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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 parabolas 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 that vary in exact location depending on the equation used to define the parabola.
 Parabolas are frequently used in physics and engineering in places such as automobile headlight reflectors and in the design 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
Or 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 y and x axes that would be used to view the parabola on a coordinate graph. The vertex of the parabola here is point P, and the diagram shows the radius r between that point and the cone's central axis, as well as the angle
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 above, 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:
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 "f" in terms of the radius and the angle:
Directrix
Lastly, 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 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: y=x^2
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
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, 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, leaving the parabola parallel to the axis of symmetry. The same effects occur 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 automobile headlight reflectors to the design of ballistic missiles. They are frequently used in physics, engineering, and many other areas.
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Key Term Reference
 Reflection
 Appears in this related concept: Inverse Trigonometric Functions
 ballistics
 Appears in these related concepts: Other Geophysical Applications and Microwaves
 circle
 Appears in these related concepts: Applications of Circles and Ellipses, Introduction to Circles, and Types of Conic Sections
 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
 distance
 Appears in these related concepts: Inequalities with Absolute Value, Linear Mathematical Models , and The Distance Formula and Midpoints of Segments
 equation
 Appears in these related concepts: Equations and Inequalities, Graphs of Equations as Graphs of Solutions, and What is an Equation?
 function
 Appears in these related concepts: Solving Differential Equations, Visualizing Domain and Range, and The Vertical Line Test
 graph
 Appears in these related concepts: Reading Points on a Graph, Graphing Equations, and Graphical Representations of Functions
 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: Defining Trigonometric Functions on the Unit Circle and Pythagorean Identities
 parabola
 Appears in these related concepts: Completing the Square, The Quadratic Formula, and Parts of a Parabola
 point
 Appears in these related concepts: The Intermediate Value Theorem, Graphing Quadratic Equations In Standard Form, and Polynomial and Rational Functions as Models
 quadratic
 Appears in these related concepts: Stretching and Shrinking, The Discriminant, and What is a Quadratic Function?
 quadratic function
 Appears in these related concepts: Standard Form and Completing the Square, Other Equations in Quadratic Form, and Scientific Applications of Quadratic Functions
 radius
 Appears in these related concepts: Interosseous Membranes, Velocity of Blood Flow, and Ulna and Radius (The Forearm)
 reflection
 Appears in these related concepts: Transformations of Functions, Reflections, and Dispersion of the Visible Spectrum
 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 of Mathematical Statements, Sequences, and Introduction to Sequences
 symmetry
 Appears in these related concepts: Symmetry of Functions, Balance, and Rhythm
 term
 Appears in these related concepts: Basics of Graphing Polynomial Functions, The 22nd Amendment, and Introduction to Variables
Sources
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Cite This Source
Source: Boundless. “Parabolas As Conic Sections.” Boundless Algebra. Boundless, 07 Jul. 2016. Retrieved 25 Jul. 2016 from https://www.boundless.com/algebra/textbooks/boundlessalgebratextbook/conicsections341/theparabola49/parabolasasconicsections2135832/