The purpose of graphing transformations is to visually represent how a function is affected by various changes to its equation. It allows us to better understand the behavior of the function and make predictions about its values.
The main types of transformations that can be applied to a graph are translations, reflections, dilations, and rotations. Translations shift the graph horizontally or vertically, reflections flip the graph across an axis, dilations stretch or shrink the graph, and rotations rotate the graph around a point.
A translation involves shifting the graph horizontally or vertically. If the translation is in the form of (a, b), the graph will shift a units to the right if a is positive, and a units to the left if a is negative. Similarly, the graph will shift b units up if b is positive, and b units down if b is negative.
A horizontal reflection flips the graph across the x-axis, while a vertical reflection flips the graph across the y-axis. This means that the x-coordinates of the points on the graph remain the same for a horizontal reflection, but the y-coordinates change. Similarly, the y-coordinates remain the same for a vertical reflection, but the x-coordinates change.
When a graph is dilated, it is either stretched or shrunk. A dilation with a scale factor greater than 1 results in a stretched graph, while a scale factor between 0 and 1 results in a shrunk graph. The center of dilation is also an important factor, as it determines the direction and amount of stretching or shrinking.