Equation of Parabola in Standard Form

A parabola is a set of all points in the plane that are equidistant from a fixed point, called the focus, and a fixed line, called the directrix. The standard form of a parabola's equation is derived from this definition and depends on the orientation of the parabola.

Standard Forms of Parabola

There are two main orientations of a parabola: vertical and horizontal. The standard form of the equation changes accordingly.

Vertical Parabolas

For a vertical parabola that opens upwards or downwards, the standard form of the equation is:

$$ y - k = a(x - h)^2 $$

where $(h, k)$ is the vertex of the parabola, and $a$ is a non-zero constant that determines how "wide" or "narrow" the parabola is and the direction it opens. If $a > 0$, the parabola opens upwards, and if $a < 0$, it opens downwards.

Horizontal Parabolas

For a horizontal parabola that opens to the right or left, the standard form of the equation is:

$$ x - h = a(y - k)^2 $$

where $(h, k)$ is the vertex of the parabola, and $a$ is a non-zero constant that determines the "width" of the parabola and the direction it opens. If $a > 0$, the parabola opens to the right, and if $a < 0$, it opens to the left.

Important Points and Differences

Feature	Vertical Parabola	Horizontal Parabola
Standard Form	$y - k = a(x - h)^2$	$x - h = a(y - k)^2$
Vertex	$(h, k)$	$(h, k)$
Axis of Symmetry	Line $x = h$	Line $y = k$
Direction of Opening	Upwards if $a > 0$, Downwards if $a < 0$	Right if $a > 0$, Left if $a < 0$
Focus	$(h, k + \frac{1}{4a})$	$(h + \frac{1}{4a}, k)$
Directrix	Line $y = k - \frac{1}{4a}$	Line $x = h - \frac{1}{4a}$
Latus Rectum Length	$	4a

Formulas

The distance between the vertex and the focus, known as the focal length, is given by $\frac{1}{4a}$ for both vertical and horizontal parabolas. The length of the latus rectum, which is a line segment perpendicular to the axis of symmetry and passes through the focus, is $|4a|$.

Examples

Example 1: Vertical Parabola

Consider the parabola with the equation $y - 3 = 2(x + 1)^2$. Here, the vertex is at $(-1, 3)$, and since $a = 2 > 0$, the parabola opens upwards. The focus is at $(-1, 3 + \frac{1}{8}) = (-1, 3.125)$, and the directrix is the line $y = 3 - \frac{1}{8} = 2.875$.

Example 2: Horizontal Parabola

Consider the parabola with the equation $x + 2 = -\frac{1}{2}(y - 4)^2$. Here, the vertex is at $(-2, 4)$, and since $a = -\frac{1}{2} < 0$, the parabola opens to the left. The focus is at $(-2 - \frac{1}{2}, 4) = (-2.5, 4)$, and the directrix is the line $x = -2 + \frac{1}{2} = -1.5$.

Understanding the standard form of a parabola's equation is crucial for graphing the parabola and determining its properties, such as the vertex, focus, directrix, axis of symmetry, and the direction in which it opens.