Last week, I had the opportunity to geek out talking math to Chris Harrow, a master mathematics teacher, and I learned so many things that I thought I’d try to share a few here. Chris maintains a great math blog at CAS Musings. Below is my best effort to summarize the firehouse of insights I took away from our conversation.

Start with the story of math—Algebra II is often presented as mastering a set of tools—the fundamental functions, that students must learn to move on to more advanced subjects. But one of the things I had not seen before is really trying to see the connections between those tools, and I can imagine a very question many students might have is “Why do we have all these different functions?”

## Everything is related, and simple at its core

Let’s start with the simplest of functions, the constant function. As an example, choose $f(x)=3$, and we see that this function never changes for all possible input values.

Suppose, instead we start with the number 3, and repeatedly add a constant quantity, say 2, creating the series 3, 5, 7, 9…. We’ve just described a linear relationship, $f(x)=3+2x$.

What if we start with the number 3, and instead add a linearly increasing quantity? For example, we could add 3, then 5, then 7 and so on. If we again start with the number 3, our series will be 3, 6, 11, 18, 27, which can be written as $f(x)=x^2+2x+3$. Now we have a quadratic relationship, which makes sense, since we know through calculus that quadratic relationships have a linear rate of change (please let me never say “we know through calculus” to my students).

We can continue this pattern. Adding another linear change to a quadratic gives us a cubic, and suddenly we have the entire family of polynomials, and can see their that they are all very similar and have addition at their core.

Let us make one simple change. Instead of adding a constant factor, let’s multiply by a constant factor. If we start with 3, and repeatedly multiply by 2, you get the series 3, 6, 12, 24, 48, an exponential relationship $f(x)=3\cdot2^x$.

I’m sure I’ve done a pretty wretched job of describing these functions and their relations, but it was interesting to me to finally release that all of these functions have very simple operations at their core, and deep connections between one another.

## Think graphically, and look for connections

Let’s take an exponential curve, say $y=2^x$. Now if we wish to slide this to the right by 4 units, we could write $y=2^{(x-4)}$. Interestingly, we would get the same exact curve if we had stretched the original vertically, as in $y=\frac{1}{16}2^x$. The fact that this must be true becomes a bit more obvious when you apply the rules of exponents:

$y=2^{(x-4)}=2^x \cdot 2^{-4}=\frac{1}{16}2^x$.

This is probably obvious to many experienced math teachers, but I thought it was cool, and a nice way of putting some of those exponent rules to use.

## What’s the co- in cosine

I’m probably the last person in the world to realize what why cosine is named cosine. Take a standard right triangle:

In this triangle the sine of alpha is $\frac{A}{C}$, and the sine of beta, the complement of alpha, is $\frac{B}{C}$. Long ago, this was the complementary sine function, but its name was shortened to cosine. Trig would have mad a lot more sense to me if I’d been told that back in high school.

And of course, cosine sucks.

## Graphing trig functions

Start with the basic trig function $y=A\sin x$.

Now draw lines at $y=+A$ and $y=-A$. The sine function describes oscillations that begin in the middle, and then bounce between the upper and lower bounds, like so:

This is not all that exciting. But what if we replace the constant A with something that changes, like $y=x \sin x$. Normally graphing or thinking about a function like this is challenging, but really, the sine function continues to oscillate between the upper (ceiling) and lower (floor) bounds, just like before:

## Polar plotting goodness

Polar plotting is another one of those topics that can often make little sense to students. But Chris showed me how to build off of this idea about the behavior of sine oscillating between a lower and upper bound to take on plotting functions like this.

$r=3+2\sin \theta$

Start with the idea of r=3, which establishes a centerline around $r=3$. Then we see that the 2 in front of the sine function gives us an amplitude of $\pm 2$ about $r=3$, so we can draw in the upper and lower bounds for the function, and just follow the pattern of sine—start in the middle, go to the upper bound, then go back to the middle, go to the lower bound, and back to the middle, all in a period of $2\pi$. Here’s a keynote animation that shows this process.

