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.
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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 a function transformation?
A function transformation refers to systematically altering a function's graph by applying specific changes such as shifting, stretching, compressing, or reflecting it. These transformations change the position or shape of the original graph without modifying the fundamental nature of the function. Understanding function transformations is essential for analyzing how algebraic modifications impact graphical representation, a key topic in mathematics supported by Vedantu’s interactive learning modules.
2. What are the 4 types of transformation?
There are four primary types of function transformations:
- Translations (Shifting): Moving the graph horizontally or vertically.
- Reflections: Flipping the graph over a line, such as the x-axis or y-axis.
- Stretches: Extending the graph away from the axis, either vertically or horizontally.
- Compressions: Squeezing the graph towards the axis, again either vertically or horizontally.
3. What is the rule for transformation of functions?
The rule for transformation of functions describes how changes to a function's equation affect its graph. Common transformation rules include:
- Vertical shifts: $f(x) + k$ moves the graph up ($k > 0$) or down ($k < 0$).
- Horizontal shifts: $f(x - h)$ shifts right ($h > 0$) or left ($h < 0$).
- Reflections: $-f(x)$ reflects over the x-axis, $f(-x)$ reflects over the y-axis.
- Stretches and Compressions: $a f(x)$ vertically stretches ($|a|>1$) or compresses ($0<|a|<1$); $f(bx)$ horizontally compresses ($|b|>1$) or stretches ($0<|b|<1$).
4. What is the functional transformation method?
The functional transformation method involves applying algebraic operations to a parent function to systematically obtain the equation and graph of its transformed version. The method typically includes:
- Identifying the base or parent function
- Applying operations in the right order (reflections/translations before stretching/compressing)
- Graphing each step to track changes visually
5. How do horizontal and vertical transformations affect a function’s graph?
Horizontal transformations (changes inside the function, such as $f(x - h)$) shift the graph right (if $h > 0$) or left (if $h < 0$). Vertical transformations (changes outside the function, such as $f(x) + k$) move the graph up (for $k > 0$) or down (for $k < 0$). Understanding these transformations enables students to predict and sketch the new position of any function, a skill covered extensively through Vedantu’s practice worksheets and video lessons.
6. What are examples of function transformations in real life?
Function transformations appear in real-life scenarios such as:
- Audio engineering: Amplifying or shifting sound waves
- Finance: Adjusting graphs of investment growth by shifting or scaling
- Physics: Reflecting position-time graphs in mechanics
7. How can Vedantu help me master graph transformations for exams?
Vedantu offers comprehensive resources such as live classes, doubt-solving sessions, detailed study material, and interactive simulations that make learning graph transformations engaging and effective. With expert tutors and customized practice sets, students can prepare thoroughly for competitive exams and school assessments involving function transformations.
8. What is the parent function, and why is it important in transformations?
The parent function is the simplest form of a function from a specific family (such as $f(x) = x^2$ for quadratic functions), serving as the basic template. All transformations (shifts, stretches, etc.) are performed relative to the parent function. Mastering the concept of parent functions helps students quickly interpret and graph transformed functions, emphasized in Vedantu’s structured mathematics courses.
9. Can function transformations be combined, and how?
Yes, multiple function transformations can be applied in sequence to a single function. For example, $g(x) = -2f(x + 1) - 3$ combines:
- Horizontal shift: left by 1 unit
- Vertical stretch: by a factor of 2
- Reflection: over the x-axis
- Vertical shift: down by 3 units
