Table of Contents
Functions and Transformations
A function is a set of ordered pairs, where each element in the set corresponds to a unique output. The function assigns a unique output to every input, and the output cannot be determined by looking at the inputs alone.
A transformation of a function is a change to the function’s graph. The transformation can be a vertical stretch or compression, a horizontal shift, a reflection across a line, or a rotation. The graph of a transformed function will look different from the graph of the original function, but the shape of the graph will be the same.
Transformation Statement Function
A transformation statement function is a function that takes a single input, which is a transformation statement, and outputs a new transformation statement. The new transformation statement is the result of applying the transformation statement input to the transformation statement function.
Types of Transformation
in Mathematics
There are three types of transformations in mathematics: geometric, linear, and affine.
Geometric transformations are transformations that preserve angles and distances. They include translations, rotations, and reflections.
Linear transformations are transformations that preserve distances but not angles. They include scalings (enlargements or reductions), reflections, and shears.
Affine transformations are transformations that preserve both angles and distances. They include translations, rotations, reflections, scalings, and shears.
Function Graph Transformation
A graph transformation is a formalism for describing the change of a graph. It is a directed graph, where the nodes represent the vertices of the graph and the edges represent the edges of the graph. The edges are labelled with a transition function, which is a function from the vertices to the set of edges. The transition function describes how the graph changes when it is applied to a given input graph. The output graph is a subgraph of the input graph, where the edges have been replaced by the edges of the transition function.
How to Graph Transformation?
To graph a transformation, first identify the equation of the original function. Next, identify the equation of the transformed function. Finally, use a graphing calculator or software to plot the two equations and find the point of intersection.
Vertical Transformation
Vertical transformation is the process of taking a two-dimensional image and transforming it into a three-dimensional object. There are many methods for accomplishing vertical transformation, but the most common is to use a 3D printer.
Horizontal Transformation
This transformation flips a figure over a horizontal line.
Reflection
A reflection flips a figure over a line, but preserves the angles.
Reflection
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Dilation
The dilation of an image is the process of making the image larger. This can be done using software, or by physically enlarging the image.
How to Do Transformations of Functions?
There are a few steps in transforming a function. The first step is to identify the function that needs to be transformed. Next, identify the new function that will result from the transformation. Finally, use the appropriate transformation rule to carry out the transformation.
Quadratic Function Transformation
A quadratic function is a polynomial function in the form of
,
where a, b, and c are real numbers and is the coefficient of x2.
The graph of a quadratic function is a parabola. The vertex of the parabola is at (h, k), where h is the height of the parabola and k is the x-coordinate of the vertex. The y-intercept of the graph is at (0, c), and the x-intercept is at (-b/2a, 0).
There are three ways to transform a quadratic function:
1. The vertex can be moved to a different location by changing the value of h.
2. The y-intercept can be moved to a different location by changing the value of c.
3. The x-intercept can be moved to a different location by changing the value of b.
Function Graph Transformation Rules
A function graph transformation rule is a mathematical statement that describes how to transform a function graph. The rule may involve adding, subtracting, multiplying, or dividing the function’s terms. It may also involve raising or lowering the function’s terms to a certain power.
Transformation of Function Examples
The following are examples of transformations of functions.
1. The function ƒ(x) = x3 is transformed into the function ƒ(x) = (x + 1)3.
2. The function ƒ(x) = x is transformed into the function ƒ(x) = (x2)1/2.
3. The function ƒ(x) = x2 is transformed into the function ƒ(x) = (x + 1)2.
