Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin.
Passes through the point (2,4); horizontal axis

Question

Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. ​
Passes through the point (2,4); horizontal axis
​

Quizplus · Accepted Answer

Since the vertex is at the origin and the parabola has a horizontal axis, the standard form of the equation of the parabola is: $y^2=4px$ where $p$ is the distance from the vertex to the focus (and also the distance from the vertex to the directrix). 
Since the vertex is at the origin, we have $p=\frac{1}{4}$ of the distance between the vertex and the point on the parabola that is closest to the directrix. Since the axis of the parabola is horizontal, this closest point is to the left of the vertex. Therefore, this point is $(2-p,4)$.
Substituting this point into the equation $y^2=4px$, we have:
$16=(2-p)^2$
Solving for $p$, we have $p=\frac{1}{4}$.
Therefore, the equation of the parabola is $y^2=4\cdot\frac{1}{4}x$, which simplifies to $y^2=x$. 
The answer is C because $y^2=x$ is only listed in option C.

Find the Standard Form of the Equation of the Parabola