The following question was raised by one of the work groups in class today: Why is the order of performing transformations different than the order of operations?
We are studying function transformations like these:
 |
| From College Algebra by Coburn & Herdlick |
Our explorations in class have supported our book's claim that function transformations must be applied in the following order:
- horizontal shifts
- reflections
- stretches/compressions
- vertical shifts
Does that sequence conflict with the order of operations? What a great question! Let’s explore it using the absolute value function
f(
x) = |
x| as our parent function and using
g(x) = -2|x-1| + 5.
to define the intended transformation. Consider what happens to a point on the graph of
y = |
x| under this transformation. Let's use the point (5,5). Where does it end up after the transformation?
To find out, we evaluate
f(5)= -2|5-1| + 8. This requires the following sequence of calculations:
- 5-1 = 4. that’s the x-1 piece; there’s the horizontal shift*
- |4| = 4. that’s |x-1|; we have just applied the parent function, |x|.
- -2*4 = 8. that’s -2|x-1|; there’s the reflection (-) and vertical stretch (by 2).
- 8+5 = 13. that’s -2|x-1|+5; there’s the vertical shift.
*But why is it a rightward shift? That's for another post.
Has the order of operations been maintained?
It is probably easiest to see if we use
GEMA rather than PEMDAS to track the order of operations. They reflect the same underlying order of operations, but GEMA seems to produce fewer
order of operations misconceptions (sounds like a PhD thesis topic to me!)
GEMA = Grouping symbols first, then Exponents, then Multiplication (and Division, from left to right), and finally Addition (and Subtraction, from left to right).
Now let’s step through GEMA:
G: Grouping symbols. The absolute value bars a type of grouping symbol (so are parentheses and brackets, square root symbols, and even the horizontal line that separates the numerator and denominator in a fraction). First, we work on the expression inside the grouping symbols (absolute value bars). There is only one operation to do in there: subtract 1 (step 1). Now we apply the absolute value bars (step 2), at which point the Grouping symbols are gone and we move on to….
E: Exponents. No exponents to deal with this time. Move on to….
M: Multiply (or Divide): With the || bars gone, the function now reads: f(4) = -2*4 + 5. We multiply by -2 next. This creates the reflection (step 3a) and stretch (3b).
A: Add (or Subtract): Only one thing left to do! (step 4).
Conclusion: It appears the transformations sequence is consistent with the order of operations.
I'm convinced. Are you
?
One way I get around that is to try to hold myself still with a problem for a bit and look for the hidden connections. With kids, I might tell them we're looking for the Hidden Wows.
Fast forward to the end of the semester. In our last week, I printed out those responses, cut them into slips of paper, and gave one slip to each student at random. They read their quote and talked about it at their table. Then they got up and found a partner ("find someone who has similar shoes") and read each other their quotes while discussing how they wanted their future students to say about their elementary math experiences.
