Find the Properties x^2=-4y

Problem

x2=−4*y

Solution

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

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

3. Solve for p

4*p=−4

p=−1

4. Find the vertex. Since there are no h or k offsets in the equation (x−h)2=4*p*(y−k) the vertex is at the origin.

Vertex=(0,0)

5. Determine the focus. For a vertical parabola, the focus is at (h,k+p)

Focus=(0,0+(−1))

Focus=(0,−1)

6. Identify the directrix. The directrix is a horizontal line given by the equation y=k−p

y=0−(−1)

y=1

7. Find the axis of symmetry. The axis of symmetry passes through the vertex and focus, which is the line x=h

x=0

8. Calculate the focal diameter (length of the latus rectum). The length is given by |4*p|

Length=|−4|

Length=4

Final Answer

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

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