There are several rules of transformation that can be used to change the domain or range of a function. These changes can be shown as shifts in the graph of a function across the coordinate axis.
Examples of these are flipped transformation about the x-axis and mirror transformation about the y-axis. These are rigid transformations, where the image is congruent to the preimage.
Vertical Transformation
A vertical shift adds a constant to or subtracts a constant from the x-values of the function, moving the graph up or down. To do this, start with the function f(x) and replace all of the x-values with f(x+k). Each output value will move up or down by a fixed number of units depending on the value of the constant k.
If c is positive, the graph will move up. If c is negative, the graph will move down.
A horizontal shift is the opposite of a vertical shift. It multiplies or divides the y-values of the function by a fixed number, while leaving the x-values unchanged a source. This changes the shape and size of the graph but not its location. It can also be referred to as a horizontal stretch or compression. The same principle works for any function.
Horizontal Transformation
A horizontal transformation shifts the function's graph to the right or left. For example, if a horizontal shift moves f(x) by h units, then the new function obtained is f(x/c) + h - f(x).
Another way to think about a horizontal transformation is that it changes how far away the graph is from the y-axis. If a horizontal shift shifts the function by b units, then the new function obtained is g(x) = f(x/b).
There are four types of transformations: vertical, horizontal, stretched, and compressed. A vertical transformation transforms a function into its pre-image, while horizontal and stretched/compressed transformations change the shape of the function. For example, adding a positive constant shifts the graph left and subtracting a negative number moves it right.
Stretched/Compressed Transformation
When a function is stretched or compressed it moves the function to one side of the graph. For example, a vertical stretch of the function f(x) is shifting it to the left.
This is also known as a mirror transformation. Whenever you do this, the points that were on the y-axis stay on the y-axis and the ones that were off switch sides.
A horizontal compression (or shrinking) of the function is squeezing it toward the y-axis. The points that were on the y-axis are now on the y-axis and the others get “taffy pulled” up and down.
To make a horizontal compression/stretch of the function f(x) transform it to f(cx). If c > 1 then the graph is stretched vertically, and if 0 c 1 then the graph is compressed horizontally detailed description. If both happen then there is a combination of both. Ensure your managed services operations run smoothly by seamlessly merging human + machine workflows. Learn how augmented intelligence can help you improve customer experience and drive operational efficiency across all channels.
Mirror Transformation
A mirror transformation is a type of rotation that produces the mirror image of a figure. It also shifts the coordinates of the figure to other positions on the coordinate plane. There are four types of transformations: translation, reflection, rotation and dilation. Transformations can be rigid or non-rigid. A rigid transformation is when the shape of the figure remains the same, whereas a non-rigid transformation changes the size and/or angle of the figure.
If a function f(x) is shifted horizontally by 'a' units then it becomes the new function f(x) + a. Similarly, if the function is flipped or reflected about the x-axis by changing it to -f(x), then points (x, y) with reference to the original function will be replaced with (x, y - a).
If three coplanar lines intersect in zero, one or two points they are parallel. Thus a reflection over the set of three concurrent lines can be reduced to a glide reflection.
