Find The Vertex, Focus, And Directrix Of The Parabola, Sketc

101 Find The Vertex Focus And Directrix Of The Parable Sketch Its

Find the vertex, focus, and directrix of the parabola. Sketch its graph, showing the focus and the directrix. Find an equation for the parabola shown in the figure. Find an equation for the parabola that satisfies the given conditions. Focus F (-3,-2), directrix y = 1. Find the vertex, focus, and directrix of the parabola, and sketch its graph, showing these features. Determine the equation of the parabola with the given focus and directrix.

Paper For Above instruction

A parabola is a symmetric curve characterized by a focus and a directrix, with the set of points equidistant from both. To analyze and graph a parabola, determining its vertex, focus, directrix, and the equation are essential steps. This paper discusses the procedures to find these components, particularly when the focus and directrix are provided, and demonstrates how to sketch the parabola accordingly.

Given the focus at F(-3, -2) and directrix y=1, we start by understanding the key features. The vertex of the parabola lies exactly midway between the focus and the directrix along the axis of symmetry. The axis of symmetry in this case is vertical, since the directrix is horizontal. The vertex's y-coordinate is the average of the y-coordinates of the focus and directrix:

yv = ( -2 + 1 ) / 2 = -0.5

The x-coordinate of the vertex is the same as that of the focus, because the parabola opens vertically:

xv = -3

Thus, the vertex is at V(-3, -0.5). The distance between the focus and vertex along the axis of symmetry is the focal length, denoted as 'p'. The focal length is the distance from the vertex to the focus:

p = | yf - yv | = | -2 - (-0.5) | = 1.5

Since the focus is below the vertex (due to yf v), the parabola opens downward. The standard form of the parabola opening vertically is:

(x - h)2 = 4p(y - k)

where (h, k) is the vertex. Substituting the known values:

(x + 3)2 = 4( -1.5 )( y + 0.5 ) = -6 ( y + 0.5 )

This is the equation of the parabola with focus at F(-3, -2) and directrix y=1.

To sketch the parabola:

  • Plot the vertex at (-3, -0.5).
  • Draw the focus at (-3, -2).
  • Draw the directrix as the line y=1.
  • Sketch the parabola opening downward, passing through points symmetric about the axis of symmetry.

In addition to this specific example, the general process involves:

  1. Identifying the focus and directrix.
  2. Finding the vertex as the midpoint between focus and directrix along the axis of symmetry.
  3. Calculating p, the focal length.
  4. Using the standard form equation with the appropriate sign for p.
  5. Sketching based on the vertex, focus, directrix, and symmetry.

Understanding the geometric relationships of the parabola's defining components allows for precise graphing and formulation of the equation, whether the parabola opens upward, downward, or sideways depending on the given parameters.

References

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