# Conics Parabolas. 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.

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Conics Parabolas

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).

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.

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

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

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

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

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

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)

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

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

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

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

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)

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

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

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

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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