 # 4.4 Conics Recognize the equations and graph the four basic conics: parabolas, circles, ellipse, and hyperbolas. Write the equation and find the focus.

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4.4 Conics Recognize the equations and graph the four basic conics: parabolas, circles, ellipse, and hyperbolas. Write the equation and find the focus of a parabola. Write the equation of a ellipse and find the foci, vertices, the length of the major and minor axis.

Parabolas Definition: A parabola is the set of all points (x, y) in a plane that are equidistant from a fixed line, called the directrix, and a fixed point, the focus, not on the line. Directrix

Parabolas Standard equation of the Parabola x2 = 4py p  0
Vertex (0, 0) Directrix y = -p Focus (0, p) Line of sym x = 0 y2 = 4px p  0 Vertex (0,0) Directrix x = -p Focus (p, 0) Line of sym y = 0

Parabola Examples Given Find the focus.
Since the squared variable is x, the parabola is oriented in the y directions. The leading coefficient is negative, so, the parabola opens down. Focus (0, p) Solve for x2 -p Solve for p. p Focus

Parabola Examples Write the standard form of the equation of the parabola with the vertex at the origin and the focus (2, 0). Note that the focus is along the x axis, so the parabola is oriented in the x axis direction, y2 = 4px. Focus (2, 0) (p, 0) p = 2 y2 = 4px = 4(2)x y2 = 8x

Classwork Page 370 problems 9 –14. Page 371 problems 17 – 28.

Ellipse Definition: An ellipse is the set of all points (x, y ) in a plane the sum of whose distances from two distinct fixed points (foci) is constant. Major Axis Focus (x, y) d1 d2 d1 + d2 = constant Center Vertex Vertex Minor Axis

Ellipse The standard form of the equation of an ellipse (Center at origin) Major axis along the x axis Major axis along the y axis where 0 < b < a c2 = a2 – b2 Major axis length = 2a Minor axis length = 2b

Ellipse Examples Given 4x2 + y2 = Find the vertices, the end points of the minor axis, the foci and center. Change the equation to the standard form. Major axis along the y-axis b2 a2 Center (0, 0) Vertices (0, 6) End points of minor axis (3, 0) (0, a) (b, 0) To find the foci use c2 = a2 – b2 c2 = 36 – 9 c2 = 27 (0, c)

Classwork Page 372 problems 35 – 40 45 – 55

Hyperbolas Definition: a hyperbola is the set of all points (x, y) the difference of whose distances from two distinct points (foci) is constant. d1 Focus (x, y) Vertex Branch d2 Transversal Axis Center Vertex Branch Focus d1 - d2 = constant

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