This section is intended to make it easy to graph functions. I'm not sure it actually helps students right away wit this, but it is important to realize that in mathematics we like to organize things that are related. In this case, functions which more or less have the same graph.

Look ar the figures 1.19 and 1.20 in the book. They show what I mean by related. The two graphs have the same shape, they are just placed in different positions on the graph. These two graphs are related and so their functions should be too.

But how?

Well, the first figure has a shift up. Basically we add to each y value, and notationally we do the same f(x)+2 is the graph compared to f(x).

The second we shift to the right. Now we do something to the x value first, then square. It is tricky but to shift right, we subtract from the x. So the graph is f(x-2).

These are vertical and horizontal shifts. Try to pay special attention to these and make sure you understand notationally and conceptually why these things are different:

f(x), f(x+c) f(x) + c.

Another one is to flip the graph upside down. Mathematically, instead of plotting y we plot -y and notationally the function we graph is -f(x). Check out example 3 for details.

There are two more to consider, but we won't in this class. You might wish to see if you can figure out what they do using your calculator. Here they are f(cx) and cf(x).