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Parabola

Parabola. A set of all points (x,y) equidistant from a fixed line called a directrix and a fixed point called the focus . Vertex is the midpoint of the focus and directrix. Axis is the line passing through the focus and vertex. Equations of a Parabola. Vertex (h, k)

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Parabola

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  1. Parabola • A set of all points (x,y) equidistant from a fixed line called a directrix and a fixed point called the focus. • Vertex is the midpoint of the focus and directrix. • Axis is the line passing through the focus and vertex.

  2. Equations of a Parabola • Vertex (h, k) • p units: distance from vertex • Vertical axis • (x - h)2 = 4p(y - k) • y = k – p directrix • (h, k + p) focus • Horizontal axis • (y - k)2 = 4p(x - h) • x = h – p directrix • (h + p, k) focus

  3. Real-Life Parabolas

  4. Ellipse • A set of all points (x, y) the sum of whose distance from two distinct points called foci is constant. • Major axis is along the chord that connects vertices and foci. • The center is the midpoint of chord along major axis. • Vertices are two points where the major axis intersect the ellipse.

  5. Equations of Ellipses • (h, k) center • 2a, 2b length of the major and minor axis, where a > b.  • c2 = a2 + b2 • Horizontal axis is major • Vertical axis is major • eccentricity e = c/a

  6. Real-Life Ellipses

  7. Hyperbolas • A set of all points (x, y), the absolute value of the difference between the distance from two distinct points (foci) is constant. • Vertices – where the line through foci intersect hyperbola • Transverse axis – segment connecting vertices • Center – midpoint of transverse axis

  8. Equations of Hyperbolas • Horizontal Transverse • Asymptotes equations in red • Vertical Transverse • Asymptotes equations in red • Eccentricity: e = c/a

  9. General Form Ax2+ Bxy + Cy2 + Dx + Ey + F = 0

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