Question

Determine which of the following statements is true about the parabola.

(1 point)
Responses

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.

The vertex and the x-intercepts share the same point.
The vertex and the x -intercepts share the same point.

There is only one x-intercept.

Answer 1

To determine the truth of the statements about the parabola, let's analyze each statement one by one:

1. **The y-intercept and the vertex share the same point.**
This statement is generally false. The y-intercept occurs when \( x = 0 \), while the vertex is determined by the vertex formula \( x = -\frac{b}{2a} \) for the quadratic equation \( y = ax^2 + bx + c \). They can be the same point in specific cases, but it is not true for all parabolas.

2. **The y-intercept and the x-intercepts share the same point.**
This statement is generally false. The y-intercept is where the parabola intersects the y-axis (again, at \( x = 0 \)), while the x-intercepts occur where the parabola intersects the x-axis (at the points where \( y = 0 \)). They can be equal in certain degenerate cases, but not generally.

3. **The vertex and the x-intercepts share the same point.**
This statement is generally false. The vertex provides the maximum or minimum point of the parabola, while the x-intercepts are the points where the parabola intersects the x-axis. In most cases, they will be different points.

4. **There is only one x-intercept.**
This statement is not universally true for all parabolas. A parabola may have zero x-intercepts (if it opens upwards and is entirely above the x-axis), one x-intercept (if it touches the x-axis at the vertex), or two x-intercepts (if it intersects the x-axis at two distinct points).

**Conclusion**: None of the statements are true in general. Therefore, if one must be selected, the most ambiguous statement among the choices provided is **"There is only one x-intercept,"** but this would need the context of a specific parabola to be true.

However, the correct answer is that **none of the given statements are universally true about a parabola.** If you have specific parameters or functions for the parabola, it could be possible to confirm one of these statements under those conditions.