Find the Properties x^2=-8y

Problem

x2=−8*y

Solution

1. Identify the form of the parabola. The equation x2=−8*y is in the standard form x2=4*p*y which represents a parabola that opens vertically.

2. Determine the value of p By comparing 4*p=−8 we solve for p

4*p=−8

p=−2

3. Find the vertex. Since the equation is in the form x2=4*p*y with no translations, the vertex is at the origin:

(h,k)=(0,0)

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

Focus=(0,0+(−2))

Focus=(0,−2)

5. Determine the directrix. The directrix is a horizontal line given by y=k−p

y=0−(−2)

y=2

6. Identify the axis of symmetry. Since the parabola opens vertically and the vertex is at (0,0) the axis of symmetry is the y-axis:

x=0

7. Determine the direction of opening. Because p<0 and the squared term is x the parabola opens downward.

Final Answer

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

Want more problems? Check here!