Rules and Graphs for Shifts Stretches and Reflections of Functions
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 Understanding Function Transformations in Algebra
1. What is a function transformation?
A function transformation is a change made to the graph of a function that shifts, stretches, compresses, or reflects it without altering its basic shape. In general, transformations modify the parent function f(x) using constants.
- f(x) + k → vertical shift
- f(x − h) → horizontal shift
- a·f(x) → vertical stretch or compression
- f(bx) → horizontal stretch or compression
- −f(x) or f(−x) → reflections
2. How do you shift a function horizontally?
A function shifts horizontally by replacing x with (x − h) inside the function. If h is positive, the graph shifts right; if h is negative, it shifts left.
- f(x − h) → shift right by h units
- f(x + h) → shift left by h units
3. How do you shift a function vertically?
A function shifts vertically by adding or subtracting a constant outside the function. The form is f(x) + k.
- If k > 0 → shift up
- If k < 0 → shift down
4. What is the difference between vertical and horizontal transformations?
The key difference is that vertical transformations change the output (y-values), while horizontal transformations change the input (x-values).
- Vertical: f(x) + k or a·f(x)
- Horizontal: f(x − h) or f(bx)
5. What is a vertical stretch or compression?
A vertical stretch or compression occurs when the function is multiplied by a constant a outside the function, written as a·f(x).
- If |a| > 1 → vertical stretch
- If 0 < |a| < 1 → vertical compression
6. What is a horizontal stretch or compression?
A horizontal stretch or compression occurs when x is multiplied inside the function, written as f(bx).
- If |b| > 1 → horizontal compression
- If 0 < |b| < 1 → horizontal stretch
7. How do reflections work in function transformations?
Reflections flip a graph across an axis using negative signs in specific positions.
- −f(x) → reflection across the x-axis
- f(−x) → reflection across the y-axis
8. What is the general form of a transformed function?
The general form of a transformed function is g(x) = a·f(b(x − h)) + k. Each parameter controls a specific transformation:
- a → vertical stretch/compression and reflection
- b → horizontal stretch/compression
- h → horizontal shift
- k → vertical shift
9. How do you describe the transformations of g(x) = 2(x − 3)² + 5?
The function g(x) = 2(x − 3)² + 5 is a transformed quadratic with stretch and shifts.
- Parent function: x²
- 2 → vertical stretch by factor 2
- (x − 3) → shift right 3 units
- +5 → shift up 5 units
10. What are common mistakes in function transformations?
The most common mistakes in function transformations involve sign confusion and mixing horizontal with vertical changes.
- Forgetting that (x − h) shifts right, not left
- Confusing a·f(x) with f(ax)
- Applying transformations in the wrong order
- Ignoring reflections when a is negative
