Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin.
Directrix: y = -1

A) $y ^ { 2 } = 4 x$
B) $x ^ { 2 } = - \frac { y } { 4 }$
C) $x ^ { 2 } = 4 y$
D) $y ^ { 2 } = - 4 x$
E) $x ^ { 2 } = - 4 y$

Question

Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. 
Directrix: y = -1

A) $y ^ { 2 } = 4 x$
B) $x ^ { 2 } = - \frac { y } { 4 }$
C) $x ^ { 2 } = 4 y$
D) $y ^ { 2 } = - 4 x$
E) $x ^ { 2 } = - 4 y$

Quizplus · Accepted Answer

EThe standard form of a parabola with its vertex at the origin and a vertical axis of symmetry is $x^2 = 4py$, where $p$ is the distance from the vertex to the focus (or equivalently, from the vertex to the directrix). Since the directrix is $y = -1$, the focus is at $y = 1$, making $p = 1$. Thus, the equation is $x^2 = 4y$. For a parabola with a horizontal axis of symmetry, the standard form is $y^2 = 4px$, and since the directrix is $y = -1$, the focus is again $y = 1$, making $p = -1$. Therefore, the equation is $y^2 = -4x$. However, the mistake in my explanation indicates that only one option should be correct based on the given directrix, which is $y = -1$, suggesting the parabola opens upwards. Hence, the correct equation reflecting a vertical orientation and opening upwards from the origin is $x^2 = 4y$, making the correct choice C.

Find the Standard Form of the Equation of the Parabola y2=4xy ^ { 2 } = 4 xy2=4x

Find the Standard Form of the Equation of the Parabola $y ^ { 2 } = 4 x$