When stretching or compressing horizontally why do we divide x by b when the format is f(bx)

Answer 1

Nathan, you are asking a great question. First, this is a subject that math teachers often disagree about. For example, some say f(x) = 4x² means that we have "stretched" the parabola f(x) = x² vertically but the exact same parabola can be written as f(x) = (2x)² which some teachers say is a "horizontal compression." You can see the confusion here--how can something be a "vertical stretch" and a "horizontal compression?"

For something like f(x) = sin(x) it is much more clear: f(x) = sin(2x) repeats twice as quickly as f(x) = sin(x), so we can agree with the claim that there is "horizontal compression"--I encourage you to graph these together and see for yourself.

If you want to see an even more confusing example, take a look at the graphs of f(x) = ln(x) and f(x) = ln(7x)-- here we have a vertical shift, not a horizontal compression.

Hope this helps, but please let me know--Andrew