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Understanding Parabolas: Definitions, Properties, and Graphing Techniques

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This guide explores the properties of parabolas, defined as the set of all points equidistant from a focus and a directrix. It covers key concepts such as the vertex, axis of symmetry, and the latus rectum. The standard form equations for both vertical and horizontal parabolas are provided, along with examples to illustrate finding vertices, focuses, directrices, and graphing techniques. Additionally, exercises challenge you to write equations based on given vertices and points on the parabolas, enhancing your understanding of these essential quadratic curves.

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Understanding Parabolas: Definitions, Properties, and Graphing Techniques

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  1. Conics: Parabolas

  2. Parabolas: • The set of all points equidistant from a fixed line called the directrix and a fixed point called the focus. • The vertex is the point midway between the focus and the directrix. • The line that connects the vertex and the focus is the axis of symmetry.

  3. The standard form equation of a parabola with vertex at (h, k) is The first equation is for a parabola that opens up or down. The second equation is for a parabola that opens left or right.

  4. If the equation is • The focus is the point . • The directrix is the line. • The axis of symmetry is parallel to the y-axis (vertical).

  5. If the equation is • The focus is the point . • The directix is the line . • The axis of symmetry is parallel to the x-axis (horizontal).

  6. Latus Rectum • The latus rectum is a line segment through the focus, parallel to the directrix. • The endpoints of the latusrectum are each a distance of 2a from the focus.

  7. Ex. 1 Find the vertex, focus and directrix of the parabola. Graph the equation. a)

  8. b)

  9. c)

  10. d)

  11. Ex.2 Write the equation of the parabola. a) Vertex: (0, 0), focus: (-4, 0)

  12. b) Directrix: y = 6, vertex: (0, 0)

  13. c) Directrix: x = -4, focus: (2, 4) Hint: start with a sketch of the graph.

  14. d) Vertex: (3, 0), focus: (3, -2)

  15. Ex. 3 Write the equation of the parabola with the given vertex and point on the parabola. • a) Vertex: (1, 0) Point: (0, 1)

  16. b) Vertex: (1, -1) Point: (0, 1)

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