depends on the specific parabola context you are considering. However, without more specific conditions given for the parabola, none of the statements can be deemed universally true. Nonetheless:
The statement **"the vertex and the x-intercepts share the same point"** can be true if the parabola has a double root (vertex on x-axis).
For a common understanding, it’s essential to recognize that:
1. **"There is only one x-intercept"** (For parabolas that touch the x-axis. Multiple conditions might suffice)
Hence, if we need to choose the most plausible consistent true scenario, I would endorse **"the vertex and the x-intercepts share the same point"**.
1. **There is only one x-intercept**: This occurs if the parabola is tangent to the x-axis (meaning it has a double root), which is a specific case but not a general property of all parabolas.
2. **The vertex and the x-intercepts share the same point**: This statement is true if the vertex is on the x-axis, which happens in the specific case where the parabola is a perfect square (has one x-intercept). Again, this is not generally true for all parabolas.
3. **The y-intercept and the vertex share the same point**: This is also a specific case where the vertex lies on the y-axis. However, not all parabolas have this property.
4. **The y-intercept and the x-intercepts share the same point**: This is generally false since the y-intercept is always at (0, f(0)) and the x-intercepts are the solutions to the equation \( y = 0 \), which usually do not coincide unless under very specific conditions.
Given these evaluations, the most consistently true statement applies to the condition where a parabola opens either upwards or downwards.
The **most general case** that includes cases with at least one x-intercept and possibly others is:
"There is only one x-intercept" is specific but can hold true in certain scenarios.
The correct and true statement from the provided options heavily depends on the specific parabola context you are considering. However, without more specific conditions given for the parabola, none of the statements can be deemed universally true. Nonetheless:
The statement **"the vertex and the x-intercepts share the same point"** can be true if the parabola has a double root (vertex on x-axis).
For a common understanding, it’s essential to recognize that:
1. **"There is only one x-intercept"** (For parabolas that touch the x-axis. Multiple conditions might suffice)
Hence, if we need to choose the most plausible consistent true scenario, I would endorse **"the vertex and the x-intercepts share the same point"**.
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