Chapter 2. Height
Let's continue our study of functions by looking at their graphs. To graph a function \( z \depends x \), we use a vertical \( z \)-axis for output and a horizontal \( x \)-axis for input. Our approach to graphing will emphasize the height of the graph at any point. This is a bit of foreshadowing. We'll have chapters on slope and bend that look at other quantities found on a graph.
Let's take a short digression to think about our everyday experience with directions and their impact on graphing. In day to day life, we tend to believe that up and down are absolute directions in a way that left and right are not. This belief is not some deep mathematical truth, but rather, it comes from a simple physical fact. The Earth's gravity always tells us which way is up and which way is down! We'll take advantage of this asymmetry when graphing. As we'll soon see, it is much easier to work with a function's output than its input. By matching output, \( z \), to the preferred direction, up, we bring significant visual intuition to our graphs.