Parabolas

• 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).

• 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.

• STD
• Opens
• Focus
• Directrix
• Axis
• Focal length
• Focal Width x²=4py up p>0
• Down p<0
(0,p) y=-p y-axis p
I4pI y²=4px right p>0 left p<0
(p,0) x=-p x-axis p
I4pI

• 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

• 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

• 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

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

• 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)

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

• 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

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

• 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

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

• 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

• STD
• Opens
• Focus
• Directrix
• Axis
• Focal length
• Focal Width
(x-h)²=4p(y-k) (y-k)²=4p(x-h) up p>0
• Down p<0
(h,k+p) y=k-p x=h p
I4pI right p>0 left p<0
(h+p,k) x=h-p y=k p
I4pI

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

• 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)