Graph the parabola and identify the vertex, directrix, and focus.
Problem 1 :
y^{2} = -12x
Solution:
y^{2} = -12x
The above equation of parabola is in standard form. The parabola is symmetric about x-axis and it opens to the left.
4a = 12
a = 3
Vertex:
(0, 0)
Equation of directrix:
x = a
x = 3
Focus:
F(-a, 0) = F(-3, 0)
Problem 2 :
x^{2} = 8y
Solution:
x^{2} = 8y
The above equation of parabola is in standard form. The parabola is symmetric about y-axis and it opens to the up.
4a = 8
a = 2
Vertex:
(0, 0)
Equation of directrix:
y = -a
y = -2
Focus:
F(0, a) = F(0, 2)
Problem 3 :
6(x + 1)^{2} + 12(y - 3) = 0
Solution:
6(x + 1)^{2} + 12(y - 3) = 0
6(x + 1)^{2} = -12(y - 3)
(x + 1)^{2} = -12/6 (y - 3)
(x + 1)^{2} = -2(y - 3)
Let X = x + 1 and Y = y - 3
X^{2} = -2Y
The above equation of parabola is in standard form. The parabola is symmetric about y-axis and it opens to the down.
4a = 2
a = 1/2
Vertex :
(0, 0)
X = 0 and Y = 0
x + 1 = 0 and y - 3 = 0
x = -1 and y = 3
The vertex is (-1, 3).
Equation of directrix:
Y = a
Y = 1/2
Y = 0.5
y - 3 = 0.5
y = 0.5 + 3
y = 3.5
Focus:
F(0, -a) = F(0, -0.5)
X = 0 and Y = -0.5
x + 1 = 0 and y - 3 = -0.5
x = -1 and y = 2.5
The focus is (-1, 2.5).
Problem 4 :
y^{2} - 12(x + 2) = 0
Solution:
y^{2} - 12(x + 2) = 0
y^{2} = 12(x + 2)
Let X = x + 2
y^{2} = 12X
The above equation of parabola is in standard form. The parabola is symmetric about x-axis and it opens to the left.
4a = 12
a = 3
Vertex:
(0, 0)
X = 0 and Y = 0
x + 2 = 0 and y = 0
x = -2 and y = 0
The vertex is (-2, 0).
Equation of directrix:
X = -a
X = -3
x + 2 = -3
x = -5
Focus:
F(a, 0) = F(3, 0)
x + 2 = 3
x = 1
The focus is (1, 0).
Problem 5 :
(x + 2)^{2} = -8(y + 2)
Solution:
(x + 2)^{2} = -8(y + 2)
Let X = x + 2 and Y = y + 2
X^{2} = -8Y
The above equation of parabola is in standard form. The parabola is symmetric about y-axis and it opens to the down.
4a = 8
a = 2
Vertex:
(0, 0)
X = 0 and Y = 0
x + 2 = 0 and y + 2 = 0
x = -2 and y = -2
The vertex is (-2, -2).
Equation of directrix:
Y = a
Y = 2
y + 2 = 2
y = 0
Focus:
F(0, -a) = F(0, -2)
X = 0 and Y = -2
x + 2 = 0 and y + 2 = -2
x = -2 and y = -4
The focus is (-2, -4).
Problem 6 :
(y - 1)^{2} = -8x
Solution:
(y - 1)^{2} = -8x
Let Y = y - 1
Y^{2} = -8x
The above equation of parabola is in standard form. The parabola is symmetric about x-axis and it opens to the left.
4a = 8
a = 2
Vertex:
(0, 0)
Y = 0
y - 1 = 0
y = 1
The vertex is (0, 1).
Equation of directrix:
x = a
x = 2
Focus:
F(-a, 0) = F(-2, 0)
x = -2 and Y = 0
y - 1 = 0
x = -2 and y = 1
The focus is (-2, 1).
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