Use the image to answer the question.

A parabola is graphed on a coordinate plane. The x-axis ranges from negative 5 to 5 in increments of 1. The y-axis ranges from negative 1 to 16 in increments of 1.

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.

1 answer

To determine which statement about the parabola is true, let's analyze the typical characteristics of a parabola and the definitions of the x-intercept, y-intercept, and vertex.

  1. X-intercepts: These are points where the parabola intersects the x-axis (y=0). A parabola can have 0, 1, or 2 x-intercepts.

  2. Vertex: This is the highest or lowest point of the parabola (depending on whether it opens upwards or downwards).

  3. Y-intercept: This is the point where the parabola intersects the y-axis (x=0).

Now let's evaluate the options:

  • There is only one x-intercept.: This could be true if the parabola is tangent to the x-axis, but it is not universally true for all parabolas.

  • The vertex and the x-intercepts share the same point.: This would only be true if the vertex is at the same point as an x-intercept, which is possible (in the case of tangency) but not guaranteed for all parabolas.

  • The y-intercept and the vertex share the same point.: This is generally true only in specific cases. For most parabolas, the y-intercept and vertex will be different.

  • The y-intercept and the x-intercepts share the same point.: This is virtually impossible unless the parabola lies on the x-axis at that intercept, which is not generally the case.

Given the analysis:

  1. If the parabola has one x-intercept, then the first option could be true.
  2. If the vertex happens to lie on the x-axis, then the second option could be true.
  3. The third and fourth options are less likely to be true for typical parabolas.

Without seeing the specific graph, it's hard to determine the correct answer definitively, but if any of the statements are generally true based on how parabolas behave, it would be the statement that there is only one x-intercept, particularly if it appears tangent at a single point on the x-axis. If multiple statements seemed plausible, one could select based on additional observations from the graph displayed.

To provide a definitive answer, I would encourage anyone working with this question to refer to the graph directly.