So, I’ve had a number of people ask about whether the x-coordinate of a turning point should be included in an interval of increase or decrease or whether the intervals should always be exclusive.
Believe it or not, I’ve been tossing this question around for awhile with my good friend Brian Battistoni at Arlington High. This is a great question and it can be argued both ways. What we really need is for NYSED to just issue a clarification, so we can all be consistent. Maybe they will, maybe they won’t. I think it is a cool question because it can be argued both ways. Here’s how…
The Exclusive Case (x-coordinates of turning points not included):
This is the convention that most people will first think of because if the turning points were included, then they would have to be included in both intervals of increase and intervals of decrease, thus seemingly making the function both increasing and decreasing at the same point. This doesn’t sit well with people. On top of that, if you look at it from a calculus perspective and define the function as increasing when its slope is positive and decreasing when its slope is negative, then turning points, where the slope is either zero (smooth) or undefined (corners), should be excluded.
The Inclusive Case (x-coordinates of turning points are included):
To make and understand this argument, we need to agree on what the question actually is. Remember, the question is always phrased like “give the interval over which f(x) is increasing.” So, let’s examine the function:
Most calculus books will define an interval of increase as follows:
In other words, bigger x’s give bigger y’s. But, in the case of the quadratic above, wouldn’t the interval:
fit this definition? Remember, the question is not whether the function is increasing at x=4. It is whether x=4 is part of an interval where bigger x’s give bigger y’s. The x-coordinate of the turning point would certainly be included in this definition and is certainly included on the Advanced Placement Calculus exam. This inclusion really gets at a fundamental geometric idea regarding intervals of increase and decrease:
An interval of increase is a stretch of the function that can be considered going uphill and an interval of decrease is a stretch of the function going downhill. Aren’t the bottom and tops of hills part of these stretches?
What’s the correct definition? Well, that’s up to the bureaucrats to decide, which is weird to say. The one sample problem Albany gave us was the exclusive use. I’m amazingly inconsistent on this because of my own struggles with two equally valid interpretations.
Anyone want to weigh in?
When that sample question (with the exclusive intervals) was released, a lot of people complained to Albany. The answer at that time was that questions would be written in a such a way that inclusive or exclusive wouldn’t be a testable issue. In other words for multiple choice, all choices would be one or the other and for free response, the rubric would accept both forms.
Personally, I’m use increasing intervals are closed unless the endpoint is not in the domain.
Another question is increasing versus strictly increasing. Your definition of increasing, y2 > y1, is sometimes referred to as strictly increasing. Some texts use y2 >= y1 as increasing which means constant functions are increasing (and decreasing). Now, your brain hurts. 🙂
Oh, I know! Right. And, of course, at this level, all of this is really just a question of semantics. There is virtually no ramification, practical or esoteric, where the inclusion or exclusion of the endpoints matter. The AP Calculus exam tries to stay away from the issue by asking questions that specifically exclude the endpoints. These questions are way more suited to sophomores in college who are math majors. Then, they can get into strictly increasing/decreasing, monotonically increasing/decreasing, etcetera.
Thanks for the clarification in the earlier report about the talk out of Albany accepting both versions. I hate the fact that they make policy by email and footnote.
I am making the argument to my students that the endpoints should not be included. I am trying to introduce the concept of instantaneous rate of change with this. I tell students to imagine they are in a roller coaster car at that point. What direction are they facing? We say that at the turning point we are neither increasing, nor decreasing. I believe that this idea helps when talking about when y is positive or negative also with the idea that the function, at some point, must be neither positive nor negative in order to change over.
Of course, when it comes to these sorts of things for most of our students, I believe the answer is largely irrelevant. What is important is that the students understand what is actually happening.
I actually told my 8th graders about this argument to help them see that math is not always a “right answer” subject!
I teach students about the discussion and expect them to understand it but do not penalize points either way.
Albany can shove their answers up their a**. They’re not math teachers, they’re appointed puppets who could care less. We have annually argued math reasoning during “grading time” and students would get penalized (getting a 2 out of 3) for doing correct math that doesn’t coincide with the rubric from Albany.
My advice is to keep pressure on Albany to let teachers run the show, not politicians who are bought, sold, and paid for by privateers.
Merry Christmas !
Adam,
Words to live by, words to live by
I do feel a bit badly for them, at least the content specialists. There are so many changes and they have to coordinate all of it K-12. Don’t get me wrong, I am angry at them as well because it is yet another half-hearted attempt at reform. Except, this time, they left much of it up to a private company (i.e. the Modules) and seem to abdicate that responsibility at every turn. I’m not sure if it is just because they are overwhelmed, or if it is something more. But, certainly in their rush to create assessments that they can measure us (the teachers) by they have forgotten long ago what the real purpose of assessment is.
This is also what I tell them.
We have to be careful not to conflate “increasing” with “having positive slope.”
If f(x) is defined on [a, b] and has positive slope on (a, b), then f(x) is said to be increasing on [a, b].
And of course this stems from the book definition of “increasing”: f(x) is increasing on [a, b] iff for all x in [a, b] with x1 < x2, f(x1) < f(x2)
Hey Jasper,
I don’t disagree with you at all. But, it is amazing how confusing this topic is to both kids and teachers. I have been torn apart before because teachers claim I am in error if I include turning points in intervals of increase and decrease, even though the definition you give, i.e. x2>x1 implies f(x2)>f(x1) is the definition of increasing, means that the endpoints should be included. It’s interesting, because in another post, I show evidence of how the state grading rubrics even try to have it both ways. Check this post out:
https://www.emathinstruction.com/intervals-of-increase-and-decrease-put-to-bed/