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
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.