Hello thisisamazingthe expression is zero when . So, instead of 'starting' at the sine graph has been moved to 'start' at ; i.e. it has been moved a distance of to the left.
(You'll see that I've put the word 'start' in quotes. That's because, of course, the graph doesn't actually start there; it extends infinitely far in each direction. But the cycle of the graph that we usually look at will start there.)
In the second example you give, :whenSo the 'basic' cosine graph has been moved to the left to 'start' at .
However, a couple of other things have happened here as well:
- is multiplied by , which means that things happen times as fast along the -axis as they will on the 'basic' graph. So instead of requiring values of from to to make a complete cycle, you'll only need values from to .
- The cosine expression has then been multiplied by . This, in turn, does two things:
Does that help to make it clear?
- The minus sign flips the graph over, reflecting it in the -axis.
- The factor of reduces the -values to one-half of their original values, 'squashing' the graph down, so that, instead of varying between and , it will now vary between -0.5 and .