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LostInParadise's avatar

Would this be a good high school math problem?

Asked by LostInParadise (31979points) December 27th, 2023

I thought of this and I think it is interesting to see how it can be solved using elementary calculations.

What is the line of symmetry of f(x) = arctan(x)+arctan(6-x)?

Seems scary? It requires no calculus. arctan works nicely because it makes a graphic picture that is easy to see, but other functions will work as well.

Here is a list of questions that lead to the solution:
How does f(6-x) compare to f(x)?

Using the previous question, given a point (x,y) on the curve, what is another point on the curve?

What is the midpoint of the two points?

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6 Answers

LostInParadise's avatar

To see the curve, go to geogebra and enter y=arctan(x)+arctan(6-x)

LostInParadise's avatar

Not looking as if there are goiing to be any answers.
Here is the solution I was looking for.

f(6-x)=arctan(6-x) + arctan(6 – (6 -x ))= arctan(6-x)+ arctan(x), which is identical to f(x), so we have f(6-x)=f(x)

Since (x,y) is on the curve, y=f(x), but f(6-x)=f(x), so (6-x, y) is also on the curve.

The midpoint of (x,y) and (6-x,y) is ((x+6-x)/2), (y+y)/2) = (3, y). The curve is made up of pairs of points equidistant from the line x=3, Therefore the line x=3 is the axis of symmetry.

Zaku's avatar

They seem like something we could have had a good way through trigonometry, which was for people good at math, and 10th or 11th grade, IIRC.

I’m not about to engage the math as an adult who doesn’t have to care, but as a captive high school trig student, I think I would have liked these because they are asking about qualities of the graph, which seems like some of the more interesting parts of trig (moreso than functions with zero context).

I wonder how easy it is to graph things these days with web sites, spreadsheets, or calculators. Let’s see…

OH @#%$!

Google will just graph it for you!

I typed in “graph f(x) = arctan(x)+arctan(6-x)” and got an (unlinkable, apparently) graph for it.

Kids have it so easy now.

LostInParadise's avatar

I hope that it is apparent from the Google graph that the axis of symmetry is the line x=3

Zaku's avatar

It is, to someone that knows what an axis of symmetry is, and what the line x=3 is.

Oh, I think this works for the link.

I used to run a Minecraft server for some US middle-schoolers (public school, not math-oriented) who were very much into the game. One thing I did was create a treasure that they were obsessing over, but I didn’t want it to be an easy giveaway, so I put it a LONG way away from where they usually played in the world, and left a series of clues in books in the game world. I thought they were a bit easy to figure out. Once you got them all (which they did, quickly, eager as they were), you got a description of where the treasure was, in the form of a couple of line equations. y = mx + b. The treasure was at the intersection of the two lines, so combine the equations, solve for x and y, then use Minecraft’s ways of telling you your current coordinates, to find the treasure, right?

Nope. No clue what this was about. So they found “the smartest kid at math at their whole school”, and showed it to him. HE had no clue what this was talking about.

In other occasions, when helping kids with math homework, I was pretty shocked and dismayed to find that:

1) They were REALLY struggling with simple math homework.
2) They were basically having fear/panic reactions to math.
3) They had almost zero conceptual understanding of what the lessons were trying to teach them.
4) When I tried to bring them from their actual level of conceptual understanding, to what the problems were saying, it was very hard to get them to approach problems in terms of understanding the concepts, at all.
5) Even when I did manage to get them to understand something, they were still afraid and angry because they were sure they were supposed to do whatever rote problem response technique they’d been taught, not something based on understanding the concepts.

I wish you much better success with your students.

LostInParadise's avatar

I am not a teacher, but I have done some math teaching, and have had similar experiences to what you have faced. One time many years ago, I had an adjunct faculty position at a community college for a pre-calculus course. I naively thought that this would be a great opportunity to demonstrate why things worked the way they do.

In the first week, one of the students told me that he appreciated what I was trying to do, but he was switching out of my class because he was only taking the course because it was a prerequisite for a biology degree, and all he wanted to know were the equations for plugging in values. I suspect that most of his classmates had a similar point of view.

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