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Conics Parabolas

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Vocabulary center: the point (h, k) at the center of a circle, an ellipse, or an hyperbola. vertex (VUR-teks): in the case of a parabola, the point (h, k) at the "end" of a parabola; in the case of an ellipse, an end of the major axis; in the case of an hyperbola, the turning point of a branch of an hyperbola; the plural form is "vertices" (VUR-tuh-seez). focus (FOH-kuss): a point from which distances are measured in forming a conic; a point at which these distance-lines converge, or "focus"; the plural form is "foci" (FOH-siy). directrix (dih-RECK-triks): a line from which distances are measured in forming a conic; the plural form is "directrices" (dih-RECK-trih-seez). axis (AK-siss): a line perpendicular to the directrix passing through the vertex of a parabola; also called the "axis of symmetry"; the plural form is "axes" (ACK-seez). major axis: a line segment perpendicular to the directrix of an ellipse and passing through the foci; the line segment terminates on the ellipse at either end; also called the "principal axis of symmetry"; the half of the major axis between the center and the vertex is the semi-major axis. minor axis: a line segment perpendicular to and bisecting the major axis of an ellipse; the segment terminates on the ellipse at either end; the half of the minor axis between the center and the ellipse is the semi-minor axis. locus (LOH-kuss): a set of points satisfying some condition or set of conditions; each of the conics is a locus of points that obeys some sort of rule or rules; the plural form is "loci" (LOH-siy).

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Parabolas Def: The set of all points (x,y) in a plane that are equidistant from a fixed line in the directrix, and a fixed point, the focus, not on the line.

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Parabolas w/ Vertex (0,0) STDx²=4pyy²=4px Opens up p>0right p>0 Down p<0left p<0 Focus(0,p)(p,0) Directrixy=-px=-p Axisy-axisx-axis Focal lengthpp Focal Width I4pI I4pI

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Vertex The midpoint between the focus and the directrix is the vertex, and the line passing through the focus and the vertex is the axis of the parabola

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Standard form of a parabola With vertex (0,0) and directrix y=-p X²=4py, p is not equal to 0 For directrix x=-p Y²=4px, p is not equal to 0

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Vertical/Horizontal Notice that a parabola can have a vertical or horizontal axis and that a parabola is symmetric with respect to its axis We can tell by what variable is squared X² Vertical axis Y² Horizontal axis

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E1 Find the focus of the parabola whose equation is y=-2x²

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E1 Answer Find the focus of the parabola whose equation is y=-2x² X is being squared so it is a vertical axis Rewrite as in form x²=4py X²=-1/2y X²=4(-1/8)y So p=-1/8 Focus (0,p) so (0,-1/8)

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E2 Find the standard form of the equation of the parabola with vertex at the orgin and the focus at (2,0).

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E2 Find the standard form of the equation of the parabola with vertex at the orgin and the focus at (2,0). Focus (p,0) so Horizontal axis Y²=4px P=2 so y²=4(2)x y²=8x

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E3 Find the focus, the directrix, and the focal width of the parabola y=-(1/2)x²

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E3 Find the focus, the directrix, and the focal width of the parabola y=-(1/2)x² -2y=x² 4p=-2 P=-1/2 Focus (0,-1/2) Directrix y=1/2 Focal width I4pl=l4(-1/2)l=2

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E4 Find the equation in STD form for the parabola whose directrix is the line x=2 and whose focus is the pt (-2,0)

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E4 Find the equation in STD form for the parabola whose directrix is the line x=2 and whose focus is the pt (-2,0) X=-p 2=-p -2=p Y²=4(-2x) Y²=-8x

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Parabolas w/ Vertex (h,k) STD (x-h)²=4p(y-k) (y-k)²=4p(x-h) Opens up p>0right p>0 Down p<0left p<0 Focus (h,k+p) (h+p,k) Directrixy=k-px=h-p Axisx=h y=k Focal lengthpp Focal Width I4pI I4pI

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E5 Find the STD form of the equation for the parabola with vertex (3,4) and focus (5,4)

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E5 Find the STD form of the equation for the parabola with vertex (3,4) and focus (5,4) h=3 k=4 h+p=5 3+p=5 so p=2 y²=4px **Because the x changed for focus (y-4) ²=8(x-3)

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EXAMPLE 3 Write an equation of a translated parabola

EXAMPLE 3 Write an equation of a translated parabola

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