Chapter 3 Functions
Section 3.4 Transformations of Functions
Objectives
1. Using Vertical Shifts to Graph Functions
2. Using Horizontal Shifts to Graph Functions
3. Using Reflections to Graph Functions
4. Using Vertical Stretches and Compressions to Graph Functions
6. Using Combinations of Transformations to Graph Functions
In this section, we will learn how to sketch the graphs of new functions using the graphs of known functions. Starting with the graph of a known function, we will “transform” it into a new function by applying various transformations.
Objective 1: Using Vertical Shifts to Graph Functions
Vertical Shifts of Functions
If [pic]is a positive real number:
The graph of [pic] is obtained by shifting the graph of [pic] vertically upward [pic]units.
The graph of [pic] is obtained by shifting the graph of [pic] vertically downward [pic]units.
Objective 2: Using Horizontal Shifts to Graph Functions
Horizontal Shifts of Functions
If [pic]is a positive real number:
The graph of [pic] is obtained by shifting the graph of [pic] horizontally to the left [pic]units.
The graph of [pic] is obtained by shifting the graph of [pic] horizontally to the right [pic]units.
For [pic], the graph of [pic]is the graph of [pic]shifted to the right [pic]units. At first glance, it appears that the rule for horizontal shifts is the opposite of what seems natural. Substituting [pic] for x causes the graph of [pic]to be shifted to the left while substituting [pic] for x causes the graph to shift to the right [pic]units.
Objective 3: Using Reflections to Graph Functions
Reflection of Functions about the x-Axis
The graph of [pic] is obtained by reflecting the graph of [pic] about the x-axis.
Reflections of Functions about the y-Axis
The graph of [pic] is obtained by reflecting the graph of [pic] about the y-axis.
Objective 4: Using Vertical Stretches and Compressions to Graph Functions
Vertical Stretches and Compressions of Functions
Suppose[pic]is a positive real number:
The graph of [pic] is obtained by the multiplying each y-coordinate of [pic] by [pic].
If [pic], the graph of [pic]is a vertical stretch of the graph of [pic]. If [pic], the
graph of [pic] is a vertical compression of the graph of [pic].
Objective 6: Using Combinations of Transformations to Graph Functions
When sketching a function that involves multiple transformations it is important to follow a certain “order of operations”. Below is the order in which each transformation will be performed in this text:
1) Horizontal Shifts
2) Horizontal Stretches/Compressions
3) Reflection about y-axis
4) Vertical Stretches/Compressions
5) Reflection about x-axis
6) Vertical Shifts
Different ordering is possible for transformations 2) through 5), but you should always perform the horizontal shift first and the vertical shift last. [pic]
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