Find the Properties x=-1/8y^2

Problem

x=−1/8*y2

Solution

1. Identify the type of conic section. Since the equation is in the form x=a*y2 it is a parabola that opens horizontally.

2. Rewrite the equation in the standard form (y−k)2=4*p*(x−h) to identify the vertex (h,k)

y2=−8*x

3. Determine the vertex (h,k) from the standard form. Since there are no constants added to x or y the vertex is at the origin.

Vertex=(0,0)

4. Calculate the value of p by setting 4*p equal to the coefficient of x

4*p=−8

p=−2

5. Find the focus. Since the parabola opens horizontally (x=a*y2 and p is negative, it opens to the left. The focus is at (h+p,k)

Focus=(0+(−2),0)=(−2,0)

6. Determine the directrix. For a horizontal parabola, the directrix is the vertical line x=h−p

x=0−(−2)

x=2

7. Identify the axis of symmetry. This is the line that passes through the vertex and focus, which is the horizontal line y=k

Axis of Symmetry:y=0

Final Answer

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

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