e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

Graphing exponential functions allows us to describe the growth of a microorganism or the exponential decay of a radioactive material. Or the decibel level of an audience's applause after performing an underwater escape from a tank full of sharks, handcuffed and blindfolded. Now that's entertainment.

If we start with an exponential in the form y = ab^x + c, then the y-intercept will be a + c (since n⁰ = 1 for any integer n). Exponential functions always increase to infinity as the domain increases to infinity. The degree to which this tendency occurs is determined by the base b. One way or another, though, it'll happen.

As x goes to negative infinity, the ab^x term will gradually be almost 0, which means that y will approach c, but never quite reach it. How sad.

The logarithm, being the inverse function to the exponential is an identical graph, mirrored over the line y = x. The logarithmic function g(x) is inverse to exponential function f(x) when f(x) = ab^x + c and the logarithm takes the form:

So if f(p) = q, then g(q) = p.

If an exponential function is pulling a rabbit out of a hat, then a logarithmic function is pulling a hat out of a rabbit. Both are impressive, but for very, very different reasons.

The trigonometric functions of sine and cosine are very similar. They take the form y = Asin/cos(Bx + C) + D. Basically, they look like waves that repeat over and over again, but A, B, C, and D all change the graph in different ways.

The amplitude A stretches the function so that the top and bottom of the wave are at a vertical distance of 2A (typically one at A and the other at -A.). The B changes how "squished together" or "spread out" the function is, or how often it repeats itself. The wave goes through one complete period every ^2π⁄_|B|. The C value shifts the graph horizontally by ^C⁄_B, and D shifts the graph up.

Tangent graphs are a little different. And by a little, we mean very.

The tangent graph is, in a word, tangential. Sorry if we're just stating the obvious. It has vertical asymptotes for every (n + 0.5)π, and crosses the x-axis at every multiple of π. Not like sine or cosine at all, right? We'd recommend introducing it to your students so that they're familiar with the tangent graph, but not so much that they start to get migraines over it.

There's a reason they call them tangents: they're difficult to keep track of. So while students should know what tangent graphs look like, they should also know that tangents aren't nearly as important as sines and cosines. Yet.