Find the equation of a parabola with its vertex at the origin and a focus of left parenthesis negative 8 comma 0 right parenthesis.

Question

(1 point)
Responses

y squared equals negative 32 x
Image with alt text: y squared equals negative 32 x

y squared equals negative 2 x
Image with alt text: y squared equals negative 2 x

x squared equals negative 32 y
Image with alt text: x squared equals negative 32 y

x squared equals negative 2 y

Answer 1

To find the equation of a parabola with its vertex at the origin and a focus at the point $(-8,0)$, we can use the standard form of the parabola's equation. Since the focus is to the left of the vertex, the parabola opens to the left.

The standard form of a horizontally oriented parabola (opening left or right) with vertex at the origin is:

$$y^2=4px$$

Here, $p$ is the distance from the vertex to the focus. In this case, the focus is at $(-8,0)$, which means $p=-8$ (since it's to the left, we use a negative value).

Substituting $p$ into the equation gives:

$$y^2=4(-8)x$$ $$y^2=-32x$$

Thus, the equation of the parabola is:

$$y^2=-32x$$

Therefore, the correct response is:

y squared equals negative 32 x.