Hello thisisamazingFirst, try thinking about the value of that makes the expression in the brackets zero. This will tell you how far the 'basic' graph has been moved. So for example, inthe 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 .
Grandad