Section 12.5 Going Deeper: Graphing Functions
Most of the time in math courses, we use the variable for the horizontal axis and for the vertical axis. This is the "standard" choice for coordinate systems when there is no specific context being applied. But a lot of mathematical ideas are applied in very specific contexts, and so we need to be able to translate our ideas from the generic situation to specific ones.
Traditionally, the \(x\) variable represents the input (or independent) variable, and \(y\) represents the output (or dependent) variable. This language is tied to how mathematicians talk about functions. We will start with the formal definition:
Definition 12.5.1. Function.
A function is a rule that assigns each object of one set to exactly one object a second set.
Conceptually, functions are often thought of as machines. On one side of the machine there's a place for you to put something in, and on the other side there's a place where something comes out.
Specifically what comes out will depend on what the machine is designed to do (the "rule" that it has been created to follow). For example, we can have an "Add 3" machine that takes whatever number we give it and add 3 to it.
If we picked a different input, we would get a different output.
In fact, if we were to give it a generic quantity, it could give us an expression that represents the appropriate output.
The mathematical shorthand for this concept is \(y = f(x)\text{.}\) This is a versatile notation that we use both to describe the function in general as well as to describe specific input-output pairs of a specific function. And while we often use the letter to represent a function, we can use other letters or symbols if we wanted. The key observation about this notation is that it defines as the independent variable and as the dependent variable. For the "Add 3" example, we would write \(y = f(x) = x + 3\text{.}\)
Notice that this is a process that gives us ordered pairs \((x,y)\text{.}\) If we were to put these values into a chart like the ones we were working with earlier in this section, we could convert this information into a graph.
There are situations where the data that you have in your table comes from measurements. For example, if you're measuring the temperature throughout the day, you will get a chart of values, but the values probably won't fit a nice formula. Even though there may not be a formula, there's still a meaningful sense that this is a function. The input variable is the time and the output variable is the measurement. And using the same ideas as above, we can put that information into a chart and get a graph of the temperature throughout the day.
Notice that we had to make some modifications to the graph. We changed the letters to better reflect what we're measuring (\(t\) for time and \(F\) for degrees Fahrenheit), and we also labeled the axes so that the values could be more easily interpreted. We also altered the time axis to correspond to clock times instead of just having numbers.
But notice how natural this is to read. Given a time, we can get an estimate of the temperature by looking at the graph. This is where a lot of the value of graphing functions starts to come into play. There is a clear relationship between the input value (time) and the output value (temperature). Even if we picked a time that isn't one of the data points, we can still come up with an estimate for the temperature.
You might have noticed that we connected the data points together with straight lines, and that makes the graph look a little artificial. That's almost certainly not how the temperature behaved! It's far more likely to have a smooth shape. This is also why we said that we could estimate the temperature instead of saying that we could determine the temperature. There's a little bit of guesswork involved in thinking through the shape of the curve in between the data points.