Section 10.2 Transformations
¶Subsection 10.2.1 Review Material
In this activity, you will review some basics from previous courses:
Vertical and Horizontal Translations
Vertical Stretches and Compressions
In the first exercise below, you will encounter your first of many WeBWorK problems. Click to open the WeBWorK exercise, and you will see a graph that you can manipulate. Answer the questions that follow, and click to submit your answers.
Checkpoint 10.2.1.
Next, recall how a graph changes when we either add or subtract from the input of the function.
Checkpoint 10.2.2.
Finally, explore the effect of multiplying the output of a function by a constant number.
Checkpoint 10.2.3.
Questions you should be able to answer:
In general, how does the graph change when you add a number to the output of a function?
In general, how does the graph change when you subtract a number from the output of a function?
Practice using these ideas in the next two exercises.
Checkpoint 10.2.4.
Checkpoint 10.2.5.
Finally, we'll review an application of using vertical and horizontal translations to quickly perform a familiar task: Finding the equation of a line
Checkpoint 10.2.6.
Checkpoint 10.2.7.
Checkpoint 10.2.8.
Subsection 10.2.2 Horizontal Stretches/Compressions
We have already seen that the graph of a function may be stretched or compressed vertically by multiplying the output of the function by a constant number \(a\text{.}\)
If \(a>1\text{,}\) then graphing \(y = af(x)\) would be the result of stretching \(y = f(x)\) vertically, away from the \(x\)–axis. All of the \(y\) values on the graph of \(f(x)\) would be multiplied by this number \(a\text{,}\) so they would now be larger in magnitude (either positive or negative). Thus, the points would be stretched farther away from the \(x\)–axis.
If \(0<a<1\text{,}\) then the graph of \(y = af(x)\) would be like compressing the graph of \(y = f(x)\) vertically, toward the \(x\)–axis. All of the \(y\) values on the graph of \(f(x)\) would be multiplied by the number \(a\text{,}\) so they would become smaller in magnitude. Thus, the points would be compressed toward the \(x\)–axis.
Finally, we know that adding or subtracting a number from the input of a function will shift the graph either left or right.
Horizontal Stretches and Compressions.
Now, we will complete the story by investigating how to stretch or compress a function horizontally.
Checkpoint 10.2.9.
Checkpoint 10.2.10.
It may have surprised you to see that multiplying the input of a function by a number larger than \(1\) actually made the graph compress horizontally.
Why did this happen? Let's “crunch some numbers” for a function and one of its transformations.
Checkpoint 10.2.11.
The table below shows some values for the function \(f(x)\) and its transformation \(f(2x)\text{.}\)
\(x\)  \(f(x)\)  \(f(2x)\) 
\(1\)  \(8\)  
\(2\)  \(13\)  \(23\) 
\(3\)  \(18\)  
\(4\)  \(23\)  
\(5\)  \(28\)  
\(6\)  \(33\) 
Complete the values for the transformation \(f(2x)\text{.}\)
For any value of the input \(x\text{,}\) evaluating \(f(2x)\) means you're looking at the output of \(f\) farther down the table, where the input is \(2x\text{.}\)
So, when \(x = 2\text{,}\) the output of \(f(2x)\) is the same as \(f(4) = 23\text{.}\)
The completed table looks like:
\(x\)  \(f(x)\)  \(f(2x)\) 
\(1\)  \(8\)  13 
\(2\)  \(13\)  \(23\) 
\(3\)  \(18\)  33 
\(4\)  \(23\)  43 
\(5\)  \(28\)  53 
\(6\)  \(33\)  63 
[The function \(f\) is linear, and its pattern can be used to determine values of the table.]
Notice that the output values for \(f(x)\text{,}\) which occur later in the table, now happen earlier for the transformation \(f(2x)\text{.}\) This makes sense graphically, as the points at certain heights now occur earlier, at smaller \(x\)–values for \(f(2x)\text{.}\)
Note 10.2.12.
It is useful to realize that a transformation like \(f(0.25x)\) may be written as:
Any transformation of the type \(f(ax)\) may be rewritten as \(f\left(\dfrac{x}{\sfrac{1}{a}}\right)\text{,}\) where one of the numbers, \(a\) or \(\frac{1}{a}\text{,}\) is greater than \(1\text{.}\)
Therefore, another way to remember what this type of transformation does is the following.
If you rewrite the transformation so that \(a>1\text{,}\) then:
 \(f(ax)\)
Compresses the graph of \(f\) by a factor of \(a\)

“Multiplying the input variable by a number greater than \(1\) will compress the graph.”
Example: \(f(3x)\) is only \(1/3\) as wide as \(f(x)\text{.}\)
 \(f\big(\frac{x}{a}\big)\)

Stretches the graph of \(f\) by a factor of \(a\)
Example? \(f\left(\frac{x}{4}\right)\) is \(4\) times as wide as \(f(x)\text{.}\)
Often, we will want to horizontally stretch or compress a function by a certain factor. Or, if we already have a transformed function's graph, we may need to determine what the stretching/compressing factor was.
Example 10.2.13.
The graph at left shows \(y = f(x)\text{.}\) This graph was horizontally stretched to make the graph \(y = f(ax)\) at right.
The objective is to find what value of \(a\) caused the horizontal stretch to the graph of \(f\text{.}\)

Since this transformation stretches points away from the \(x\)–axis, fix your attention on the point \((2, 0)\) on the graph of \(f\text{.}\)
Its corresponding point \((5, 0)\) on the graph of \(g\) has been stretched by what factor away from the \(x\)–axis?^{ 1 }That is, how many times larger is the \(x\)–value of the stretched point?
Using the idea from Note 10.2.12, determine the value of \(a\) if the stretched graph is \(f\left(\frac{x}{a}\right)\text{.}\)
The points on \(g(x)\) are \(\frac{5}{2}=2.5\) times as far from the \(x\)–axis as those on the graph of \(f(x)\text{.}\)
Since the graph was stretched, then we know that the transformation is \(f\left(\frac{x}{a}\right)\) where \(a = 2.5\text{.}\) So, the transformation could be written as either:
Or
Stretching or Compressing a standard function.
You are already familiar with many basic functions:
\(y = x\) 
\(y = x^2\) 
\(y = e^x\) 
\(y = log_{10}(x)\) 
\(y = \frac{1}{x}\) 
If you know these basic shapes, you can create infinite variations by using appropriate transformations.
In the next exercise, you will be given a function and its formula. Then use what you know about transformations in order to choose the correct formula for the transformed graph.