Parabolas
by aoum, Mar 16, 2025, 10:20 PM
Parabolas: The Geometry of Quadratics
A parabola is a fundamental curve in mathematics defined by a quadratic equation. Parabolas arise naturally in algebra, geometry, and physics—especially in the motion of projectiles and the shape of satellite dishes. This article explores the properties, equations, and applications of parabolas in detail.
1. Definition of a Parabola
A parabola is the set of all points equidistant from a fixed point called the focus and a fixed line called the directrix.
Formally, if
is the focus and 
is the directrix, a point 
lies on the parabola if:
![\[
\text{Distance to the focus} = \text{Distance to the directrix}.
\]](//latex.artofproblemsolving.com/4/9/f/49f6d534bc8103f8711097fd30141d6b7963dc6d.png)
This geometric property leads to the standard equation of a parabola.
2. The Standard Equation of a Parabola
If the vertex of the parabola is at the origin
and the focus is at 
, the parabola opens upwards and satisfies the equation:
![\[
y = \frac{1}{4p}x^2,
\]](//latex.artofproblemsolving.com/1/1/e/11e3c3dccafc22571d0d41b196bda6c88cb42373.png)
where:
More generally, the parabola's equation can be written as:
![\[
y = ax^2 + bx + c,
\]](//latex.artofproblemsolving.com/9/0/0/900b5922a9483a08e5068b5a8676c00661b4b20c.png)
where:
3. Orientation of Parabolas
Depending on the direction the parabola opens, the equation changes:
For a parabola opening rightward with focus
:
![\[
x = \frac{1}{4p}(y - k)^2 + h.
\]](//latex.artofproblemsolving.com/1/2/8/12810c7235a6000be43063b0d820ac68658dbd5e.png)
4. Important Features of a Parabola
Each parabola has several key components:
5. Deriving the Equation of a Parabola
Using the definition of a parabola:
Let the focus be and the directrix be 
. For any point on the parabola:
![\[
\sqrt{x^2 + (y - p)^2} = |y + p|.
\]](//latex.artofproblemsolving.com/1/a/0/1a05694d61bae7ac5ceb1e66db738a202e7b3da5.png)
Square both sides:
![\[
x^2 + (y - p)^2 = (y + p)^2,
\]](//latex.artofproblemsolving.com/2/c/b/2cb7b0837c68a352793e9f4bed7cfb468a78950f.png)
Expand and simplify:
![\[
x^2 = 4py,
\]](//latex.artofproblemsolving.com/1/8/4/1844b950d551e492ed5ec25def54ef65cff454c6.png)
which is the standard form for a parabola opening upward.
6. Parametric Form of a Parabola
The parabola
can also be described parametrically:
![\[
(x, y) = (t, at^2),
\]](//latex.artofproblemsolving.com/2/c/9/2c98b4ce5420455bd8e0d67dfed3af90edea5f4e.png)
where
is the parameter corresponding to horizontal distance.
For a parabola with focus
and directrix 
:
![\[
(x, y) = (h + 2pt, k + pt^2).
\]](//latex.artofproblemsolving.com/7/c/6/7c6f7b476ed186fcda6708c2304eef5bd2f44c5a.png)
7. Reflective Property of Parabolas
A parabola exhibits a unique reflective property:
8. Applications of Parabolas
Parabolas are not just theoretical curves—they have practical uses across multiple fields:
9. Example Problems
Example 1: Find the equation of a parabola with focus
and vertex at the origin.
Since the parabola opens upwards with
:
![\[
y = \frac{1}{4p}x^2 = \frac{1}{12}x^2.
\]](//latex.artofproblemsolving.com/f/f/c/ffc783199370adf3e2ff4a1943c38b56fda13660.png)
Example 2: What is the vertex, focus, and directrix of the parabola
?
Complete the square:
![\[
y = 2(x^2 - 2x) + 3 = 2((x - 1)^2 - 1) + 3,
\]](//latex.artofproblemsolving.com/a/8/5/a8509972be59f35eda83602e0d425b9e6aa18071.png)
Simplify:
![\[
y = 2(x - 1)^2 + 1.
\]](//latex.artofproblemsolving.com/2/c/7/2c766d1fc44bf4914634a2279034e72688ec32e7.png)
Thus, the vertex is
, and the focus is 
, with directrix 
.
10. Generalization: Paraboloids
In three dimensions, the analog of a parabola is a paraboloid. The standard form of a paraboloid opening along the
-axis is:
![\[
z = ax^2 + by^2.
\]](//latex.artofproblemsolving.com/0/5/1/0515574e7fc901e6a2c233d4e02d5ec8e0873f60.png)
If
, the paraboloid is circular; otherwise, it is elliptical.
11. Conclusion
Parabolas are a cornerstone of mathematics, connecting algebra, geometry, and real-world applications. Their reflective properties and unique shape make them indispensable in science and engineering.
References
A parabola is a fundamental curve in mathematics defined by a quadratic equation. Parabolas arise naturally in algebra, geometry, and physics—especially in the motion of projectiles and the shape of satellite dishes. This article explores the properties, equations, and applications of parabolas in detail.
1. Definition of a Parabola
A parabola is the set of all points equidistant from a fixed point called the focus and a fixed line called the directrix.
Formally, if
is the focus and 
is the directrix, a point 
lies on the parabola if:![\[
\text{Distance to the focus} = \text{Distance to the directrix}.
\]](http://latex.artofproblemsolving.com/4/9/f/49f6d534bc8103f8711097fd30141d6b7963dc6d.png)
This geometric property leads to the standard equation of a parabola.
