Table of Contents
Geometric definition of a parabola
A parabola is the set of all points in a plane that are the same distance from a fixed point (called the focus) and a fixed line (called the directrix).
More precisely, a point $P(x,y)$ lies on a parabola if its distance to the focus and its distance to the directrix are equal.
If the focus is $F$ and the directrix is a line $d$, then
$$
\text{distance}(P,F) = \text{distance}(P,d)
$$
is the defining condition of the parabola.
From this geometric definition, the algebraic equations of parabolas are derived.
Standard equations of parabolas
We focus on simple “standard positions” where the parabola is centered around the origin and opens along one of the coordinate axes.
Parabola opening up or down
Consider a parabola whose vertex is at the origin $(0,0)$ and opens upward or downward. Let its focus be on the $y$-axis and its directrix be a horizontal line. The standard equation can be written as
$$
x^2 = 4py,
$$
where $p$ is a nonzero constant.
- The vertex is at $(0,0)$.
- The focus is at $(0,p)$.
- The directrix is the line $y = -p$.
- If $p>0$, the parabola opens upward.
- If $p<0$, the parabola opens downward.
The distance $|p|$ is the distance from the vertex to the focus (and also to the directrix).
Parabola opening right or left
Similarly, if the vertex is at the origin and the parabola opens to the right or to the left along the $x$-axis, the standard equation is
$$
y^2 = 4px.
$$
- The vertex is at $(0,0)$.
- The focus is at $(p,0)$.
- The directrix is the line $x = -p$.
- If $p>0$, the parabola opens to the right.
- If $p<0$, it opens to the left.
Again, $|p|$ is the distance from the vertex to the focus (and to the directrix).
Vertex form and translations
A parabola does not have to be centered at the origin. If we translate the parabola so that its vertex moves to $(h,k)$, the equations shift accordingly.
Vertical parabolas (axis parallel to $y$-axis)
Parabolas that open up or down with vertex $(h,k)$ have the form
$$
(x - h)^2 = 4p(y - k).
$$
- Vertex: $(h,k)$.
- Focus: $(h, k + p)$.
- Directrix: $y = k - p$.
- If $p>0$, opens upward.
- If $p<0$, opens downward.
Horizontal parabolas (axis parallel to $x$-axis)
Parabolas that open right or left with vertex $(h,k)$ have the form
$$
(y - k)^2 = 4p(x - h).
$$
- Vertex: $(h,k)$.
- Focus: $(h + p, k)$.
- Directrix: $x = h - p$.
- If $p>0$, opens to the right.
- If $p<0$, opens to the left.
In both cases, the line through the focus and vertex (and perpendicular to the directrix) is called the axis of symmetry of the parabola.
Focus, directrix, and parameter $p$
The parameter $p$ plays a geometric role:
- $|p|$ is the distance from the vertex to the focus.
- $|p|$ is also the distance from the vertex to the directrix.
- The axis of symmetry is the line passing through the vertex and focus, perpendicular to the directrix.
For the four principal forms:
- $x^2 = 4py$
- Vertex: $(0,0)$
- Focus: $(0,p)$
- Directrix: $y = -p$
- Axis: $x = 0$
- $y^2 = 4px$
- Vertex: $(0,0)$
- Focus: $(p,0)$
- Directrix: $x = -p$
- Axis: $y = 0$
- $(x-h)^2 = 4p(y-k)$
- Vertex: $(h,k)$
- Focus: $(h, k+p)$
- Directrix: $y = k-p$
- Axis: $x = h$
- $(y-k)^2 = 4p(x-h)$
- Vertex: $(h,k)$
- Focus: $(h+p, k)$
- Directrix: $x = h-p$
- Axis: $y = k$
Latus rectum and width of a parabola
An important segment of a parabola is the latus rectum.
- For a parabola, the latus rectum is the line segment through the focus, perpendicular to the axis of symmetry, whose endpoints lie on the parabola.
For the standard vertical form $x^2 = 4py$ with $p>0$:
- Focus: $(0,p)$.
- Axis of symmetry: $x = 0$.
- The latus rectum is horizontal through $(0,p)$.
To find its endpoints, substitute $y = p$ into $x^2 = 4py$:
$$
x^2 = 4p(p) = 4p^2.
$$
So $x = \pm 2p$. The endpoints are $(-2p, p)$ and $(2p, p)$.
Thus:
- The length of the latus rectum is $4|p|$.
- This length can be viewed as a measure of how “wide” the parabola is near the focus.
Analogous conclusions hold for horizontal parabolas of the form $y^2 = 4px$.
Parabolas and quadratic functions
For parabolas whose axis is vertical (opening up or down), the equation can also be written as a quadratic function of $x$:
$$
y = ax^2 + bx + c,
$$
with $a \neq 0$.
In analytic geometry of parabolas, one often moves between this form and the vertex form.
From quadratic form to vertex form
The vertex form of a vertical parabola is
$$
y = a(x - h)^2 + k,
$$
where $(h,k)$ is the vertex.
The connection between
$$
y = ax^2 + bx + c
\quad\text{and}\quad
y = a(x - h)^2 + k
$$
is obtained by completing the square (handled elsewhere). The key geometric facts:
- Vertex:
$$
h = -\frac{b}{2a}, \qquad k = f(h) = a h^2 + b h + c.
$$ - If $a>0$, the parabola opens upward; if $a<0$, it opens downward.
- The axis of symmetry is the vertical line $x = h$.
Once the equation is in the form
$$
(x - h)^2 = 4p(y - k),
$$
the focus, directrix, and $p$ can be read off as before. The relationship between $a$ and $p$ here is
$$
a = \frac{1}{4p}
\quad \text{for vertical parabolas in vertex form}.
$$
Graphing parabolas from their equations
Graphing a parabola efficiently uses its geometric features.
For a vertical parabola in vertex form
$$
(x - h)^2 = 4p(y - k) \quad \text{or} \quad y = a(x - h)^2 + k,
$$
one can:
- Identify the vertex $(h,k)$.
- Determine whether it opens up or down from the sign of $p$ (or $a$).
- Find the focus and directrix using $p$.
- Optionally, plot the endpoints of the latus rectum to gauge the shape near the focus.
- Sketch the symmetric curve with respect to the axis of symmetry.
For a horizontal parabola in the form
$$
(y - k)^2 = 4p(x - h),
$$
the procedure is similar, but openings are left or right, and the axis of symmetry is horizontal.
Parabolas as conic sections
Within the broader study of conic sections, parabolas correspond to the case where the plane cuts the cone parallel to a generating line of the cone. Analytically, parabolas can be characterized by their eccentricity:
- The eccentricity of a parabola is $e = 1$.
This fits into the general pattern (for conics):
- Ellipses: $0 < e < 1$.
- Parabolas: $e = 1$.
- Hyperbolas: $e > 1$.
In the parabola case, the equality of distances to the focus and directrix exactly matches the eccentricity being $1$.
Applications of parabolas (geometric viewpoint)
Several physical and geometric phenomena naturally involve parabolas:
- Reflective property: A ray coming in parallel to the axis of symmetry of a parabola reflects off the parabolic curve and passes through the focus (for ideal reflections). This is used in:
- Satellite dishes.
- Parabolic mirrors in telescopes.
- Car headlights and flashlights.
- Projectile motion (idealized): In a uniform gravitational field with no air resistance, the path of a projectile thrown with an initial velocity is a parabola in the $x$–$y$ plane.
These applications rely heavily on the geometric and analytic properties of parabolas developed in this chapter.