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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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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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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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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

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9.3 Parabolas