## Introducing functions

One of the things that I think students stubble with a lot early on in Algebra is function notation. They don’t seem to see it’s purpose, and it feels like a lot of excess baggage. Here, Chris gave me another excellent method for introducing this topic.

Start with a function on the board, say $y=x+4$. Ask, if $x=2$, what is y? Students have no trouble with this. Now add a second function to the board $y=2x$. This time, ask what y is when $x=4$? Here there should be a bit of confusion. Which y are you referring to? Really we need a way to name functions other than just y, and so we can name functions whatever we want, f, r, fred or rita. This turns out to be very handy in physics. Suppose someone walked into the room just as you shouted $y=6$. This person didn’t hear the input, $x=2$, and so they don’t know which function you’re referring to. Wouldn’t it be nice if we could also capture the input when writing out a function? And so we get function notation

$f(2)=6$ is really just a short hand way for saying “the ordered pair (2,6) on the curve f.”

## Digging into graphing

Start with a simple example, like $y=x^2$. Are there any points in this function that are invariant, where the input = output? This would require:

$x=x^2$, which can be rewritten as

$0=x^2-x=x(x-1)$

which tells us the invariant points are $x=0\; \textrm{or}\; 1$

If we plot the graph y=x, we know that that we’ve got the two invariant points at 0 and 1 that don’t change when we when we square. Between 0 and 1, the output values must get smaller, and beyond 1, they must get bigger.

All of this is to build up a sort of graphical intuition for functions, which enhances any work you do with having students graph functions with technology (which we both think is essential).

Let’s consider the function $y=\frac{1}{x}$. Where are the invariant points here?

$x=\frac{1}{x}$ can be rearranged to:
$x^2=1$ which tells us that there are invariant points at $x=\pm 1$

This function has one other wrinkle, you can’t have an input of $x=0$, so there must be an asymptote at $x=0$. Finally, we can think about the behavior of the function by thinking about what happens when you invert numbers that are small (between 0 and 1)—-they should get big, and inverting numbers that are big should get very small. So we should expect the function to go off toward positive infinity as inputs approach 0 from from the right, and it should approach zero for very large positive inputs, making it pretty easy to explain this graph.

Chris reminded me of the idea that asymptotes can be odd (where the function diverges oppositely to positive and negative infinity on either side of the asymptote), or even (where the function diverges in the same way on both sides of the asymptote, to either positive or negative infinity). Knowing this, it becomes easy to think about sketches of even very complex functions, like

$y=\frac{1}{(x-1)^101(x+4)^98}$

In this function, we have an odd asymptote at $x=1$, since the power on that term is odd, and an even asymptote at $x=4$, since the power on that term is even. We can also see that for very large positive values, the function is positive, and approaches zero, and this gives us all the info we need to sketch the function:

which is pretty close to what Desmos gives us:

Here’s, one last intuitive graphing challenge. Let’s look at the graph of $y=\sin^2 x$

We can see that this function looks a lot like a sine wave, except it stats at its lower bound, zero and travels to its upper bound of 1, which is the opposite of the behavior of cosine. We can also see it has an amplitude of $\frac{1}{2}$. It’s also shifted vertically by $\frac{1}{2}$ and finally, the repeats itself with a period of $\pi$, so it the we need to change the term in front of the input to double the frequency of the function.

Put all this together and we can write:

$y=-\frac{1}{2}\cos^2(2x)+\frac{1}{2}$

A perfect match, and we’ve also inspected our way to the trig identity: $\sin^2 x=-\frac{1}{2}\cos^2(2x)+\frac{1}{2}$.

## Some final thoughts

I’m not sure how many of these insights I’m going to be able to turn into lessons for my Algebra II students, but if I can capture just the tiniest fraction of enthusiasm for exploration and mathematical play that Chris showed when explaining these ideas to me, I think I’m going to be just fine.