Functions and Their Graphs

Creating the graph of a function given by an equation is similar to creating one from a table of values - we need to plot enough points $(x,y)$ where $y = f(x)$ so we can be confident of the shape and location of the graph of the entire function. We can find a point $(x,y)$ which satisfies $y = f(x$) by picking a value for $x$ and then calculating the value for $y$ by evaluating $f(x)$. Then we can enter the $(x,y)$ value in a table or simply plot the point $(x,y)$.

If you recognize the form of the equation and know something about the shape of graphs of that form, you may not have to plot many points. If you do not recognize the form of the equation then you will have to plot more points, maybe 10 or 20 or 234: it depends on how complicated the graph appears and on how important it is to you (or your boss) to have an accurate graph. Evaluating $y = f(x)$ at a lot of different values for $x$ and then plotting the points $(x,y)$ is usually not very difficult, but it can be very time–consuming. Fortunately, there are now calculators and personal computers which will do the evaluations and plotting for you.