Chapter 3 Functions
3.6 Transformation of Functions
Learning Objectives
In this section, you will:
- Graph functions using vertical and horizontal shifts.
- Graph functions using reflections about the [latex]\,x\text{-axis}\,[/latex] and the [latex]\,y\text{-axis}.[/latex]
- Determine whether a function is even, odd, or neither from its graph.
- Graph functions using compressions and stretches.
- Combine transformations.
We all know that a flat mirror enables us to see an accurate image of ourselves and whatever is behind us. When we tilt the mirror, the images we see may shift horizontally or vertically. But what happens when we bend a flexible mirror? Like a carnival funhouse mirror, it presents us with a distorted image of ourselves, stretched or compressed horizontally or vertically. In a similar way, we can distort or transform mathematical functions to better adapt them to describing objects or processes in the real world. In this section, we will take a look at several kinds of transformations.
Graphing Functions Using Vertical and Horizontal Shifts
Often when given a problem, we try to model the scenario using mathematics in the form of words, tables, graphs, and equations. One method we can employ is to adapt the basic graphs of the toolkit functions to build new models for a given scenario. There are systematic ways to alter functions to construct appropriate models for the problems we are trying to solve.
Identifying Vertical Shifts
One simple kind of transformation involves shifting the entire graph of a function up, down, right, or left. The simplest shift is a vertical shift, moving the graph up or down, because this transformation involves adding a positive or negative constant to the function. In other words, we add the same constant to the output value of the function regardless of the input. For a function [latex]\,g\left(x\right)=f\left(x\right)+k,\,[/latex] the function [latex]\,f\left(x\right)\,[/latex] is shifted vertically [latex]\,k\,[/latex] units. See Figure 2 for an example.
To help you visualize the concept of a vertical shift, consider that [latex]\,y=f\left(x\right).\,[/latex] Therefore,[latex]\,f\left(x\right)+k\,[/latex] is equivalent to [latex]\,y+k.\,[/latex] Every unit of [latex]\,y\,[/latex] is replaced by [latex]\,y+k,\,[/latex] so the y-value increases or decreases depending on the value of [latex]\,k.\,[/latex] The result is a shift upward or downward.
Vertical Shift
Given a function [latex]f\left(x\right),[/latex] a new function [latex]g\left(x\right)=f\left(x\right)+k,[/latex] where [latex]\,k[/latex] is a constant, is a vertical shift of the function [latex]f\left(x\right).[/latex] All the output values change by [latex]k[/latex] units. If [latex]k[/latex] is positive, the graph will shift up. If [latex]k[/latex] is negative, the graph will shift down.
Adding a Constant to a Function
To regulate the temperature in a green building, airflow vents near the roof open and close throughout the day. Figure 3 sh