Types of Function Transformations with Easy Visuals
A function f from domain X to domain Y is represented as f: X → Y. A function is defined as a map that maps each element in the domain to exactly one element in the codomain. Euler was the first to use modern representation f(x)( read “ f of x) to determine the value return by a function given by an argument x. Suppose the function f maps x ∊ Y to y ∊ Y. Then we can express it as y = f(x).
Here, we will look at some of the important concerts related to function and transformation of functions. Most probably, you must have encountered each of their terms earlier, but here we will merge the concepts together.
Transformation of Function
In Mathematics, a transformation of a function is a function that turns one function or graph into another, usually related function or graph. For example, translating a quadratic graph (parabola) will move the axis of symmetry and vertex but the overall shape of the parabola stays the same.
There are four types of transformation namely rotation, reflection, dilation, and translation In this article, we will discuss how to do transformation of a function, function graph transformation, and how to graph transformation function.
Transformation Statement Function
In Mathematics, transformation statement function is a function f that maps set A to itself i.e. f : A→A. Transformation in other areas of Mathematics simply refers to any function, regardless of domain and codomain.
Transformation can be an invertible function from set A to itself, or from set A to another set B. The alternatives of the term transformation may simply imply that the geometric features of functions are considered ( for example in terms to variants).
Types of Transformation
The four types of transformation of function are :
Rotation Transformation - Rotation Transformation rotates or turns the curve around an axis without changing the size and shape of an object.
Reflection Transformation - Reflection Transformation flips the object across a line by keeping it size or shape constant.
Dilation Transformation - Dilation transformation enlarges or shortens the object by keeping its shape or orientation the same. This is known as resizing.
Translation Transformation - Translation transformation slides or moves the object in the space by keeping its size and orientation the same.
Function Graph Transformation
Function graph transformation is a process through which a graphed equation or existing graph is modified to obtain the variation of the preceding graph. Function graph transformation is an usual kind of problem in algebra, specifically the modification of algebraic equations.
Sometimes graphs are stretched, rotated, translated, or moved about the xy plane. Many issues arise in the form of stretching the function f(x) by c units, shifting the function f(x) by c units, or rotating the function f(x) by x units about x- axis,y- axis, or z-axis. In each of these situations, transformation affects the basic function in certain ways that can be calculated
How to Graph Transformation?
Function transformations are mathematical operations that cause change in the shape of a graph. When a graph of a function is changed in appearance or location, we call it a transformation. Here, we will discuss how to graph transformations.
Vertical Transformation
The first transformation is vertical transformation. This type of transformation shifts the graph up or down relative to the parent graph. The graph will shift up if we add positive constant to each y- coordinate whereas the graph will shift down if we add negative constant.
Horizontal Transformation
This type of transformation shifts the left or right relative to the parent graph. This takes place when we add or subtract coordinates from the x- axis before the function is applied.
Reflection
A reflection is a transformation in which a mirror image is obtained about an x-axis. The graph of a function is reflected about the x-axis if each coordinate of y-axis is multiplied by -1 whereas the graph of a function is reflected about the y-axis if each coordinate of x-axis is multiplied by -1 proper applying the function.
Dilation
Functions that are multiplied by a real number apart from 1, depending upon the real number, appear to vertically or horizontally stretch. This form of non rigid transformation is known as dilation.
How to Do Transformations of Functions?
Here are the rules on how to do transformation of function that can be used to graph a function.
Quadratic Function Transformation
Transformation rules can be applied to graphs of function.
Here is the graph of function that represents the transformation of reflection.
The red curve represents the graph of function f(x) = x³.
The transformation g(x) = -x³ is completed and it obtains the reflection of f(x)about the x - axis.
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Now lets us learn the transformation of translation
The red curve represents the graph of function f(x) = x².
The quadratic function transformation f(x) = (x + 2)² will shift the parabola 2 steps to the right side.
[Image will be uploaded soon]
Now lets us learn the transformation of rotation
To rotate the graph 90º: (x, y) →(-y, x)
To rotate the graph 180º: (x, y) →(-x, -y)
To rotate the graph 270º: (x, y) →(y, -x)
Here, we can observe that the preimage is rotated to 180º.
Function Graph Transformation Rules
Here are some rules to transform the given graph of function.
f(x + a)horizontally shift the graph of f(x)left by a units
f(x - a)horizontally shift the graph of f(x) right by a units
f(x)+ a vertically shift the graph of f(x) upward by a units
f(x)- a vertically shift the graph of f(x) downwards by a units
af(x) vertically stretches the graph of f(x) by a factor of a units
1a f(x) vertically shrink the graph of f(x) by a factor of a units
f(ax) horizontally shrink the graph of f(x) by a factor of a units
fxa horizontally stretch the graph of f(x)by a factor of a units
-f(x)represents the reflection of the graph of f(x) over the x axis.
f(-x)represents the reflection of the graph of f(x) over the y axis
Transformation of Function Examples
Here are a few transformation of function examples to make you understand the concepts better.
1. What Does the Transformation Given Below Do to the Graph?
f(x)→f(x)- 2
f(x)→f(x - 2)
Solution:
f(x)→f(x)- 2
The y- coordinates encounter the change by 2 units.
Hence, the transformation here is translation 2 units down.
f(x)→f(x - 2)
The x- coordinates encounter the change by 2 units.
