# Transformations of Functions

## Transformations alter a function while maintaining the original characteristics of that funcction.

#### 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).

#### Terms

• The three-dimensional space of Euclidean geometry. The term “Euclidean” distinguishes these spaces from the curved spaces of non-Euclidean geometry and Einstein's general theory of relativity.

#### Figures

The black curve indicates the original function while the red and blue curves represent two vertical translations of that function.

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 $T_v(p) = p + v$. A graphical representation of vertical translations can be viewed in Figure 1.

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 three-dimensional 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.

#### Key Term Glossary

algebraic
or function}} Containing only numbers, letters and arithmetic operators.
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center
The point in the interior of a circle or sphere that is equidistant from all points on the circumference.
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constant
An identifier that is bound to an invariant value.
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degree
the sum of the exponents of a term; the order of a polynomial.
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distance
The amount of space between two points, measured along a straight line
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euclidean space
The three-dimensional space of Euclidean geometry. The term “Euclidean” distinguishes these spaces from the curved spaces of non-Euclidean geometry and Einstein's general theory of relativity.
##### Appears in these related concepts:
factor
To find all the factors of (a number or other mathematical object) (the objects that divide it evenly).
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function
A relation in which each element of the input is associated with exactly one element of the output.
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geometric
increasing or decreasing in a geometric progression, i.e. multiplication by a constant.
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linear
Of or relating to a class of polynomial of the form y = ax + b .
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matrix
A rectangular arrangement of numbers or terms having various uses such as transforming coordinates in geometry, solving systems of linear equations in linear algebra and representing graphs in graph theory.
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operation
A procedure for generating a value from one or more other values.
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origin
The point at which the axes of a coordinate system intersect
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point
An entity that has a location in space or on a plane, but has no extent
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reflection
a mapping from a Euclidean space to itself that is an isometry with a hyperplane as set of fixed points
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set
A collection of zero or more objects, possibly infinite in size, and disregarding any order or repetition of the objects that may be contained within it.
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term
any value (variable or constant) or expression separated from another term by a space or an appropriate character, in an overall expression or table.
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transformation
The replacement of the variables in an algebraic expression by their values in terms of another set of variables; a mapping of one space onto another or onto itself; a function that changes the position or direction of the axes of a coordinate system.
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vector
A directed quantity with both magnitude and direction.