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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
Interactive Graph: Basic Quadratic Equation
Graph of three different y=x^{2} equations. The blue curve indicates the original function while the red and green curves represent two vertical translations of that function. Try to move the graph vertically. For more of a challenge, try to move the graph horizontally.
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: Inequalities with Absolute Value, Symmetry, and The Distance Formula and Midpoints of Segments
 factor
 Appears in these related concepts: Randomized Design: SingleFactor, The Perceptual Process, and Solving Quadratic Equations by Factoring
 function
 Appears in these related concepts: Inverse Functions, Solving Differential Equations, and Modal Mixture
 geometric
 Appears in these related concepts: Addition, Subtraction, and Multiplication, Sums and Series, and The Cartesian System
 linear
 Appears in these related concepts: Trinomials of the Form ax^2 + bx + c, Where a is Not Equal to 1, Exponential Growth and Decay, and Graphs of Linear Inequalities
 matrix
 Appears in these related concepts: The Inverse of a Matrix, Matrix Equations, and Matrix Multiplication
 operation
 Appears in these related concepts: Outsourcing, Designing the Operation, and Integer Exponents
 point
 Appears in these related concepts: The Intermediate Value Theorem, Quadratic Functions of the Form f(x) = ax^2 + bx + c, Where a is not Equal to 0, and Graphing Equations
 reflection
 Appears in these related concepts: Reflections, Dispersion of the Visible Spectrum, and The Ray Aspect of Light
 set
 Appears in these related concepts: Sequences, Introduction to Sequences, and Expressions and Sets of Numbers
 term
 Appears in these related concepts: Arithmetic Sequences, Basics of Graphing Polynomial Functions, and The 22nd Amendment
 transformation
 Appears in these related concepts: Bacterial Transformation, Horizontal Gene Transfer, and Prokaryotic Reproduction
 vector
 Appears in these related concepts: Force, Angular vs. Linear Quantities, and Superposition of Electric Potential
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Source: Boundless. “Transformations of Functions.” Boundless Algebra. Boundless, 21 Jul. 2015. Retrieved 22 Jul. 2015 from https://www.boundless.com/algebra/textbooks/boundlessalgebratextbook/graphsfunctionsandmodels2/transformations21/transformationsoffunctions1175543/