Learning Outcomes
- Graph Functions Using Compressions and Stretches.
Adding a constant to the inputs or outputs of a function changed the position of a graph with respect to the axes, but it did not affect the shape of a graph. We now explore the effects of multiplying the inputs or outputs by some quantity.
We can transform the inside (input values) of a function or we can transform the outside (output values) of a function. Each change has a specific effect that can be seen graphically.
Vertical Stretches and Compressions
When we multiply a function by a positive constant, we get a function whose graph is stretched or compressed vertically in relation to the graph of the original function. If the constant is greater than 1, we get a vertical stretch; if the constant is between 0 and 1, we get a vertical compression. The graph below shows a function multiplied by constant factors 2 and 0.5 and the resulting vertical stretch and compression.
A General Note: Vertical Stretches and Compressions
Given a function , a new function , where is a constant, is a vertical stretch or vertical compression of the function .
- If , then the graph will be stretched.
- If , then the graph will be compressed.
- If , then there will be combination of a vertical stretch or compression with a vertical reflection.
How To: Given a function, graph its vertical stretch.
- Identify the value of .
- Multiply all range values by .
- If , the graph is stretched by a factor of .
If , the graph is compressed by a factor of . If , the graph is either stretched or compressed and also reflected about the -axis.
Example: Graphing a Vertical Stretch
A function models the number of fruit flies in a population over time, and is graphed below.
A scientist is comparing this population to another population, , whose growth follows the same pattern, but is twice as large. Sketch a graph of this population.
Try It
How To: Given a tabular function and assuming that the transformation is a vertical stretch or compression, create a table for a vertical compression.
- Determine the value of .
- Multiply all of the output values by .
Example: Finding a Vertical Compression of a Tabular Function
A function is given in the table below. Create a table for the function .
2 | 4 | 6 | 8 | |
1 | 3 | 7 | 11 |
Try It
A function is given below. Create a table for the function .
2 | 4 | 6 | 8 | |
12 | 16 | 20 | 0 |
Example: Recognizing a Vertical Stretch
The graph is a transformation of the toolkit function . Relate this new function to , and then find a formula for .
Try It
Write the formula for the function that we get when we vertically stretch (or scale) the identity toolkit function by a factor of 3, and then shift it down by 2 units.
Check your work with an online graphing tool.
Horizontal Stretches and Compressions
Now we consider changes to the inside of a function. When we multiply a function’s input by a positive constant, we get a function whose graph is stretched or compressed horizontally in relation to the graph of the original function. If the constant is between 0 and 1, we get a horizontal stretch; if the constant is greater than 1, we get a horizontal compression of the function.
Given a function , the form results in a horizontal stretch or compression. Consider the function . The graph of is a horizontal stretch of the graph of the function by a factor of 2. The graph of is a horizontal compression of the graph of the function by a factor of 2.
A General Note: Horizontal Stretches and Compressions
Given a function , a new function , where is a constant, is a horizontal stretch or horizontal compression of the function .
- If , then the graph will be compressed by .
- If [latex]0
- If , then there will be combination of a horizontal stretch or compression with a horizontal reflection.
How To: Given a description of a function, sketch a horizontal compression or stretch.
- Write a formula to represent the function.
- Set where for a compression or [latex]0
Example: Graphing a Horizontal Compression
Suppose a scientist is comparing a population of fruit flies to a population that progresses through its lifespan twice as fast as the original population. In other words, this new population, , will progress in 1 hour the same amount as the original population does in 2 hours, and in 2 hours, it will progress as much as the original population does in 4 hours. Sketch a graph of this population.
Example: Finding a Horizontal Stretch for a Tabular Function
A function is given below. Create a table for the function .
2 | 4 | 6 | 8 | |
1 | 3 | 7 | 11 |
Example: Recognizing a Horizontal Compression on a Graph
Relate the function to .
Try It
Write a formula for the toolkit square root function horizontally stretched by a factor of 3.
Use an online graphing tool to check your work.
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Candela Citations
- Revision and Adaptation. Provided by: Lumen Learning. License: CC BY: Attribution
- Question ID 112707, 112726. Authored by: Lumen Learning. License: CC BY: Attribution. License Terms: IMathAS Community License CC-BY + GPL
- College Algebra. Authored by: Abramson, Jay et al.. Provided by: OpenStax. Located at: http://cnx.org/contents/9b08c294-057f-4201-9f48-5d6ad992740d@5.2. License: CC BY: Attribution. License Terms: Download for free at http://cnx.org/contents/9b08c294-057f-4201-9f48-5d6ad992740d@5.2
- Question ID 74696. Authored by: Meacham,William. License: CC BY: Attribution. License Terms: IMathAS Community License CC-BY + GPL
- Question ID 60791, 60790. Authored by: Day, Alyson. License: CC BY: Attribution. License Terms: IMathAS Community License CC-BY + GPL