When you want your expression to produce a sweeping arc, when you want your expression to imitate gravity, when you want your expression to portray a meteor streaming through the sky, the tool of choice is transcendental Mathematics because this set includes everything from cones to parabolas, to ellipses and hyperbolas.
In our everyday life this encompasses simple things like acceleration (a bird taking flight), tossing a rock into a pond (gravity), or more precisely in our design work, the sweeping motion of a camera or a light.
How often we see a camera span the area of interest whether it be landscape or audience, or have a light span the arena whether it be a galaxy or the Rolling Stones.
This behavior is even more simple in a daily metaphor because that is how we evaluate our presence, how we take in a new scene, at the park or at the office.
We view the landscape zooming in on this and that object, we scan the area assigning familiarity.
Long before Ken Burns was credited with this technique of 'pan and scan', it has been implemented by 'Everyman'.
The ability to imitate this behavior in animation really is mathematical and while it's not quite as easy as your multiplication tables, it is well documented and very straightforward.
Once you get comfortable creating expressions drawing on these basic shape definitions you will be well on your way to applying them to the Life imitating nature of your compositions.
Using Math in After Effects expressions is quite easy but some of the definitions, functions, and presets are not intuitive.
After Effects expressions use radians instead of degrees which requires a conversion.
After Effects uses a 'Javascript' style expression language, 'Camel style' spelling of variables.
The built in functions could be better documented.
An affinity for Mathematics is helpful but the endless possibility of definition draws on the infinite rhythms of these basic functions defined and refined by ancient Greek mathematicians, such as Pythagoras.
What is so special about these functions? It is their rhythms and repetition and their non-linear flow.
They precisely define sweeping possibility of motion and change as we see daily in nature drawing on some of the oldest basics of Geometry and Trigonometry.
When you view the curve of the sine wave beginning at 0 to 1 to -1 and back, it's points follow an ever increasing, decreasing, increasing rhythm that could translate to waves in the sea, planetary motion, electricity or music.
To introduce yourself to expressions, begin at the beginning, creating a single object, a ball or a box.
Define an expression using the alt key.
For example, choose position for your object, click the position stopwatch while holding alt key and this will open the expression definition window.
Key in two numbers enclosed in a bracket: [100,100].
Keying in two numbers instead of a function will simply place your object there.
You are defining position.
Try a couple different numbers just to see how basic this is.
Now, define you x and y position with variables.
Again, something very simple would be: x=(time*50); y=(time*25); With this definition you have equated the value of x to the amount of seconds your animation has been running times 50.
After Effects sees 'time' in these expressions as seconds if you do not convert it.
So the definition above would have x=50, y=25, at second 1, x=100,y=50 at second 2 and so on.
The point here is that you have tied these positional coordinates now to something in your animation that has it's own life, in this case, time, how long it's been running.
I chose 50 and 25 because of the size of the display grid I'm working with (640,480) to be sure my object is visible in the short animation run.
Starting with something simple like this can be a great way to get the feel for expressions and then get a bit more ambitious.
Using transcendental functions like a sin wave plotted against your position, will give your animation the grace of tidal waves.
Starting with some simple definitions will provide the building blocks necessary to graduate to these loftier expressions.