Activity
Vertical Stretches and Compressions
On the same coordinate grid graph the following linear functions. You do not need to label the lines in the graphing tool.
Line Label | Function |
Line | |
Line | |
Line |
Compare your work:
2. What do you notice about the slopes of lines , , and as the number of the coefficient gets larger?
Compare your answer:
As the coefficient increases, the line gets steeper.
3. Now, graph lines and on the same graph as .
Line Label | Function |
Line | |
Line |
Compare your work:
4. How does Line , representing , compare to Line , representing ?
Compare your answer:
As the coefficient gets smaller, the line is less steep.
5. Examine lines and . What can you say about the lines when ?
Compare your answer:
As the coefficient gets smaller, the line is less steep.
6. Now graph lines , , and on the same coordinate grid.
Line Label | Function |
Line | |
Line | |
Line |
Compare your work:
7. Compare the equations of lines to the graphs of lines . What changed with the equations of these functions?
Compare your answer:
There is a negative coefficient in lines while in lines , they are positive.
8. Compare the graphs of lines to the graphs of lines . How did the change in their equations affect their graphs?
Compare your answer:
Since there is a negative coefficient, the graphs of lines are reflections of the positive versions of the lines over the -axis. The graphs were vertically “flipped.”
In the equation , the is acting as the vertical stretch or vertical compression of the parent function. It dilates the graph. When is negative, there is also a vertical reflection of the graph.
- Multiplying the equation of by vertically stretches or expands the graph of by a factor of units if .
- The vertically compresses the graph of by a factor of units if .
- This means the larger the absolute value of , the steeper the slope.
In , the acts as the vertical dilation (stretch or compression factor). This is a type of transformation to the graph of .
Vertical Dilations of a Function
A vertical stretch or compression “transforms” the parent function into another function by vertically dilating the graph by “a” units.
Vertical Dilation →
If the value is negative, the graph is reflected over the -axis. If the value is greater than 1, the graph is stretched by scale factor of . If the a value is between 0 and 1, the graph is compressed by a scale factor of .
Video: Identifying Vertical Stretches and Compressions
Watch the following video to learn more about vertical stretches and compressions.
Self Check
Additional Resources
Vertical Transformations of Linear Functions
Given the equation of a linear function, use transformations to graph the linear function in the form .
Example
Graph using transformations.
Solution
Step 1 - Graph the parent function, .
Step 2 - Vertically stretch or compress the graph by a factor . The equation for the function shows that , so the identity function is vertically compressed by .
This graph shows two functions on an coordinate plane. One shows an increasing function of divided by 2 that runs through the points and . The second shows an increasing function of and runs through the points and .
Step 3 - Shift the graph up or down units.
The equation for the function also shows that , so the parent function is vertically shifted down 3 units.
Now, show the vertical shift:
This graph shows two functions on an coordinate plane. The first is an increasing function of divided by 2 and runs through the points and . The second shows an increasing function of divided by 2 minus 3 and passes through the points and . An arrow pointing downward from the first function to the second function reveals the vertical shift.
Try it
Vertical Transformations of Linear Functions
Name the transformations to go from the parent function, , to .
Here is how to determine the transformations:
First, the slope tells that there is a vertical stretch of 2.
Next, the -intercept is 4, so there is a vertical shift up 4.