Question

Which equation represents a parabola opening downward with a vertex of the origin and a focus at (0,-1)?

y= -1/4x^2
y=1/4x^2
y=1/12x^2
y=-1/12x^2

Answer 1

The equation that represents a parabola opening downward with a vertex of the origin and a focus at (0,-1) is:

y = -1/4x^2

The standard form of the equation of a parabola with vertex at the origin and focus (0, p) is given by:

y^2 = 4px

In this case, p = -1, so the equation becomes:

y^2 = -4x

Solving for y, we get:

y = ± √(-4x)

Since the parabola opens downward, we take the negative square root:

y = -√(-4x)

Simplifying, we get:

y = -2√(x)

Multiplying by a constant, we get:

y = -1/4(4√(x))^2

Therefore, the equation that represents a parabola opening downward with a vertex of the origin and a focus at (0,-1) is:

y = -1/4x^2.

So, the answer is "y = -1/4x^2."