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9.3 Parabolas9.3 Parabolas
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A directrix is a fixedline that serves as a
guide in creating aparabola
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A focusfocusis the pointused to determine
the parabola'sopenness and
distance from the
directrix
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A parabola is the setof all points that are
equidistant from apoint called a focus
and a line called adirectrix.
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The distance from the
point on the parabolato the focus is equalto the distance fromthe same point to thedirectrix.(See page 412)
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ocus
Directrix
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Focus lies on axis ofsymmetryVertex is midwaybetween focus and
directrixFocus is ( a) units
away from the vertexDirectrix is ( a) units
away from the vertex
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Ex.Ex. If the focus is (0,4)If the focus is (0,4)
and directrix is the lineand directrix is the liney = -6 what is they = -6 what is the
vertex?vertex?Use the midpoint
formula:
),(
,
10
264
200 ),( 60
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GeneralGeneral
Equations ofEquations of
different types ofdifferent types of
parabolas withparabolas withvertex at originvertex at origin
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yy22 = 4cx with c>0)= 4cx with c>0)
*Opens right
*Focus (c,0)
*Directrix:
x = -c
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yy22 = 4cx with c
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xx22 = 4cy with c>0)= 4cy with c>0)
*Opens up
*Focus (0,c)
*Directrix:
y = -c
y=-c(0,c)
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xx22 = 4cy with c
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Ex. Find the vertex,Ex. Find the vertex,
focus, directrix andfocus, directrix andwhether the graph openswhether the graph opens
up, down, left or right.up, down, left or right.1. y2 = 6x 2. y = 4x2yx4
12 =
V: (0,0) V: (0,0)F: (3/2,0) F: (0,1/16)
D: x = -3/2 D: = 1/16 E Fi d hE Fi d h
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Ex. Find the vertex,Ex. Find the vertex,
focus, directrix andfocus, directrix andwhether the graph openswhether the graph opens
up, down, left or right.up, down, left or right.1. y2 = 6x 2. y = 4x2yx4
12 =
V: (0,0) V: (0,0)F: (3/2,0) F: (0,1/16)
D: x = -3/2 D: = 1/16
Opensright Opensup
E Fi d hE Fi d h
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Ex. Find the vertex,Ex. Find the vertex,
focus, directrix andfocus, directrix andwhether the graph openswhether the graph opens
up, down, left or right.up, down, left or right.1. y2 = 6x 2. y = 4x2yx4
12 =
V: (0,0) V: (0,0)F: (3/2,0) F: (0,1/16)
D: x = -3/2 D: = 1/16
Rewrite into
x2=4cy E Fi d hE Fi d h
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Ex. Find the vertex,Ex. Find the vertex,
focus, directrix andfocus, directrix andwhether the graph openswhether the graph opens
up, down, left or right.up, down, left or right.1. y2 = 6x 2. y = 4x2
Rewrite into
x2=4cy
yx4
12 =
V: (0,0) V: (0,0)F: (3/2,0) F: (0,1/16)
D: x = -3/2 D: = 1/16
To find c, set the
coefficients e ual to 4c
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Other equationsOtherequationsof parabolas:ofparabolas:
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y k = 1 (x - h)2
Opens up if 1/4c ispos. ordown if neg.Focus is |c| unitsfrom vertex
Focus: (h k+c)
4c
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(y - k)2 = 4c(x - h)
Opens right if 1/4c ispos. orleft if neg.
Focus is |c| unitsfrom vertex
Focus: (h+c k)
x - h= 1 (y k)2
4c
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PODPODFind an equation forFind an equation for
the parabola withthe parabola with
V(-1,1) and directrixV(-1,1) and directrix
y = -5y = -5
Fi d th t
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Find the vertex,
focus, directrix of x2
- 4x 4y = 0
1st) Place x & y onopposites sides of =
2nd) Complete thesquare according
to the s uared term 3rd)R it ti
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3rd)Rewrite equation
into either:
x - h= 1 (y k)2
4c
y k = 1 (x - h)2
4cor
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HW: Page 416HW: Page 416(17-26 all)(17-26 all)
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PODPODFind an equation forFind an equation for
the parabola withthe parabola withV(-2,5) and directrixV(-2,5) and directrix
x = -5x = -5