Hence, the transformation here is translation 2 units right.
2. If We Have the Graph of y = x², then How Will the Graph of y = x² - 6x + 9 Be Obtained?
Solution:
Let f(x) = x²
Completing the square on y = x² - 3x + 9 , we get (x - 3)² + 6. We identify this as f(x - 2)+ 3.
Therefore, to obtain the graph of y = x² - 3x + 9, we need to shift it to the right by 3 and then shift it up by 6.
3. How is the Graph of y= (x - 4)- 5 Related to the Graph of y = f(x)?
Solution:
When the graph of y = f(x)is moved right by 4 units, we get y = f(x - 4)
When the graph of y = f(x - 4)is moved down by 5 units, we get y= (x - 4)- 5 .
Hence, the graph of y= (x - 4)- 5 is located 4 units right, 5 units down of the graph of y = f(x). Hence, the point (x, y) moves to ( x + 4, y - 5).
FAQs on Function Transformation Explained: Step-by-Step Guide
1. What is meant by the transformation of functions in mathematics?
In mathematics, a function transformation is a process that alters an existing function, called the parent function, to create a new, modified function. This alteration changes the graph of the parent function by shifting its position, changing its shape, or adjusting its orientation. Common transformations include translations (shifts), scaling (stretching or compressing), and reflections (flipping).
2. What are the main types of function transformations explained with examples?
The main types of transformations applied to a function f(x) are:
Translation (Shifting): This moves the graph without changing its shape. A vertical shift is represented by f(x) + c (up) or f(x) - c (down). A horizontal shift is represented by f(x - c) (right) or f(x + c) (left).
Scaling (Stretching/Compressing): This changes the shape of the graph. Vertical scaling is c * f(x), which stretches the graph if |c| > 1 and compresses it if 0 < |c| < 1. Horizontal scaling is f(c * x), which compresses the graph if |c| > 1 and stretches it if 0 < |c| < 1.
Reflection (Flipping): This flips the graph across an axis. A reflection across the x-axis is -f(x), and a reflection across the y-axis is f(-x).
3. How does transforming the input variable 'x' differ from transforming the entire function 'f(x)'?
Transforming the input variable 'x' (e.g., f(x + c) or f(cx)) primarily affects the graph's horizontal properties. These are often considered 'inside' transformations and can seem counter-intuitive (e.g., adding 'c' to x shifts the graph left). In contrast, transforming the entire function 'f(x)' (e.g., f(x) + c or c*f(x)) affects the graph's vertical properties. These 'outside' transformations are generally more direct (e.g., adding 'c' to f(x) shifts the graph up).
4. What is the correct order to apply multiple transformations to a function?
When a function involves multiple transformations, such as in the form y = a * f(b(x - c)) + d, the recommended order is crucial for accuracy. A reliable method is to apply them in the following sequence:
Horizontal Shift (Translation): Apply the shift determined by 'c'.
Horizontal Stretch/Compression and Reflection: Apply the scaling and/or reflection determined by 'b'.
Vertical Stretch/Compression and Reflection: Apply the scaling and/or reflection determined by 'a'.
Vertical Shift (Translation): Apply the final shift determined by 'd'.
Essentially, you work from the 'inside' of the function parenthesis outwards.
5. Why does the transformation f(x + c) shift a graph to the left, not to the right?
This common point of confusion arises from thinking about the variable 'x' instead of the input to the function. For the new function g(x) = f(x + c) to produce the same output as the original function f(x) at a certain point, the new x-value must be 'c' units smaller. For example, to get the output f(0), you must input x = -c into g(x), because g(-c) = f(-c + c) = f(0). This means every point on the original graph is now achieved at an x-value that is 'c' less than before, resulting in a horizontal shift to the left.
6. Can you provide an example of transforming the parent function y = x²?
Certainly. Let's transform the parent function f(x) = x² to get the new function g(x) = -2(x - 3)² + 5.
Parent Function: f(x) = x², a parabola with its vertex at (0, 0).
Step 1 (Horizontal Shift): The term (x - 3) shifts the graph 3 units to the right. The vertex is now at (3, 0).
Step 2 (Vertical Stretch & Reflection): The factor of -2 reflects the graph across the x-axis (due to the negative sign) and stretches it vertically by a factor of 2, making it narrower.
Step 3 (Vertical Shift): The '+ 5' shifts the entire graph 5 units up. The final vertex is at (3, 5).
7. What is the importance of understanding function transformations?
Understanding function transformations is fundamentally important in mathematics because it allows you to quickly visualise and sketch the graphs of complex functions based on simpler, known parent functions (like y=x², y=sin(x), or y=|x|). This skill is essential in calculus for analysing limits, derivatives, and integrals, and in physics and engineering for modelling real-world phenomena that are variations of basic patterns.
8. How can you identify the parent function and the transformations from a given equation?
To identify the parent function and transformations from an equation, first look for the most basic form of the function. For example, in y = -3√(x+2) - 4, the core function is the square root, so the parent function is f(x) = √x. Then, identify the transformations by comparing them to the general form a * f(b(x - c)) + d:
a = -3: A reflection across the x-axis and a vertical stretch by a factor of 3.
b = 1: No horizontal stretch or compression.
c = -2: A horizontal shift 2 units to the left (from x - (-2) = x + 2).
d = -4: A vertical shift 4 units down.
