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Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations.

$ y = 1 + \sin \pi x $

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Our goal is to figure out what the graph of this function looks like, but we're going to start with the graph of one of the standard functions. So let's start with the graph of y equals sine X. Okay, so goes up to a height of one down to a height of negative one. It has a period of two pi. We can draw a couple of the cycles here, and it looks like this. That's the basic Siggraph. Now what do we want to do to it? Well, the pie in here that represents a horizontal shrink. So it's going to be pi times narrower than it used to be, and the old period was two pi. If we divide that by pi, the new period is going to be too. So it's going to be roughly three times narrower, So we'll graph y equals sign of Pi X. I'm not going to make it using the exact same scale as I used on the last one going to change my scale because now that the period is too, I'm just going to put a two out here and a negative to out there, okay, so still goes up to a height of one and down to a height of negative one. But it's narrow. Where now? Okay, finally, we want to add one to it and adding one to any graph will shift it up one. So we want to take what we just drew and shifted up one. So it was going up to a height of one and down to a height of negative one. Now it's going up to a height of two and down to a height of zero. So we have or Siggraph looking like this shifted up one.