5.6: Combining Transformations

Prior Lesson | Next Lesson

Objectives

• Combine transformations of functions
• Practice the material in Chapter 5

Lesson Guide

Warm-Up

To continue reviewing the material from this section, have students do some sort of review of all transformations used at this point. A table of possible transformations (similar to the one in the Section 5.4 lesson guide) is useful for students to fill in with their groups, but make sure you are able to check these/go over them on the board so they are correct! Test their knowledge by asking them to fill the table out without notes to start and then once they can't fill in any more, they can use their notes.

Have students do Problem 1. If students have any problems you can refer them to Problem 4 from Worksheet 5.5 (this is basically the same problem).

Combine transformations of functions

Have students complete Problem 2, which discusses having multiple vertical transformations. This part is usually more intuitive, so make sure you discuss the formulas students came up with for part (c).

Have students complete Problem 3, which discusses having multiple horizontal transformations. This part is much less intuitive. Part (c) only asks that they write two possible functions (thinking about order of operations), so you should discuss which formula goes with which order.

It may be good to do another example of horizontal transformations.

• Example:

Describe how we can transform the graph of $y=m(x)$ to get the graph of $y=m(3x-6)$. First off, we know that the following transformations are involved:

• Horizontal compression
• Horizontal shift

Since there are multiple horizontal transformations, we know that the order matters (c.f., Problem 3), so we will need to figure out what the order is. There are several ways that you can teach this, which are described below (with pros and cons). We will not be testing students on which method they use, but rather just that they can use a correct method. It may be best to choose one method to use for each example, rather than showing students all three methods.

Method 1: Order of Operations

When there are multiple vertical transformations involved, we follow the order of operations. When there are multiple horizontal transformations involved, we follow the reverse order of operations.

If we follow order of operations, we should multiply by 3 first, then subtract 6. However, since these are horizontal transformations, we need to follow the reverse order of operations. Therefore, the order of transformations is:

• Horizontally shift right by $6$ units
• Horizontally compress by a factor of $3$

Pros: Easy to teach and memorize Cons: Does not explain why, easy to mix-up

Method 2: Standard Form of Transformations

For constants $A$, $B$, $h$ and $k$, the graph of the function \[y=Af(B(x-h))+k\] is obtained by applying the transformations to the graph of $f(x)$ in the following order:

• Horizontal stretch/compression by a factor of $|B|$
• Horizontal shift by $h$ units
• Vertical stretch/compression by a factor of $|A|$
• Vertical shift by $k$ units

If $A<0$, follow the vertical stretch/compression by a reflection across the $x$-axis. If $B<0$, follow the horizontal stretch/compression by a reflection about the $y$-axis.

To apply this to our problem, we need to rewrite $y=m(3x-6)$ in the standard form of transformations: \[y=m(3(x-2)).\] Now we can see that $A=1$, $B=3$, $h=2$, and $k=0$. Therefore, the order of transformations is:

• Horizontally compress by a factor of $3$
• Horizontally shift right by $2$ units

Pros: Just requires memorization, how the book teaches it Cons: Does not explain why, easy to mix-up

Method 3: Bubble Diagram

If we think about the function $m(3x-6)$ as a composition of several simple functions (refer back to \S5.1: Function Composition), we can represent it using the following function diagram:

It's tempting to think that this diagram implies that first we would compress, then shift. However, recall that horizontal transformations occur in the opposite way than you might think. To explain why, suppose that $m(a)=b$. To see how this point transforms to become a point on $m(3x-6)$, we do the following:

Therefore, the order of transformations is:

• Horizontally shift right by $6$ units
• Horizontally compress by a factor of $3$

Pros: Visual representation, explains why Cons: Hard to teach well

NOTE: Since each transformation in Problem 4 involves at most one vertical and at most one horizontal transformation, note that order doesn't matter yet.

Have students work on Problems 4 and 5. It would be beneficial to have students share answers to 5(c) and (d) at least.

Note that the point of Problem 5 is to help students to recognize that order matters if there are multiple vertical or horizontal transformations.

Do one part of Problem 6 to show students how to draw the graph of a transformation of $f(x)$ given the graph for $y=f(x)$. Have students do the rest of Problem 6.

Make sure that students realize that order does matter when applying multiple vertical or horizontal transformations.

Comments

• I would only recommend using the bubble diagrams if you have used this throughout other sections (inverse functions, function composition, etc.)