5. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
There's an old story about three blind men who encounter an elephant for the first time and try to describe it. The first finds himself in contact with the elephant's tail, and says with certainty, "An elephant is just like a rope!" The second comes into contact with the elephant's side, and says with equal certainty, "An elephant is just like a wall!" The third comes into contact with the elephant's trunk, and claims, "An elephant is just like a snake!"
Of course, the story ends there. We never find out what happens when the elephant gets tired of being poked and prodded by these three guys. We're guessing it wasn't pretty. Anyway, the moral of the story has to do with understanding your limited perspective. What each of us sees (or feels, as the case may be) isn't the whole picture.
As useful as that may be, it doesn't really help us out with math. So we're going to offer up a slightly different (but not inaccurate) moral to the story. Sometimes—and, in fact, most of the time—one or two characteristics don't do a great job of describing anything well.
Graphs are the same way. There are many ways students can describe the same graph. Using their knowledge of functions, equations, and graphs and they should be able to describe a graph qualitatively and be able to draw a graph if given a qualitative description. When they come across a graph, they should consider a few specific things.
- Is the graph a function? (In other words, does it pass the vertical line test?)
- Does the graph increase or decrease? Both? Neither? For which intervals? (An increasing graph goes up from left to right, while a decreasing graph goes down.)
- Is the graph continuous? (Can they trace the entire graph without lifting their pencil from the paper?) Is it linear, quadratic, absolute value, or something else?
- Does it have any turning points? (Those are exactly what they sound like: points where the graph changes direction—like the bottom point of a U-shaped parabola.)
These are just some of the many characteristics that can be used to describe what a graph looks like. And, like those blind guys and the elephant, each gives just a hint as to the entire description of the graph.