What can we say about a parabola with its vertex at the origin (0,0) and its focus located in the negative part of the y-axis?

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Understanding the Parabola

A parabola is a curve formed by the intersection of a cone with a plane parallel to its side. The vertex of the parabola is a crucial point, and in this case, we’ve identified it as the origin (0,0).

Characteristics of the Given Parabola

When the focus of the parabola is positioned on the negative part of the y-axis, it indicates that the parabola opens downwards. The standard form of such a parabola can be expressed as:

        y = -a(x^2)

Where a is a positive constant that determines the width and steepness of the parabola. The larger the value of a, the narrower the parabola becomes.

Identifying True Statements

  • The focus lies at (0, -p): The focus of the parabola will be placed at the point (0, -p), where p is the distance from the vertex to the focus. Since the focus is below the vertex, p is a positive number.
  • The directrix is above the vertex: Corresponding to the negative position of the focus, the directrix of this parabola will be a horizontal line located at y = p (where p is a positive value).
  • The axis of symmetry is vertical: The axis of symmetry for this parabola is the y-axis (x=0). This means that for every point on one side of the parabola, there is a corresponding point on the opposite side.
  • Points on the parabola will have negative y-values: Since the parabola opens downwards, the outputs (y-values) will be negative for most of the x-values as we move away from the vertex.
  • This is a standard vertical parabola: The parabola can be categorized as a standard vertical parabola due to its orientation and the position of its focus.

Conclusion

In summary, a parabola with its vertex at the origin (0,0) and a focus located on the negative part of the y-axis has distinct characteristics: it opens downward, and its points, focus, directrix, and axis of symmetry follow specific relationships that define its unique geometric properties. Understanding these aspects is essential for further studies in quadratic functions and their applications.

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