Let's return to the graph of y and get into an issue that will be very important throughout our study of calculus. Again, execute:
Plot[y, {x,-4,4}, PlotRange -> All]
A recuring theme in calculus can be paraphrased: "most functions are `almost linear' if you look closely enough". To illustrate this point, you can "zoom" in on the graph at x near 2:
1. Modify the plot command below so that the domain of x is 1.9 to 2.1.
(If necessary, you may scroll back to review the directions "Modifying
a command.")
2. Execute the plot command.
You should see a smaller section of the graph and note that it is starting to straighten out a bit. To further illustrate the point, modify the x domain in the plot command below to 1.99 to 2.01 and execute the command, and notice that the graph is almost linear.
Plot[y, {x, 1.99, 2.01}, PlotRange -> All]
Is there anything special about the point (2, -5)? No. You may want to experiment on your own by modifying the command with any value for the domain of x that you choose, and "zoom in" as we have done. Remember the fundamental point being made:
"most functions are `almost linear' if you look closely enough".
We will revisit this notion often in Calculus.