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Parabolas
Parabolas are common in algebra as the graphs of quadratic functions, and they have many important real world applications.
Learning Objective

Identify the focus, vertex, and axis of symmetry of a parabola given its formula
Key Points
 Parabolas are frequently encountered as graphs of quadratic functions, including the very common equation y=x^2.
 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.
Terms

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, the distance from which a point on a conic has a constant ratio to that from the focus

vertex
A point on the curve with a local minimum or maximum of curvature.
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. Another way to generate a parabola is to examine a point (the focus) and a line (the directrix), as can be visualized in . The locus of points in that plane that are equidistant from both the line and point is a parabola. In algebra, parabolas are frequently encountered as graphs of quadratic functions, such as:
Interactive Graph: Parabola with Focus and Directrix
Graph of the parabola where . Parabolic curve showing directrix (L) and focus (F). The distance from any point on the parabola to the focus (PnF) equals the perpendicular distance from the same point on the parabola to the directrix (PnQn).
The line perpendicular to the directrix and passing through the focus, that is, the line that splits the parabola through the middle, is called the axis of symmetry. The point on the axis of symmetry that intersects the parabola is called the "vertex", and it is the point where the curvature is greatest. The distance between the vertex and the focus, measured along the axis of symmetry, is the "focal length". 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 xcoordinate of the vertex, cast the equation for y in terms of
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
 ballistics
 Appears in these related concepts: Other Geophysical Applications and Microwaves
 conic section
 Appears in this related concept: Applications and ProblemSolving
 conical
 Appears in these related concepts: Quadratic Functions of the Form f(x) = ax^2 + bx + c, Where a is not Equal to 0, Applications and ProblemSolving, and Nonlinear Systems of Equations and ProblemSolving
 distance
 Appears in these related concepts: Inequalities with Absolute Value, Symmetry, and The Distance Formula and Midpoints of Segments
 equation
 Appears in these related concepts: A General Approach, Equations and Inequalities, and Equations and Their Solutions
 function
 Appears in these related concepts: Inverse Functions, Solving Differential Equations, and Functions and Their Notation
 graph
 Appears in these related concepts: Graphing on Computers and Calculators, Reading Points on a Graph, and Graphing Functions
 parabola
 Appears in these related concepts: Standard Form and Completing the Square, The Quadratic Formula, and Standard Equations of Hyperbolas
 point
 Appears in these related concepts: The Intermediate Value Theorem, Circles, and Graphing Equations
 quadratic
 Appears in these related concepts: Solving Quadratic Equations by Factoring, The Discriminant, and Quadratic Functions of the Form f(x) = a(xh)^2 + k
 quadratic function
 Appears in these related concepts: Stretching and Shrinking, Zeroes of Polynomial Functions with Real Coefficients, and Quadratic Equations and Quadratic Functions
 reflection
 Appears in these related concepts: Reflections, Dispersion of the Visible Spectrum, and The Ray Aspect of Light
 term
 Appears in these related concepts: Arithmetic Sequences, Basics of Graphing Polynomial Functions, and The 22nd Amendment
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Cite This Source
Source: Boundless. “Parabolas.” Boundless Algebra. Boundless, 21 Jul. 2015. Retrieved 22 Jul. 2015 from https://www.boundless.com/algebra/textbooks/boundlessalgebratextbook/conicsections7/theparabola49/parabolas2135832/