Find the Properties x^2=2y

Problem

x2=2*y

Solution

1. Identify the type of conic section. The equation is in the form x2=4*p*y which represents a parabola that opens vertically.

2. Determine the value of p By comparing x2=2*y to the standard form x2=4*p*y we set 4*p=2

4*p=2

p=1/2

3. Locate the vertex. Since there are no horizontal or vertical shifts (h,k) the vertex is at the origin.

Vertex=(0,0)

4. Find the focus. For a parabola opening upward, the focus is located at (0,p)

Focus=(0,1/2)

5. Determine the directrix. The directrix is a horizontal line located at y=−p

y=−1/2

6. Identify the axis of symmetry. Since the x term is squared, the parabola is symmetric about the yaxis.

x=0

Final Answer

Vertex: *(0,0), Focus: *(0,1/2), Directrix: *y=−1/2, Axis of Symmetry: *x=0

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