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Question 480717: Graph each hyperbola. Include labels indicating the coordinates of the foci and vertices and the equations of the asymptotes.
(y-3)^2/9 - (x+2)^2/16 = 1
thanks :)
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website! Graph each hyperbola. Include labels indicating the coordinates of the foci and vertices and the equations of the asymptotes.
(y-3)^2/9 - (x+2)^2/16 = 1
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Given equation is that of a hyperbola with vertical transverse axis of the standard form:
(y-k)^2/a^2-(x-h)^2/b^2=1, (h,k) being the (x,y) coordinates of the center.
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For given equation:
center: (-2,3)
Vertices:
a^2=9
a=3
length of transverse axis=2a=6
vertices: (-2,3±a)=(-2,3±3)=(-2,6) and (-2,0)
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Foci:
b^2=16
b=4
c^2=a^2+b^2=9+16=25
c=5
Foci: (-2, 3±c)=(-2,3±5)=(-2,8) and (-2,-2)
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Asymptotes:
slope,m,=±a/b=±3/4
Equations of asymptotes: y=±3x/4+b
solving for b using coordinates of center thru which straight line asymptotes pass.
y=3x/4+b
3=3*-2/4+b
3=-6/4+b
b=3+6/4=12/4+6/4=18/4
Equation: y=3x/4+18/4
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y=-3x/4+b
3=-3*-2/4+b
3=6/4+b
b=3-6/4=12/4-6/4=6/4=3/2
Equation: y=-3x/4+3/2
See graph below as a visual check on answers
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y=±(9+9(x+2)^2/16)^.5+3
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