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Transformations of Functions
Transformations alter a function while maintaining the original characteristics of that funcction.
Learning Objective

Differentiate between three common types of translations: reflections, rotations, and scaling
Key Points
 Transformations are ways that a function can be adjusted to create new functions.
 Transformations often preserve the original shape of the function.
 Common types of transformations include rotations, translations, reflections, and scaling (also known as stretching/shrinking).
Term

euclidean space
The threedimensional space of Euclidean geometry. The term "Euclidean" distinguishes these spaces from the curved spaces of nonEuclidean geometry and Einstein's general theory of relativity.
Full Text
A transformation could be any function mapping a set, X, on to another set or on to itself. However, often the set X has some additional algebraic or geometric structure and the term "transformation" refers to a function from X to itself that preserves this structure.
Examples include translations, reflections, rotations, and scaling. These can be carried out in Euclidean space, particularly in dimensions 2 and 3. They are also operations that can be performed using linear algebra and described explicitly using matrices.
A translation, or translation operator, is an affine transformation of Euclidean space which moves every point by a fixed distance in the same direction. It can also be interpreted as the addition of a constant vector to every point, or as the shifting of the origin of the coordinate system. In other words, if v is a fixed vector, then the translation Tv will work as
A reflection is a map that transforms an object into its mirror image. In geometry a "mirror" is a hyperplane of fixed points. For example, a reflection of the small English letter p in respect to a vertical line would look like q. In order to reflect a planar figure one needs the "mirror" to be a line (axis of reflection or axis of symmetry), while for reflections in the threedimensional space one would use a plane (the plane of reflection or symmetry) for a mirror.
A rotation is a transformation that is performed by "spinning" the object around a fixed point known as the center of rotation. You can rotate your object at any degree measure but 90° and 180° are two of the most common.
Uniform scaling is a linear transformation that enlarges or diminishes objects. The scale factor is the same in all directions; it is also called a homothety or dilation. The result of uniform scaling is similar (in the geometric sense) to the original. Scaling can also be referred to as "stretching" or "shrinking" a function.
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Key Term Reference
 constant
 Appears in these related concepts: Inverse Variation, Quadratic Functions of the Form f(x) = a(xh)^2 + k, and Direct Variation
 degree
 Appears in these related concepts: Experimental Probabilities, Polynomials: Introduction, Addition, and Subtraction, and Partial Fractions
 distance
 Appears in these related concepts: Symmetry, Linear Mathematical Models , and The Distance Formula and Midpoints of Segments
 factor
 Appears in these related concepts: Randomized Design: SingleFactor, Finding Factors of Polynomials, and Solving Quadratic Equations by Factoring
 function
 Appears in these related concepts: Inverse Functions, Solving Differential Equations, and Functions and Their Notation
 geometric
 Appears in these related concepts: Addition, Subtraction, and Multiplication, Sums and Series, and Introduction to Sequences
 linear
 Appears in these related concepts: SecondOrder Linear Equations, Linear Approximation, and Graphs of Linear Inequalities
 matrix
 Appears in these related concepts: The Identity Matrix, The Inverse of a Matrix, and IQ Tests
 operation
 Appears in these related concepts: Designing the Operation, Integer Exponents, and Converting between Exponential and Logarithmic Equations
 point
 Appears in these related concepts: The Intermediate Value Theorem, Graphing Equations, and Introduction: Polynomial and Rational Functions and Models
 reflection
 Appears in these related concepts: The Law of Reflection and Its Consequences, Dispersion of the Visible Spectrum, and The Ray Aspect of Light
 set
 Appears in these related concepts: Sequences of Statements, Sequences, and Expressions and Sets of Numbers
 term
 Appears in these related concepts: Basics of Graphing Polynomial Functions, The 22nd Amendment, and Democracy
 transformation
 Appears in these related concepts: Cancer Viruses, Putting Foreign DNA into Cells, and Reflections
 vector
 Appears in these related concepts: Vectors in the Plane, Electric Field Lines: Multiple Charges, and Introduction to Memory Storage
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Cite This Source
Source: Boundless. “Transformations of Functions.” Boundless Algebra. Boundless, 21 Jul. 2015. Retrieved 29 Apr. 2016 from https://www.boundless.com/algebra/textbooks/boundlessalgebratextbook/graphsfunctionsandmodels2/transformations21/transformationsoffunctions1175543/