Q.Find the equation of the parabola that satisfies the given conditions: Vertex $(0, 0)$ passing through $(2, 3)$ and axis is along $x$-axis.

Concept: Standard form of a parabola with vertex at the origin and horizontal axis.

When a parabola has vertex at $(0, 0)$ and its axis along the $x$-axis, it opens either to the right or to the left. The standard equation is:

where $a > 0$ means the parabola opens rightward, and $a < 0$ means it opens leftward.

Since the parabola passes through $(2, 3)$, substitute $x = 2$ and $y = 3$:

$a = \frac{9}{8}$ …

When a parabola has its vertex at the origin and opens along one of the coordinate axes, it takes on a particularly clean standard form. The key insight here is recognizing which standard form applies.

Since the axis of the parabola is along the $x$-axis, the parabola opens either to the right or to the left. The standard equation for such a parabola with vertex at $(0, 0)$ is:

where $a$ is the focal parameter. If $a > 0$, the parabola opens to the right; if $a < 0$, it opens to the left. The focus is at $(a, 0)$ and the directrix is the line $x = -a$.

This form differs from the more familiar $x = ay^2$ precisely because we're treating $x$ as a function of $y$ rather than the reverse—the parabola is "horizontal" rather than "vertical."

Now we determine the value of $a$ using the given point.

1. Start with the standard form. Since the vertex is at the origin and the axis is the $x$-axis, we have:

2. Substitute the point $(2, 3)$. The parabola passes through this point, so when $x = 2$, we must have $y = 3$:

$9 = 8a$ …