What is the equation of a parabola with its vertex at the origin and the directrix at x = 4.75?

The equation of a parabola can be expressed in the vertex form, which for a parabola with its vertex at the origin (0,0) is either:

y² = 4px (if the parabola opens to the right or left) or x² = 4py (if the parabola opens up or down).

The letter 'p' represents the distance from the vertex to the focus of the parabola or, equivalently, from the vertex to the directrix. The sign in front of 'p' tells you the direction the parabola opens. Since in this case the directrix is vertical and given as x = 4.75, it means the parabola opens toward the left, away from the directrix. Hence, 'p' will be negative because the focus is to the left of the vertex at the origin.

Since the directrix is at x = 4.75, the distance from the origin to the directrix is 4.75 units. Therefore, p = -4.75.

Now we can write the equation of the parabola as:

y² = 4px y² = 4 * (-4.75) * x y² = -19x

So the equation of the parabola with its vertex at the origin and the directrix at x = 4.75 is y² = -19x.