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

Question

(1 point)
Responses

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 x-intercepts share the same point.
The y -intercept and the x -intercepts share the same point.

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

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

Answer 1

To determine which statement is true about a parabola, we should clarify what each statement means:

1. The vertex and the x-intercepts share the same point.
   This means that the vertex is also one of the x-intercepts. This occurs if the parabola only touches the x-axis at the vertex (i.e., it has one x-intercept), which is the case if the parabola opens upward or downward and the vertex is on the axis.

2. The y-intercept and the x-intercepts share the same point.
   This means that the y-intercept (point where the parabola crosses the y-axis, found by evaluating the function at x=0) is also one of the x-intercepts. This is generally not the case for most parabolas.

3. There is only one x-intercept.
   This would imply that the parabola touches the x-axis at exactly one point (the vertex, as mentioned earlier). This is true for parabolas that are just tangent to the x-axis.

4. The y-intercept and the vertex share the same point.
   This means that the vertex is also the y-intercept, which again does not generally hold for all parabolas.

Based on these explanations, the true statement generally pertains to the parabolas that touch the x-axis only at the vertex. Therefore, the statement that reflects this condition is:

There is only one x-intercept.