1 / 20

10.2 Parabolas

10.2 Parabolas. Objective. To determine the relationship between the equation of a parabola and its focus, directrix, vertex, and axis of symmetry. To graph a parabola. Definition . A set of points equidistant from a fixed point (focus) and a fixed line (directrix).

willoughby
Download Presentation

10.2 Parabolas

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 10.2 Parabolas

  2. Objective • To determine the relationship between the equation of a parabola and its focus, directrix, vertex, and axis of symmetry. • To graph a parabola

  3. Definition • A set of points equidistant from a fixed point (focus) and a fixed line (directrix).

  4. The midpoint between the focus and the directrix is called the vertex. • The line passing through the focus and the vertex is called the axis of the parabola. • A parabola is symmetric with respect to its axis.

  5. p is the distance from the vertex to the focus and from the vertex to the directrix.

  6. Vertical

  7. General Form If p > 0 opens up, if p < 0 opens down

  8. Vertex: (h, k) • Focus: (h, k + p) • Directrix: y = k – p • Axis of symmetry: x = h • If the vertex is at the origin (0, 0), the equation is:

  9. Horizontal parabola

  10. General Form If p > 0 opens right, if p < 0 opens left

  11. Vertex: (h, k) • Focus: (h + p, k) • Directrix: x = h-p • Axis of symmetry: y = k

  12. Example 1 • Find the standard equation of the parabola with vertex (3, 2) and focus (1, 2)

  13. Example2Finding the Focus of a Parabola • Find the focus of the parabola given by

  14. Example 3Finding the Standard Equation of a Parabola • Find the standard form of the equation of the parabola with vertex (1, 3) and focus (1, 5)

  15. Example 4 • opens: p = • vertex focus • directrix axis of symmetry

  16. Application • A line segment that passes through the focus of a parabola and has endpoints on the parabola is called a focal chord. The focal chord perpendicular to the axis of the parabola is called the latus retum.

  17. A line is tangent to a parabola at a point on the parabola if the line intersects, but does not cross, the parabola at the point. • Tangent lines to parabolas have special properties related to the use of parabolas in constructing reflective surfaces.

  18. Reflective Property of a Parabola • The Tangent line to a parabola at a point P makes equal angles with the following two line: • The line passing through P and the focus • The axis of the parabola.

  19. Example 5Finding the Tangent Line at a point on a Parabola • Find the equation of the tangent line to the parabola given by • At the point (1, 1)

More Related