# Trigonometric Graphs and the Box Method

Graphing trigonometric functions is a crucial skill, but there is a lot of reasoning involved in the process. I learned the method that follows from a colleague in my first years of teaching in the Bronx, and used it later on when I actually taught Algebra 2 and PreCalculus.

Here's an example for f(x) = 2 cos(2x) + 3:

Students first (lightly) draw a box that is one period wide, twice the amplitude tall, and centered vertically on the midline of the sinusoid. The process of doing this isolates the reasoning about transformations of the function from the actual drawing of the graph, which also takes some skill. I usually ask students to state the period, amplitude, and average value anyway, but this method implicitly requires them to find these quantities anyway. We use the language of the amplitude, period, and average value to describe this box and the transformations of the parent function.

Once the box is in the right place, then we can focus on the graphing details. Is it a cosine or sine? How does the graph of each fit into the box we have drawn? Where does the curve cross the midline? This conversation is separate from the location and dimension conversations, and this is a good thing. The shape of the curve merits a separate line of reasoning, and encouraging that separation through this method reduces the cognitive demand along the way. I have also seen that keeping this shape conversation separate has reduced the quantity of the pointy sawtooth graphs that students inevitably produce.

I have considered doing this for other parent functions, but haven't been convinced of the potential payoff yet compared with the perfect fit thus far of the trigonometric family of functions. Thoughts?