Order of function transformations

Question

Here is a question that arose during class today (taken from: https://madasmaths.com/archive/maths_booklets/standard_topics/various/transformations_of_graphs_exam_questions.pdf)

Three geometric transformations are defined as follows:

• $R$ is a reflection in the $y$-axis.
• $S$ is a stretch parallel to the $x$-axis, by scale factor of $\frac{1}{2}$.
• $T$ is a translation by 3 units in the negative $x$ direction.

The transformation $T$, followed by $S$, followed by $R$ is applied to the graph of the curve with equation $y = \sqrt{x}$.

(a) Determine a simplified equation for the transformed graph.

(b) Determine a sequence of transformations in terms of $R$, $S$, and $T$ only, which map the graph of $y = \sqrt{x}$ onto the graph of $y = \sqrt{4x + 3}$.

For part (a), the author of the document stated the answer to part (a) is $y=\sqrt{3-2x}$.

The argument: $\sqrt{x} \mapsto \sqrt{x+3} \mapsto \sqrt{2x+3} \mapsto \sqrt{-2x+3}$.

But here is another 'method': if we let $T(x)=x+3$, $S(x)=2x$ and $R(x)=-x$, then applying '$T$ then $S$ then $R$' would be doing $R(S(T(x)))=-2x-6$. Seemingly, I should get $\sqrt{-2x-6}$ instead.

What is the best way to explain this in discrepancy to students?

EDIT: it is my opinion that the 'correct way' to use function composition would be to define the following:

$T(f(x))=f(x+3)$

$S(f(x))=f(2x)$

$R(f(x))=f(-x)$

Then compute as follows: $R(S(T(\sqrt(x))))=R(S(\sqrt{x+3})=R(\sqrt{2x+3})=\sqrt{-2x+3}$

Answer 1

It helps to let $u=x+3$ and draw a new coordinate system, overlapping the first if possible. Then in your second method, $T(x)=x+3=u$, and $S(u)=2u$. This last operation stretches the plane about the line $u=0$, or $x=-3$, not $x=0$ like you want. Not only that, but then your $R$ applied to $S(u)$ reflects the graph across the line $u=0$, which again you don't want. This is one of the more subtle points, I think. There is no one equation for $R$ that will do the job for all types of problems like this, because we're not reflecting about "the origin." We're reflecting about the original $y$ axis, which is a fixed line that could be ten billion units away from our new origin whenever we finally want to do the reflection!

Something similar happens if you try the sequence of transformations $x\mapsto -x\mapsto -x+3$. If you think that last $+3$ shifts the graph to the left (with respect to the original axes at least), think again.

The lesson is that phrases like "shifts the graph" or "reflects the graph" are somewhat sloppy, even though we're all guilty of saying them. The shifts and reflections take place with respect to certain coordinate systems and if we introduce new coordinates we have to keep track of which is which.

(Also if you're not asking about part (b) you may as well delete it from your question.)

Answer 2

For me this issue always came up at the beginning, usually with how reflection about the $x$-axis followed by a vertical shift differs from the same vertical shift followed by reflection about the $x$-axis. For example, if beginning with $y = x^2$ and the vertical shift is "up by $4$ units", then the former gives $y = -x^2 + 4$ and the latter gives $y = -(x^2 + 4) = -x^2 - 4.$ I told students that it was simpler to do the reflection before the vertical shift, because otherwise you'll have to anticipate in advance how reflection (i.e. multiplying an expression by $-1)$ alters an already-put-in-place additive constant and adjust accordingly in advance. For more examples, see this 19 October 2009 sci.math post. See also this Mathematics SE answer and the comments to it.

In what follows, I almost always (if not always) disallowed the use of calculators. I mainly wanted to prevent graphing calculators from being used, but since nothing in what follows was arithmetically challenging in the least, it was simplest to not allow any calculators. When these were on major tests in which calculators otherwise could be used, the types of questions below would be on separate pages that they were to work on first, then hand in to me, then I'd give them the rest of the test on which they were free to use their calculator.

I taught this stuff a lot from the mid 1990s to the mid 2000s (also a lot in the 1980s, but it wasn't until the mid 1990s that I came up the the types of questions I discuss below), and my instructions were always to give a labeled sequence of transformations (and I explicitly specified the types of "basic transformations" that could be used, and said that only one type was to be used at a time) and they had to show both a sketch of the current graph after each transformation and write down an equation for that graph. There was no single correct answer. Also, there were many ways a student could get (minor) point deductions along the way, despite having the final graph be correct.

In the case of something like sketching a graph of $y = -|x+2| - 3,$ I did this by hand-drawing $4$ pairs of coordinate axes on the original test page (that photocopies were later made from) for them to sketch graphs on and, under each pair of coordinate axes, I drew a blank line on which they were to write an equation for their sketch. I also said that in some cases, it might take fewer than $4$ transformations. In their sketches they were to identify what I called the "basic points" (for lack of anything better to call them) for the original "starting graph" (I called these basic graphs), and they were to show where each of these points went after each of the transformations, doing this by drawing appropriate largish dots on their sketches and labeling both coordinates of that point. The basic points depended on the basic graph, but unless I'm overlooking something, it was always the points in the basic graph corresponding to some or all (depending on domain) points where $x=-1,\,0,\, 1.$ So, for example, in the case of $y = -|x+2| - 3,$ in the first pair of axes they should have a sketch of $y = |x|$ and, in the blank line for the equation, write $y = |x|.$ After this their answers could vary, but most did an $x$-axis reflection next, then horizontal shift left by $2$ units, then vertical shift down by $3$ units (or they would have these 2nd and 3rd steps interchanged). Trig. graphs were NOT included when I did this, by the way, but the ideas were of course used when we later got to trig. graphs.

I also had matching questions, where I'd have $10$ to $15$ equations such as $y = -\frac{1}{\sqrt[3]{x+2}}$ (the question numbers were labeled $1$ through whatever, and I drew a short blank just to the right of each question number) listed on the left side of the paper and $3$ or $4$ MORE smallish rough hand-drawn sketches on the right side of the page (labeled A, B, C, until the last of the graphs), with instructions saying to write in each blank the appropriate letter for that equation OR write "no graph" (when none of the graphs remotely corresponded to the equation). I said in the instructions that not all equations will have a matching graph and not all graphs will be used -- this aspect made these $15$ to $25$ minute short quizzes especially challenging. I often avoided, or at least greatly shortened, this type of matching task on the major tests (3 to 4 in a semester, not including the final exam) due to time limitations.

Still another type of question I often gave involved using a computer algebra system to produce a neat looking graph of an implicitly defined curve and then asking various graph transformation questions based on it. For example, in an actual Sept. 2004 precalculus worksheet of mine I gave a nice looking graph of $\;x^3 + y^2 \, = \, 6xy\;$ and asked $10$ questions about it, two of which are:

If this graph is reflected about the $y$-axis, then shifted up by $2$ units, then reflected about the $x$-axis, and then shifted left by $5$ units, what is an equation for the shifted graph?

Describe an ordered sequence of shifts and/or reflections that when applied to the above graph will result in the graph of $\;-(x-1)^3 + y^2 \; = \; 6(x-1)y.$

Note, by the way, the phrasing I used -- "an equation" and "an ordered sequence". There is no single correct answer, and thus it would not be appropriate to say "the equation" or "the ordered sequence".