Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

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 $a$	$f (x) = 3 x$
Line $b$	$f (x) = 2 x$
Line $c$	$f (x) = x$

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Compare your work:

A graph showing three lines: solid blue for f of x equals 3 times x, dotted green for f of x equals 2 times x, and dashed orange for f of x equals x, each passing through the origin with different slopes. The legend matches each line to its equation.

2. What do you notice about the slopes of lines $c$ , $b$ , and $a$ as the number of the coefficient gets larger?

Compare your answer:

As the coefficient increases, the line gets steeper.

3. Now, graph lines $d$ and $e$ on the same graph as $a - c$ .

Line Label	Function
Line $d$	$f (x) = \frac{1}{2} x$
Line $e$	$f (x) = \frac{1}{3} x$

Compare your work:

A graph comparing five linear functions: f of x equals 3 times x is a blue line, f of x equals 2 times x is a green dotted line, f of x equals x is an orange dashed line, f of x equals one-half times x is a black line, and f of x equals one-third times x is a red line, all passing through the origin with a positive slope.

4. How does Line $d$ , representing $f (x) = \frac{1}{2} x$ , compare to Line $b$ , representing $f (x) = 2 x$ ?

Compare your answer:

As the coefficient gets smaller, the line is less steep.

5. Examine lines $d$ and $e$ . What can you say about the lines when $0 < m < 1$ ?

Compare your answer:

As the coefficient gets smaller, the line is less steep.

6. Now graph lines $f$ , $g$ , and $h$ on the same coordinate grid.

Line Label	Function
Line $f$	$f (x) = - 2 x$
Line $g$	$f (x) = - x$
Line $h$	$f (x) = - \frac{1}{2} x$

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Compare your work:

Three lines are shown on a graph: a green dotted line for f of x equals negative 2 times x, a brown dashed line for f of x equals negative x, and a solid black line for f of x equals negative one-half times x. All lines have negative slopes.

7. Compare the equations of lines $f - h$ to the graphs of lines $b - d$ . What changed with the equations of these functions?

Compare your answer:

There is a negative coefficient in lines $f - h$ while in lines $b - d$ , they are positive.

8. Compare the graphs of lines $f - h$ to the graphs of lines $b - d$ . How did the change in their equations affect their graphs?

Compare your answer:

Since there is a negative coefficient, the graphs of lines $f - h$ are reflections of the positive versions of the lines over the $x$ -axis. The graphs were vertically “flipped.”

In the equation $f (x) = a x$ , the $a$ is acting as the vertical stretch or vertical compression of the parent function. It dilates the graph. When $a$ is negative, there is also a vertical reflection of the graph.

Multiplying the equation of $f (x) = x$ by $a$ vertically stretches or expands the graph of $f$ by a factor of $a$ units if $a > 1$ .
The $a$ vertically compresses the graph of $f$ by a factor of $a$ units if $0 < a < 1$ .
This means the larger the absolute value of $a$ , the steeper the slope.

This graph shows seven versions of the function, f of x = x on an x, y coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. Seven multi-colored lines run through the point (0, 0). Starting with the lines in the top right quadrant and moving clockwise, the first line is f of x equals 3 times x and has a slope of 3, the next line is f of x equals 2 times x which has a slope of 2, the next line is f of x equals x which has a slope of 1, the next line is f of x equals x divided by 2 which has a slope of .5. The last line in this quadrant is f of x equals x divided by 3 which has a slope of one third x. In the bottom right quadrant moving clockwise, the first line is f of x equals negative x divided by 2, which has a slope of negative one half, the middle line is f of x equals negative x which has a slope of negative 1, and the last line is f of x equals negative 2 times x which has a slope of negative 2.

In $f (x) = m x + b$ , the $m$ acts as the vertical dilation (stretch or compression factor). This is a type of transformation to the graph of $f (x) = x$ .

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 → $a f (x)$

If the $a$ value is negative, the graph is reflected over the $x$ -axis. If the $a$ value is greater than 1, the graph is stretched by $a$ scale factor of $a$ . If the a value is between 0 and 1, the graph is compressed by a scale factor of $a$ .

Video: Identifying Vertical Stretches and Compressions

Watch the following video to learn more about vertical stretches and compressions.

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Self Check

How does the graph of $f(x)=\frac14 x -3$ change from the parent function, $f(x)=x$ ?

Vertical stretch of $\frac14$ , left 3
Vertical compression of $\frac14$ , up 3
Vertical compression of $\frac14$ , down 3
Vertical stretch of $\frac14$ , down 3

Additional Resources

Vertical Transformations of Linear Functions

Given the equation of a linear function, use transformations to graph the linear function in the form $f (x) = m x + b$ .

Example

Graph $f (x) = \frac{1}{2} x - 3$ using transformations.

Solution

Step 1 - Graph the parent function, $f (x) = x$ .

Step 2 - Vertically stretch or compress the graph by a factor $m$ . The equation for the function shows that $m = \frac{1}{2}$ , so the identity function is vertically compressed by $\frac{1}{2}$ .

This graph shows two functions on an x, y coordinate plane. One shows an increasing function of y equals x divided by 2 that runs through the points (0, 0) and (2, 1). The second shows an increasing function of y equals x and runs through the points (0, 0) and (1, 1).

This graph shows two functions on an $(x, y)$ coordinate plane. One shows an increasing function of $y = x$ divided by 2 that runs through the points $(0, 0)$ and $(2, 1)$ . The second shows an increasing function of $y = x$ and runs through the points $(0, 0)$ and $(1, 1)$ .

Step 3 - Shift the graph up or down $b$ units.

The equation for the function also shows that $b = - 3$ , so the parent function is vertically shifted down 3 units.

Now, show the vertical shift:

This graph shows two functions on an x, y coordinate plane. The first is an increasing function of y equals x divided by 2 and runs through the points (0, 0) and (2, 1). The second shows an increasing function of y equals x divided by 2 minus 3 and passes through the points (0, 3) and (2, negative 2). An arrow pointing downward from the first function to the second function reveals the vertical shift.

This graph shows two functions on an $(x, y)$ coordinate plane. The first is an increasing function of $y = x$ divided by 2 and runs through the points $(0, 0)$ and $(2, 1)$ . The second shows an increasing function of $y = x$ divided by 2 minus 3 and passes through the points $(0, 3)$ and $(2, - 2)$ . 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, $f (x) = x$ , to $f (x) = 4 + 2 x$ .

Here is how to determine the transformations:

First, the slope $m = 2$ tells that there is a vertical stretch of 2.

Next, the $y$ -intercept is 4, so there is a vertical shift up 4.

4.11.4 Vertical Stretches and Compressions

Activity

Vertical Stretches and Compressions

Video: Identifying Vertical Stretches and Compressions

Self Check

Additional Resources

Vertical Transformations of Linear Functions

Vertical Transformations of Linear Functions