Section 7.3 Graphs of Functions
Subsection 7.3.1 Functions and Coordinates
This section relies a bit on the concepts review in Chapter 11, where we discuss the idea of a function.
If we have a function \(y = f(x)\) where \(x\) is the independent variables and \(y\) is the dependent variables, we can visualize the function using coordinates. The choice of \(x\) and \(y\) here intentionally matches with the choices of the axis of the plane. Our convention is to put the independent variable on the horizontal axis and the dependent variable on the vertical axis. Let's use an example function to show how we display the function with coordinates. I'll choose the square root function.
Given any \(x\) value, say \(x = 9\text{,}\) the function outputs a \(y\) value which is the square root of the input, in this case \(y = \sqrt{9} = 3\text{.}\) That gives us a points in coorinates: \((9,3)\text{.}\) All values of the function are pairs like this: an input and an ouput. All of these pairs can be thought of as coordinates. These coordinate produce a shape in the plane, which we call the graph of the function. Figure 7.3.2 shows the graph of the square root function with several points labeled.
In this way, we can draw a picture of any function using coordinates. These pictures give us a very useful way to see what the function is doing. A lot of the qualitative analyais of functions is done throught their graphs. For the square root function, we can talk about how the function starts at the origin, increases steeply at first, but over time bends down and becomes a nearly horizontal.
Computers (including calculations) are very good at graphing function. When I encounted a new function that I would like to understand, I'll usually ask a computer for a graph of the function, to give me an idea of what I'm working with. I encourage you to make use of this advantage of technology.