Function transformations are fundamental operations that change the graph of a function by shifting, stretching, compressing, or reflecting it. The main types include translations which shift the graph, reflections which flip it across an axis, and stretches or compressions which change its size. These transformations can be combined to create more complex changes while preserving the basic shape of the function.
Translations or shifts move the graph of a function without changing its shape. A vertical shift is represented by y equals f of x plus k, where k greater than zero shifts the graph up and k less than zero shifts it down. A horizontal shift is represented by y equals f of x minus h, where h greater than zero shifts the graph right and h less than zero shifts it left. Let's see these transformations in action with a parabola.
Reflections flip the graph of a function across an axis. Reflection across the x-axis is represented by y equals negative f of x, which flips the graph vertically. Every point above the x-axis moves below it and vice versa. Reflection across the y-axis is represented by y equals f of negative x, which flips the graph horizontally. Every point to the right of the y-axis moves to the left and vice versa.
Stretches and compressions change the size of the graph. Vertical scaling is represented by y equals a times f of x. When the absolute value of a is greater than one, we get a vertical stretch. When a is between zero and one, we get vertical compression. If a is negative, it includes a reflection. Horizontal scaling is represented by y equals f of b times x. When the absolute value of b is greater than one, we get horizontal compression. When b is between zero and one, we get horizontal stretch.
To summarize what we've learned about function transformations: They are powerful tools that change the position, size, or orientation of function graphs. Translations shift graphs up, down, left, or right. Reflections flip graphs across axes. Stretches and compressions change the dimensions of graphs. These transformations can be combined to create complex effects while preserving the fundamental shape of the original function.