Read: Transform Linear Functions
Learning Objectives
- Define and use a vertical stretch or compression to graph a linear function
- Define and use a vertical shift to graph a linear function
- Combine transformations to graph a linear function
Graphing a Linear Function Using Transformations
Another option for graphing is to use transformations of the identity function . A function may be transformed by a shift up, down, left, or right. A function may also be transformed using a reflection, stretch, or compression.Vertical Stretch or Compression
In the equation , the m is acting as the vertical stretch or compression of the identity function. When m is negative, there is also a vertical reflection of the graph. Notice in the figure below that multiplying the equation of by m stretches the graph of f by a factor of m units if and compresses the graph of f by a factor of m units if . This means the larger the absolute value of m, the steeper the slope.
Vertical Shift
In , the b acts as the vertical shift, moving the graph up and down without affecting the slope of the line. Notice in the figure below that adding a value of b to the equation of shifts the graph of f a total of b units up if b is positive and units down if b is negative.
How To: Given the equation of a linear function, use transformations to graph the linear function in the form .
- Graph .
- Vertically stretch or compress the graph by a factor |m|.
- Shift the graph up or down b units.
Example
Describe the transformations to the identity for the function , and draw a graph.Answer:
In this case, , so this is a vertical compression since .
The graph of is plotted below with the identity:
Note how the identity is compressed because the rate of change is "slowed" by the vertical compression of
Example
Describe the transformations to the identity for the function , and draw a graph.Answer: In this case, , so this is a vertical stretch since . The graph of is plotted below with the identity:

Example
Describe the transformations to the identity for the function , and draw a graph.Answer: In this case, , so this is a vertical stretch since , the negative sign reflects the graph across the y axis. The graph of is plotted below with the identity:

Example
Graph using transformations.Answer:
The equation for the function shows that so the identity function is vertically compressed by . The equation for the function also shows that so the identity function is vertically shifted down units. First, graph the identity function, and show the vertical compression.
The function , compressed by a factor of .
Then show the vertical shift.
The function , shifted down units.
Q& A
In the example above, could we have sketched the graph by reversing the order of the transformations? No. The order of the transformations follows the order of operations. When the function is evaluated at a given input, the corresponding output is calculated by following the order of operations. This is why we performed the compression first.Summary
- Vertical compressions of the identity happen when the slope is between and
- Vertical stretches of the identity happen when the slope is greater than
- Reflections happen when the slope is negative
- Vertical shifts happen when the intercept is not equal to
- Transformations can be combined