2. The Standard Equation of a Parabola
If the vertex of the parabola is at the origin
and the focus is at 
, the parabola opens upwards and satisfies the equation:![\[
y = \frac{1}{4p}x^2,
\]](http://latex.artofproblemsolving.com/1/1/e/11e3c3dccafc22571d0d41b196bda6c88cb42373.png)
where:
is the distance from the vertex to the focus.- The directrix is the horizontal line .
is the distance from the vertex to the focus.More generally, the parabola's equation can be written as:
![\[
y = ax^2 + bx + c,
\]](http://latex.artofproblemsolving.com/9/0/0/900b5922a9483a08e5068b5a8676c00661b4b20c.png)
where:
determines the parabola's "width" and direction (up if 
, down if 
).
and 
shift the parabola horizontally and vertically.
determines the parabola's "width" and direction (up if 
,
)
and 
shift the parabola horizontally and vertically.3. Orientation of Parabolas
Depending on the direction the parabola opens, the equation changes:
- Opens Up/Down:
(vertex at ) - Opens Left/Right:
(vertex at )
(vertex at 
(vertex at For a parabola opening rightward with focus
:![\[
x = \frac{1}{4p}(y - k)^2 + h.
\]](http://latex.artofproblemsolving.com/1/2/8/12810c7235a6000be43063b0d820ac68658dbd5e.png)
4. Important Features of a Parabola
Each parabola has several key components:
- Vertex (
): The point where the parabola changes direction. - Focus (): The point inside the parabola where reflected rays converge.
- Directrix: A line perpendicular to the axis of symmetry.
- Axis of Symmetry: The vertical or horizontal line through the vertex, about which the parabola is symmetric.
- Latus Rectum: The line segment through the focus perpendicular to the axis of symmetry. Its length is
.
)
.5. Deriving the Equation of a Parabola
Using the definition of a parabola:
Let the focus be
and the directrix be 
. For any point on the parabola:![\[
\sqrt{x^2 + (y - p)^2} = |y + p|.
\]](http://latex.artofproblemsolving.com/1/a/0/1a05694d61bae7ac5ceb1e66db738a202e7b3da5.png)
Square both sides:
![\[
x^2 + (y - p)^2 = (y + p)^2,
\]](http://latex.artofproblemsolving.com/2/c/b/2cb7b0837c68a352793e9f4bed7cfb468a78950f.png)
Expand and simplify:
![\[
x^2 = 4py,
\]](http://latex.artofproblemsolving.com/1/8/4/1844b950d551e492ed5ec25def54ef65cff454c6.png)
which is the standard form for a parabola opening upward.
6. Parametric Form of a Parabola
The parabola
can also be described parametrically:![\[
(x, y) = (t, at^2),
\]](http://latex.artofproblemsolving.com/2/c/9/2c98b4ce5420455bd8e0d67dfed3af90edea5f4e.png)
where
is the parameter corresponding to horizontal distance.For a parabola with focus
and directrix 
:![\[
(x, y) = (h + 2pt, k + pt^2).
\]](http://latex.artofproblemsolving.com/7/c/6/7c6f7b476ed186fcda6708c2304eef5bd2f44c5a.png)
7. Reflective Property of Parabolas
A parabola exhibits a unique reflective property:
- Rays parallel to the axis of symmetry reflect through the focus.
- This property explains the shape of satellite dishes and parabolic microphones.
8. Applications of Parabolas
Parabolas are not just theoretical curves—they have practical uses across multiple fields:
- Physics: Describes the trajectory of projectiles under gravity.
- Engineering: Used in designing reflective surfaces like telescope mirrors and headlights.
- Architecture: Parabolic arches provide structural stability (e.g., the Gateway Arch).
- Optics: Satellite dishes use parabolas to focus signals.
9. Example Problems
Example 1: Find the equation of a parabola with focus
and vertex at the origin.Since the parabola opens upwards with
:![\[
y = \frac{1}{4p}x^2 = \frac{1}{12}x^2.
\]](http://latex.artofproblemsolving.com/f/f/c/ffc783199370adf3e2ff4a1943c38b56fda13660.png)
Example 2: What is the vertex, focus, and directrix of the parabola
?Complete the square:
![\[
y = 2(x^2 - 2x) + 3 = 2((x - 1)^2 - 1) + 3,
\]](http://latex.artofproblemsolving.com/a/8/5/a8509972be59f35eda83602e0d425b9e6aa18071.png)
Simplify:
![\[
y = 2(x - 1)^2 + 1.
\]](http://latex.artofproblemsolving.com/2/c/7/2c766d1fc44bf4914634a2279034e72688ec32e7.png)
Thus, the vertex is
, and the focus is 
, with directrix 
.10. Generalization: Paraboloids
In three dimensions, the analog of a parabola is a paraboloid. The standard form of a paraboloid opening along the
-axis is:![\[
z = ax^2 + by^2.
\]](http://latex.artofproblemsolving.com/0/5/1/0515574e7fc901e6a2c233d4e02d5ec8e0873f60.png)
If
, the paraboloid is circular; otherwise, it is elliptical.11. Conclusion
Parabolas are a cornerstone of mathematics, connecting algebra, geometry, and real-world applications. Their reflective properties and unique shape make them indispensable in science and engineering.
References
- Wikipedia: Parabola
- Stewart, J. Calculus: Early Transcendentals (8th Edition).
- AoPS Wiki: Parabola
- Coxeter, H. Introduction to Geometry (2nd Edition).
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