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 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. The y-intercept and the vertex share the same point.

Answer 1

To determine which of the statements about the parabola is true, let's analyze each one in the context of a typical parabola represented by the equation \( y = ax^2 + bx + c \).

1. **There is only one x-intercept.**
This is true for parabolas that touch the x-axis at exactly one point, which happens when the discriminant \( b^2 - 4ac = 0 \). This case describes a parabola that has a vertex on the x-axis.

2. **The y-intercept and the x-intercepts share the same point.**
The y-intercept occurs when \( x = 0 \), so \( y = c \). This point will not generally equal the x-intercepts unless the parabola has a very specific form (such as being horizontally aligned with the y-axis at 0).

3. **The vertex and the x-intercepts share the same point.**
This can only happen in the case where the vertex lies on the x-axis, which again occurs when \( b^2-4ac = 0 \). But it's not true for all parabolas.

4. **The y-intercept and the vertex share the same point.**
This statement is not generally true as the y-intercept is at \( (0, c) \) while the vertex has coordinates given by \( \left( -\frac{b}{2a}, f\left(-\frac{b}{2a}\right) \right) \), which will not usually coincide with the y-intercept.

**Conclusion:**
The statement that is most universally true is **"There is only one x-intercept,"** which occurs in the case where the parabola touches the x-axis at the vertex. Other statements either require specific conditions to hold or are generally not true across all parabolas.

Hence, the most accurate answer here is:

**There is only one x-intercept.**