Families of Curves

When I put out my call for help with Project Based Learning, I got a wonderful email from @gelada (a.k.a. Edmund Harriss of the blog Maxwell’s Demon) with a few things he’s done in his classes. And he — I am crossing my fingers tight — is going to put those online at some point for everyone. To just give you a taste of how awesome he is, I will just say that he was in NYC a few years ago and agreed to talk to my classes about what it’s like to be a real mathematician (“like, does a mathematician just like sit in a like room all day and like solve problems?”), and have kids think about and build aperiodic tilings of the plane.

Anyway, he sent me something about families of curves, and that got my brain thinking about how I could incorporate this in my precalculus class. Students studied function transformations last year in Algebra II, and we reviewed them and applied them to trig functions. But I kinda want to have kids have some fun and make some mathematical art.

First off, I should say what a family of curves is.


That’s from Wikipedia. A simple family of curves might be y=kx which generates all the lines that go through the origin except for the vertical line.


I made this in Geogebra with one command:

Sequence[k*x, k, -10, 10, 0.5]

This tells geogebra to graph y=kx for all values of k from -10 to 10, increasing each time by 0.5.

Okay, pretty, but not stunning. Let’s mix things up a bit.

lines 2

Sequence[k*x+k^2, k, -10, 10, 0.1]

Much prettier! And it came about by a simple modification of the geogebra command. Now for lines with a steep slope, they are also shifted upwards by k^2. This picture is beautiful, and gives rise to the question: is that whitespace at the bottom a parabola?

Another one?



And finally, just one more…



Just kidding! I can’t stop! One more!


Sequence[k sec(x)+(1/k)*x,k,-10,10,0.25]

What I like about these pictures is…





And then, if you’re me, they raise some questions… Why do they look like they do? What is common to all the curves (if anything)? Does something special happen when k switches from negative to positive? What if I expanded the range of k values? What if I plotted the family of curves but with an infinite number of k values? Do the edges form a curve I can find? Can I make a prettier one? Can I change the coloring so that I have more than one color? What would happen if I added a second parameter into the mix? What if I didn’t vary k by a fixed amount, but I created a sequence of values for k instead? Why do some of them look three-dimensional? On a scale of 1 to awesomesauce, how amazingly fun is this?

You know what else is cool? You can just plot individual curves instead of the family of curves, and vary the parameter using a slider. Geogebra is awesome. Look at this .gif I created which shows the curves for the graph of the tangent function above… It really makes plain what’s going on… (click the image to see the .gif animate!)

sec animation

Okay, so I’m not exactly sure what I’m going to do with this… but here’s what I’ve been mulling over. My kids know how to use geogebra. They are fairly independent. And I don’t want to “ruin” this by putting too much structure on it. So here’s where I’m at.

We’re going to make a mathematical art gallery involving families of curves.

1. Each student submits three pieces to the gallery.

2. Each piece must be a family of curves with a parameter being varied — but causing at least two transformations (so y=kx^2 won’t count because it just involves a vertical stretch, but y=k(x-k)^2 would be allowed because there is a vertical stretch and horizontal shift).

3. At least one of the three pieces must involve the trig function(s) we’ve learned this year.

4. The art pieces must be beautiful… colors, number of curves in the family of curves, range for the parameter, etc., must be carefully chosen.

Additionally, accompanying each piece must be a little artists statement, which:

0. Has the title of the piece

1. States what is going on with each curve which allows the whole family of curves to look the way they do, making specific reference to function transformations.

2. Has some plots of some of individual curves in the family of curves to illustrate the writing they’ll be doing.

3. Has a list of things they notice about the graph and things they wonder about the graph.

At the end, I’ll photocopy the pieces onto cardstock and make a gallery in the room — but without the artist’s names displayed. I’ll give each student 5 stickers and they’ll put their stickers next to the pieces they like the most (that are not their own). I’ll invite the math department, the head of the upper school, and other faculty to do the same. The family of curves with the most stickers will win something — like a small prize, and for me to blow their artwork into a real poster that we display at the school somewhere. And hopefully the creme de la creme of these pieces can be submitted to the math-science journal that I’m starting this year.

Right now, I have a really good feeling about this. It’s low key. I can introduce it to them in half a class, and give them the rest of that class to continue working on it. I can give them a couple weeks of their own time to work on it (not using class time). And by trying to suss out the family of curves and why it looks the way it does, it forces them to think about function transformations (along with a bunch of reflections!) in a slightly deeper way. It’s not intense, and I’ll make it simple to grade and to do well on, but I think that’s the way to do it.

What’s also nice is when we get to conic sections, I can wow them by sharing that all conics are generated by r(\theta)=\frac{k}{1+k\cos(\theta)}. In other words, conic sections all can be generated by a single equation, and just varying the parameter k. Nice, huh?

PS. Since I am not going to do this for a few weeks, let me know if you have any additional ideas/thoughts to improve things!



  1. Great ideas here! What about asking students to try and replicate a couple of these without knowing the function used. Of course I am thinking about functions with which students are fairly familiar. Thanks for kickstarting my brain this morning.

    1. Haha, honestly I tried to remember and I can’t! I think it had to do with differentiating with respect to k, like how a slight change in the k value changes the y? Sorry I’m a doofus and can’t remember.

  2. I love Geogebra art and have used it to show my students (they’re generally Algebra I or 8th grade, though sometimes Algebra II) the power of the variable. Another fascinating aspect to add is to use dynamic colors – the RGB triad can be varied with your slider or based on another parameter. (For example, put a point P on on of your curves and then let n=x(P) and alter the Red, Green and/or Blue value based on n. Sometimes you get muddy colors (if the R,G and B are close to each other) and sometimes you get some beautiful shades. Here’s a file I just created: http://www.geogebratube.org/student/m29129

      1. Very nice! Do your students use scripting when they use Geogebra? I’m not good at it myself, and my students have never really seen me use it. I’m thinking about taking my Algebra I’s down to the computer room to make some line art this week now. Also, I’m about to do your Matrix Multiplication activity, introducing it with a food web then transitioning to a social network. Great ideas!

        -Sue Russo

        Quoting Continuous Everywhere but Differentiable Nowhere

      2. Hi Sue,

        No, they don’t know how to script. In fact, I don’t either — my colleague is the GGB expert and he is figuring all these cool things out. My kids are just learning how to graph, change axes, things like that. I recently realized we could do matrices in GGB, so maybe next year. I’m sad that we had to limit our matrix unit, because we didn’t get to do any of the fun stuff that you are doing!


      3. I’m actually not a big fan of teaching matrices in Algebra 2 (or in high school, for that matter, when there’s so much other material we could cover, but this is the first year I’ve had any fun with them. Not sure if the kids are enjoying them, but I’m seeing them as less of a waste of time this year. If only we hadn’t had 2 snow days and so much frivolity (Spirit Week) recently – we could have made good progress. BUT we had “Penny Wars” and now I’m itching to have the kids figure out how many of each type coin each class turned in using a 5×5 system of equations!

        Thanks again for the ideas. I’m looking foward to this week – not an easy thing to say in February.


        Quoting Continuous Everywhere but Differentiable Nowhere

  3. This is brilliant. It gets explicitly at a really important conceptual leap: from “what does the graph of this function do?” to “what does the graph of this type of function do?”. Going from specific numbers to variables. But it does it in such a concrete, visually appealing way, that allows for experiment and keeps things from getting too abstract. I am so stealing this.

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