4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

We don't know if you've noticed, but there are a lot of books out there. Some are poetry books, others are murder mysteries, and some are even both. (This haiku is quite the nail-biter: "Mister Green? Hall? Rope? / Keep looking and find the clues / To know who done it.") There are even graphic novels, short stories, and more genres of books out there than we can list. And still, as many as there are, whatever story they contain, they're all books.

In the same way, functions are functions regardless of all the different ways we can express them. We can describe a function using words or draw it out on the x-y plane. If we really want, we can find the equation that describes the function or make a table of values to describe it. Students should know that they can use these different methods to describe the same function. They should also know how to do so.

Students should also know that expressing functions in different ways has its advantages. For instance, if we want to describe f in terms of x as a second order polynomial, we can write f(x) = x² + x + 1. While this would allow us to determine the value f(x) for any value x, it might be more difficult to imagine behavior over all integers. In order to see how x behaves over the grand scheme of things, we could use a graph.

With a graph, we can get a better sense for the relationship at a larger scale. The end behavior of the function is much more easily extracted from a visual standpoint. In other words, we're able to see that as x increases, f(x) increases as well. It's also easy to identify the minimum of the function from the figure. All these aspects of the function aren't as clear when all we have is an equation.

Students should know that we can identify the zeros by manipulating the equation (when x = 0 or f(x) = 0) or by examining the graph. While it might be easier to use the graph to find the intercepts (or zeros), students should know that this involves a certain amount of error. Don't judge a book by its cover, and don't judge a function by its graph.

Students should also know that a minimum or maximum occurs when the slope of the function is 0. If this is a strange or confusing concept for them, have them draw the tangent lines of a function. Confirm that whenever the tangent lines are perfectly horizontal, the function has a maximum or minimum.

Given a verbal description of a function, students should be able to draw the function. It doesn't have to be more than a stick-figure equivalent to the Mona Lisa, but if you ask for the Mona Lisa and you get a Picasso, you'll know there's a problem. Sometimes these verbal descriptions will be straightforward and other times, they should ask students to apply these functions to real-life scenarios (such as a ball being thrown into the air traveling along a parabola).

When students can switch back and forth between equations, graphs, and verbal descriptions, they'll be able to appreciate functions from lingual, symbolic, and visible perspectives. After that, all that's left is getting them to appreciate murder mystery haikus, but you can let their English teacher take care of that one.