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1). Vertex at origin, Focus: 
(0,(1)/(4))
A) 
y=x^(2)
B) 
y=4x^(2)
C) 
y=-(x+2)^(2)-1
D) 
y=-(x+2)^(2)+1
E) 
y=-x^(2)

11). Vertex at origin, Focus: (0,14) \left(0, \frac{1}{4}\right) \newlineA) y=x2 y=x^{2} \newlineB) y=4x2 y=4 x^{2} \newlineC) y=(x+2)21 y=-(x+2)^{2}-1 \newlineD) y=(x+2)2+1 y=-(x+2)^{2}+1 \newlineE) y=x2 y=-x^{2}

Full solution

Q. 11). Vertex at origin, Focus: (0,14) \left(0, \frac{1}{4}\right) \newlineA) y=x2 y=x^{2} \newlineB) y=4x2 y=4 x^{2} \newlineC) y=(x+2)21 y=-(x+2)^{2}-1 \newlineD) y=(x+2)2+1 y=-(x+2)^{2}+1 \newlineE) y=x2 y=-x^{2}
  1. Identify Parabola Orientation: Since the vertex is at the origin (0,0)(0,0) and the focus is at (0,14)(0,\frac{1}{4}), the parabola is vertical.
  2. Calculate Distance to Focus: The distance from the vertex to the focus, pp, is 14\frac{1}{4}.
  3. Use Standard Form Equation: The standard form of a vertical parabola is y=4px2y = 4px^2.
  4. Substitute pp Value: Substitute p=14p = \frac{1}{4} into the standard form equation to get y=4(14)x2y = 4\left(\frac{1}{4}\right)x^2.
  5. Simplify Equation: Simplify the equation to y=x2y = x^2.

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