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Lesson 9.1 Parabolas

Write any parts of a parabola that you know:. Lesson 9.1 Parabolas. Conic Sections Many shapes and curves can be classified as a conic section These shapes can be written algebraically as Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 or in relation to a locus (collection) of points

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Lesson 9.1 Parabolas

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  1. Write any parts of a parabola that you know: Lesson 9.1Parabolas

  2. Conic Sections Many shapes and curves can be classified as a conic section These shapes can be written algebraically as Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 or in relation to a locus (collection) of points (x – h)2 + (y – k)2 = r2

  3. Parabolas (Locus Definition) Definition: Set of all points equidistant from a fixed point (focus) and line (directrix). The midpoint between the focus and directrix is the vertex focus vertex directrix

  4. Standard Equation of a Parabola (Locus) Here, p is the distance from the focus to the vertex Traditional → Visual → I like → Same width as y = x2 p = ¼ → p = larger → p = smaller → Wider than y = x2 Narrower than y = x2

  5. Or, we can look at the 4p 4●p is the focal length: the width of the curve through the focus

  6. Some things to know: • A parabola can go Up/Down or Left/Right • Why do “I like” the last equation? • You can see the effects of p or 4p on the “slope” – a x-quantity squared → Up/Down y-quantity squared → Left/Right Large p→ large 4p → small a → low slope or flatter parabola Small p→ small 4p →large a → high slope or steeper parabola

  7. Example Find a standard form equation of a parabola with a vertex at the origin and focus at (0, 8)

  8. Example Find the vertex, focus, and equation of the directrix of the parabola. Click for more examples

  9. At your tables draw and label the following: A large parabola with vertex, V The focus, F and directrix The axis of symmetry A tangent line to the parabola at point, P A line passing through P and F An angle, , formed by the tangent and An angle, , formed by the tangent and the axis Compare your drawing with your neighbors to make a conjecture about the angles and .

  10. Reflective Property of Parabolas A tangent to a parabola at point P makes equal angles to: 1) A line passing through P and the focus and 2) The axis of the parabola Then these sides are congruent If these angles are congruent…

  11. Example Find the equation of the tangent line to y = x2at the point (2, 4).

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