Parabolas As Conic Sections
Parabolas are one of the four shapes known as conic sections, and they have many important real world applications.
Learning Objective

Describe the parts of a parabola as parts of a conic section
Key Points
 A parabola is formed by the intersection of a plane and a right circular cone.
 All parabolas contain a focus, a directrix, and an axis of symmetry. These vary in exact location depending on the equation used to define the parabola.
 Parabolas are frequently used in physics and engineering for things such as the design of automobile headlight reflectors and the paths of ballistic missiles.

Parabolas are frequently encountered as graphs of quadratic functions, including the very common equation
$y=x^2$ .
Terms

focus
A point inside the parabolic section defined by forming a right triangle with the axis of symmetry and the cone's horizontal radius.

axis of symmetry
A line that divides the parabola into two equal halves and also passes through the vertex of the parabola.

vertex
The point where the plane intersects the exterior surface of the right circular cone, forming one end of the parabola.

ballistic
Relating to projectiles moving under their own momentum, air drag, gravity, and sometimes rocket power

directrix
A line used to define a curve or surface, especially a line from which any point on a parabola curve has a distance equal to the distance from the focus.
Full Text
In mathematics, a parabola is a conic section, created from the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface. In other words, the plane is at the same angle as the outside surface of the cone.
Parabolic conic section
This diagram shows how a parabola is generated by the intersection of a plane with a right circular cone. It also shows the
All parabolas have the same set of basic features. The axis of symmetry is a line that is at the same angle as the cone and divides the parabola in half.
Vertex
The vertex is the point where the plane intersects the exterior surface of the cone. It forms the rounded end of the parabola. The vertex is therefore also a point on the cone, and the distance between that point and the cone's central axis is the radius of a circle.
Focus
In the diagram showing the parabolic conic section, a red line is drawn from the center of that circle to the axis of symmetry, so that a right angle is formed. The point on the axis of symmetry where the right angle is located is called the focus. By doing this, a right triangle is created.
Right triangle
A right triangle is formed from the focal point of the parabola.
The focal length is the leg of the right triangle that exists along the axis of symmetry, and the focal point is the vertex of the right triangle. Using the definition of sine as opposite over hypotenuse, we can find a formula for the focal length "
Directrix
All parabolas have a directrix. The directrix is a straight line on the opposite side of the parabolic curve from the focus. The parabolic curve itself is the set of all points that are equidistant (equal distances) from both the directrix line and the focus.
Features of Parabolas
On a coordinate plane, parabolas are frequently encountered as graphs of quadratic functions, such as:
Parabolas can open up, down, left, right, or in some other arbitrary direction. Any parabola can be repositioned and rescaled to fit exactly on any other parabola—that is, all parabolas are similar.
To locate the
For example, in the parabola
Parabolas have the property that, if they are made of material that reflects light, then light which enters a parabola traveling parallel to its axis of symmetry is reflected to its focus. This happens regardless of where on the parabola the reflection occurs. Conversely, light that originates from a point source at the focus is reflected, or collimated, into a parallel beam. The light leaves the parabola parallel to the axis of symmetry. The same effect occurs with sound and other forms of energy. This reflective property is the basis of many practical uses of parabolas.
The parabola has many important applications, from the design of automobile headlight reflectors to calculating the paths of ballistic missiles. They are frequently used in physics, engineering, and other sciences.
Key Term Reference
 Parabola
 Appears in these related concepts: Conics in Polar Coordinates, Nonlinear Systems of Inequalities, and Parts of an Ellipse
 Reflection
 Appears in these related concepts: Symmetry of Functions and Inverse Trigonometric Functions
 ballistics
 Appears in these related concepts: Other Geophysical Applications and Microwaves
 circle
 Appears in these related concepts: Introduction to Circles, Types of Conic Sections, and Applications of Circles and Ellipses
 conic section
 Appears in these related concepts: Circles as Conic Sections, Applications of Hyperbolas, and What Are Conic Sections?
 conical
 Appears in these related concepts: Applications of the Parabola and Nonlinear Systems of Equations and ProblemSolving
 direction
 Appears in this related concept: Parallel and Perpendicular Lines
 distance
 Appears in these related concepts: Inequalities with Absolute Value, The Distance Formula and Midpoints of Segments, and Linear Mathematical Models
 equation
 Appears in these related concepts: Equations and Inequalities, Graphs of Equations as Graphs of Solutions, and What is an Equation?
 focal point
 Appears in these related concepts: Nearsightedness, Farsidedness, and Vision Correction, Refraction Through Lenses, and Thin Lenses and Ray Tracing
 function
 Appears in these related concepts: Functions and Their Notation, The Vertical Line Test, and Solving Differential Equations
 graph
 Appears in these related concepts: Graphical Representations of Functions, Graphing Equations, and Reading Points on a Graph
 hypotenuse
 Appears in these related concepts: Arc Length and Speed, Trigonometric Functions, and Right Triangles and the Pythagorean Theorem
 legs
 Appears in these related concepts: Slope and Defining Trigonometric Functions on the Unit Circle
 parabola
 Appears in these related concepts: The Quadratic Formula, Parts of a Parabola, and Completing the Square
 point
 Appears in these related concepts: The Intermediate Value Theorem, Polynomial and Rational Functions as Models, and Graphing Quadratic Equations In Standard Form
 quadratic
 Appears in these related concepts: Stretching and Shrinking, What is a Quadratic Function?, and The Discriminant
 quadratic function
 Appears in these related concepts: Other Equations in Quadratic Form, Scientific Applications of Quadratic Functions, and Standard Form and Completing the Square
 radius
 Appears in these related concepts: Interosseous Membranes, Introduction to the Polar Coordinate System, and Ulna and Radius (The Forearm)
 reflection
 Appears in these related concepts: Dispersion of the Visible Spectrum, Transformations of Functions, and Reflections
 right triangle
 Appears in these related concepts: How Trigonometric Functions Work, Sine, Cosine, and Tangent, and Finding Angles From Ratios: Inverse Trigonometric Functions
 set
 Appears in these related concepts: Sequences, Introduction to Sequences, and Sequences of Mathematical Statements
 term
 Appears in these related concepts: The 22nd Amendment, Basics of Graphing Polynomial Functions, and Introduction to Variables
 vertices
 Appears in these related concepts: Octahedral Complexes, Introduction to Hyperbolas, and Parts of a Hyperbola
Sources
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