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Graphing

Conic Sections:

Ellipse:

standard form of the equation of an ellipse:

where b²= a² - c²

center    (0,0)    (0,0)
major axis  on x-axis, length 2a  on y-axis, length 2a
minor axis  on y-axis, length 2b  on x-axis, length 2b
foci   (±c,0)    (0,±c)
vertices  (±a,0)    (0,±a)
covertices  (0,±b)    (±b,0)

a is always larger than b; and a,b, and c are related by c²= a² - b²

ex.
Graph 9x²+ 4y²= 36

which is an ellipse.
a² = 9 ; b² = 4
center (0,0)
major axis: y-axis
vertices: (0, ± 3)
covertices: (± 2,0)

Graph 9x²+ 25y²= 225

a²= 25 ; b²= 9
center (0,0)
major axis: x-axis
vertices: (± 5,0)
covertices (0,± 3)

c²= a² - b²
c²= 25 - 9
c² = 16

foci: (± 4,0)

Ellipse:

standard equations for ellipse:

where b² = a² - c²

center:  (h,k)    (h,k)
major axis:  x = h, length 2a  y = k, length 2a
minor axis:  y = k, length 2b  x = h, length 2b
foci:   (h ± c,k)   (h, k ± c)
vertices:  (h ± a,k)    (h, k ± a)
covertices:  (h, k ± b)   (h ± b,k)

a is always larger than b; and a,b, and c are related by c²= a² - b²

ex.
graph 16x²+ 25y²- 64x-200y + 64 = 0
convert to standard form
16x²- 64x + 25y²-200y = - 64
complete the square for the x and y terms

(16x²- 64x ) + (25y²-200y ) = - 64
16(x²- 4x ) + 25(y²- 8y ) = -64 complete the square
16(x²- 4x + 4) + 25(y²- 8y + 16) = -64 + 64 + 400
16(x - 2)² + 25(y - 4)² = 400

a = 5; b = 4
center: (2,4)
major axis: x = 2, length 10
minor axis: y = 4, length 8
c²= a²- b²
c² = 25 - 16
c² = 9
c = 3

foci: (2 ± 3,4)  (5,4) and (-1,4)
vertices: (2 ± 5,4)   (7,4) and (-3,4)
covertices: (2, 4 ± 4)   (2,8) and (2,0)

ex.
graph 25x²+ 9y²+ 200x + 54y + 256 = 0
convert to standard form
25x²+200x + 9y²+ 4y = -256
complete the square for the x and y terms

(25x²+200x )+ (9y²+ 54y ) = - 256
25(x²+ 8x ) + 9(y²+ 6y ) = -256
25(x² + 8x + 16) + 9(y²+ 6y + 9) = -256 + 400 + 81
25(x + 4)² + 9(y + 3)² = 225

a = 5; b = 3
center: (-4,-3)
major axis: y = -3, length 10
minor axis: x = -4, length 6
c²= a²- b²
c² = 25 - 9
c² = 16
c = 4

foci:   (-4, -3 ± 4)  (-4,1) and (-4,-7)
vertices:  (-4, -3 ± 5)  (-4,2) and (-4,-8)
covertices:  (-4 ± 3,-3)  (-1,-3) and (-7,-3)

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