Q. For the parabola defined by the equation x^2 = -12y, what is the direction in which it opens?
A.
Upwards
B.
Downwards
C.
Left
D.
Right
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Solution
The equation x^2 = -12y indicates that the parabola opens downwards.
Correct Answer:
C
— Left
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Q. For the parabola defined by the equation x^2 = 16y, what is the distance from the vertex to the focus?
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Solution
In the equation x^2 = 4py, we have 4p = 16, thus p = 4. The distance from the vertex to the focus is 4.
Correct Answer:
B
— 4
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Q. For the parabola defined by the equation x^2 = 16y, what is the length of the latus rectum?
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Solution
The length of the latus rectum for the parabola x^2 = 4py is 4p. Here, p = 4, so the length is 8.
Correct Answer:
B
— 8
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Q. For the parabola defined by the equation y = -x^2 + 4x - 3, what is the y-intercept?
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Solution
To find the y-intercept, set x = 0. The equation becomes y = -0^2 + 4(0) - 3 = -3.
Correct Answer:
A
— -3
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Q. If a parabola has its vertex at (3, -2) and opens downwards, what is the general form of its equation?
A.
y + 2 = a(x - 3)^2
B.
y + 2 = -a(x - 3)^2
C.
y - 2 = a(x + 3)^2
D.
y - 2 = -a(x + 3)^2
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Solution
For a downward-opening parabola with vertex (h, k), the equation is y - k = -a(x - h)^2. Here, h = 3 and k = -2.
Correct Answer:
B
— y + 2 = -a(x - 3)^2
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Q. If a parabola opens to the left, which of the following is its standard form?
A.
y^2 = -4px
B.
x^2 = -4py
C.
y^2 = 4px
D.
x^2 = 4py
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Solution
The standard form of a parabola that opens to the left is y^2 = -4px.
Correct Answer:
A
— y^2 = -4px
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Q. If a parabola opens to the right and has its vertex at the origin, what is the general form of its equation?
A.
y^2 = 4px
B.
x^2 = 4py
C.
y = mx + c
D.
x = ay^2
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Solution
The general form of a parabola that opens to the right is y^2 = 4px.
Correct Answer:
A
— y^2 = 4px
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Q. If the focus of a parabola is at (0, 2) and the directrix is y = -2, what is the equation of the parabola?
A.
x^2 = 8y
B.
x^2 = 4y
C.
y^2 = 8x
D.
y^2 = 4x
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Solution
The distance from the focus to the directrix is 4, so the equation is x^2 = 8y.
Correct Answer:
A
— x^2 = 8y
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Q. The equation of a parabola with vertex at (0, 0) and directrix y = -3 is?
A.
x^2 = -12y
B.
y^2 = -12x
C.
x^2 = 12y
D.
y^2 = 12x
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Solution
The distance from the vertex to the directrix is 3, so the equation is x^2 = -12y.
Correct Answer:
A
— x^2 = -12y
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Q. The equation of a parabola with vertex at (0, 0) and focus at (0, 3) is?
A.
x^2 = 12y
B.
y^2 = 12x
C.
x^2 = 6y
D.
y^2 = 6x
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Solution
The distance from the vertex to the focus is 3, so the equation is x^2 = 12y.
Correct Answer:
A
— x^2 = 12y
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Q. The parabola y^2 = 12x opens in which direction?
A.
Upwards
B.
Downwards
C.
Left
D.
Right
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Solution
The equation y^2 = 12x indicates that the parabola opens to the right since it is in the form y^2 = 4px.
Correct Answer:
D
— Right
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Q. The vertex of the parabola given by the equation y = 2x^2 - 4x + 1 is located at which point?
A.
(1, -1)
B.
(2, 0)
C.
(1, 0)
D.
(0, 1)
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Solution
To find the vertex, use the formula x = -b/(2a). Here, a = 2, b = -4, so x = 1. Plugging x = 1 into the equation gives y = -1.
Correct Answer:
A
— (1, -1)
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Q. What is the axis of symmetry for the parabola given by the equation y = -3(x - 2)^2 + 5?
A.
x = 2
B.
y = 5
C.
y = -3
D.
x = -2
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Solution
The axis of symmetry for a parabola in vertex form y = a(x - h)^2 + k is x = h. Here, h = 2.
Correct Answer:
A
— x = 2
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Q. What is the axis of symmetry for the parabola given by the equation y^2 = 6x?
A.
x-axis
B.
y-axis
C.
y = x
D.
x = 0
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Solution
The axis of symmetry for the parabola y^2 = 4px is the x-axis.
Correct Answer:
B
— y-axis
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Q. What is the directrix of the parabola defined by the equation y^2 = 20x?
A.
x = -5
B.
x = 5
C.
y = 5
D.
y = -5
Show solution
Solution
For the equation y^2 = 4px, p = 5. The directrix is given by x = -p, which is x = -5.
Correct Answer:
A
— x = -5
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Q. What is the equation of the directrix of the parabola x^2 = 12y?
A.
y = 3
B.
y = -3
C.
y = 6
D.
y = -6
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Solution
The directrix of the parabola x^2 = 4py is given by y = -p. Here, p = 3, so the directrix is y = -3.
Correct Answer:
B
— y = -3
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Q. What is the focus of the parabola defined by the equation y^2 = 20x?
A.
(5, 0)
B.
(0, 5)
C.
(0, 10)
D.
(10, 0)
Show solution
Solution
In the equation y^2 = 4px, we have 4p = 20, thus p = 5. The focus is at (5, 0).
Correct Answer:
A
— (5, 0)
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Q. What is the focus of the parabola given by the equation y^2 = 20x?
A.
(5, 0)
B.
(0, 5)
C.
(0, -5)
D.
(10, 0)
Show solution
Solution
For the parabola y^2 = 4px, here 4p = 20, so p = 5. The focus is at (5, 0).
Correct Answer:
A
— (5, 0)
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Q. What is the latus rectum of the parabola given by the equation y^2 = 12x?
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Solution
The latus rectum of a parabola y^2 = 4px is given by 4p. Here, 4p = 12, so p = 3, and the latus rectum is 4p = 12.
Correct Answer:
C
— 6
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Q. What is the value of p for the parabola defined by the equation y^2 = 20x?
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Solution
In the equation y^2 = 4px, we have 4p = 20, thus p = 5.
Correct Answer:
A
— 5
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Q. What is the vertex of the parabola given by the equation y = 2x^2 - 4x + 1?
A.
(1, -1)
B.
(1, 0)
C.
(2, 1)
D.
(0, 1)
Show solution
Solution
To find the vertex, use the formula x = -b/(2a). Here, a = 2, b = -4, so x = 1. Substitute x = 1 into the equation to find y = -1.
Correct Answer:
A
— (1, -1)
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Q. Which of the following points lies on the parabola y = x^2 - 4?
A.
(2, 0)
B.
(0, -4)
C.
(1, -3)
D.
(3, 5)
Show solution
Solution
Substituting x = 1 into the equation gives y = 1^2 - 4 = -3, so the point (1, -3) lies on the parabola.
Correct Answer:
C
— (1, -3)
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Q. Which of the following points lies on the parabola y^2 = 8x?
A.
(2, 4)
B.
(1, 2)
C.
(4, 4)
D.
(2, 2)
Show solution
Solution
To check if a point lies on the parabola, substitute the x-coordinate into the equation. For (2, 4), 4^2 = 16 and 8*2 = 16, so it lies on the parabola.
Correct Answer:
A
— (2, 4)
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