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Question 618495: How do you find the vertex, focus, directrix, and axis of symmetry of the parabola?
x^2-2x+8y+9=0
Found 3 solutions by ewatrrr, lwsshak3, John10:
Answer by ewatrrr(24785)   (Show Source):
You can put this solution on YOUR website!
Hi, re TY, note ommission on the minus.
x^2-2x+8y+9=0
 V(1,-1), 4p=8, p = 2 F(1,-3)
 a = (-1/8) < 0 Parabola opens downward
Directrix y = 1 and x = 1 is the axis of symmetry
See below descriptions of various conics
Standard Form of an Equation of a Circle is 
where Pt(h,k) is the center and r is the radius
Standard Form of an Equation of an Ellipse is  where Pt(h,k) is the center. (a positioned to correspond with major axis)
a and b are the respective vertices distances from center and ± are the foci distances from center: a > b
Standard Form of an Equation of an Hyperbola opening right and left is:
 where Pt(h,k) is a center with vertices 'a' units right and left of center.
Standard Form of an Equation of an Hyperbola opening up and down is:
 where Pt(h,k) is a center with vertices 'b' units up and down from center.
the vertex form of a parabola opening up or down,  where(h,k) is the vertex.
The standard form is  , where the focus is (h,k + p)
the vertex form of a parabola opening right or left,  where(h,k) is the vertex.
The standard form is  , where the focus is (h +p,k )
Answer by lwsshak3(11628)  (Show Source):
You can put this solution on YOUR website! How do you find the vertex, focus, directrix, and axis of symmetry of the parabola?
x^2-2x+8y+9=0
complete he square
(x^2-2x+1)+8y+9-1=0
(x-1)^2=-8y-8
(x-1)^2=-8(y+1)
This is an equation of a parabola that opens downwards
Form of equation: (x-h)^2=-4p(y-k), (h,k)=(x,y) coordinates of vertex
For given equation:
vertex:(1,-1)
axis of symmetry: x=1
4p=8
p=2
focus: (1,-1-p)=(1,-1-2)=(1,-3) (p units below vertex on axis of symmetry)
directrix: y=-1+p=-1+2=1 (p units above vertex on axis of symmetry)
Answer by John10(297)  (Show Source):
You can put this solution on YOUR website! Hint:
Convert your equation in form: (x - h)^2 = 4p(y - k)
where
(h,k) is the vertex
(h, k + p) is the focus
y = k - p is the directrix
x = h is the axis of symmetry
I'll help you to convert it:
x^2 - 2x + 8y +9 = 0
x^2 - 2x + 1 = -8y - 8
(x - 1)^2 = -8( y + 1)
You can follow up from here. Good luck:)
John10
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