Uploaded April 2010 | Updated September 2026, 3 weeks ago
Given a sketch of y = f(x), does y = f(x-3) then correspond to a shift to the left or to the right of that graph?
I never remember, and I never will as I simply refuse to think in those terms! Here's how to deal with such basic graph transformations with ease.
Given a sketch of y = f(x), does y = f(x-3) then correspond to a shift to the left or to the right of that graph?
I never remember, and I never will as I simply refuse to think in those terms! Here's how to deal with such basic graph transformations with ease.








![Fractions take up No Space: Rationals Have Measure Zero (TANTON Mathematics)
How much space do the fractions take up on the number line? NONE!
In this video we add up pieces of ribbon to prove that the fractions have measure zero. It is all very easy and very perturbing because it is easy!
[This video does assume familiarity with the geometric series formula - so you might want to watch a video on that too. I do that with paper tearing!] Fractions take up No Space: Rationals Have Measure Zero (TANTON Mathematics)](https://i.ytimg.com/vi/j2QAhOCdm_o/mqdefault.jpg)

