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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.$y = 2 \cos 3x$

## $y=2 \cos 3 x:$ Start with the graph of $y=\cos x,$ compress horizontally by a factor of $3,$ and then stretch vertically by a factor of 2.

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our goal list. A graph y equals to co sign of three x So let's start by thinking about why equals co sign of X to go from y equals co sign of X to y equals to co sign of three X. There's going to be a vertical stretch by a factor of two, so two times is tall and there's going to be a horizontal shrink by a factor of three, so three times narrower. Okay, let's take this step by step. So let's first start by thinking about how why equals co sign of X looks. It goes upto one down to negative one. It has a period of two pi. So the graph of Y equals co sign of X looks like this. Okay, now let's take it one transformation at a time. Now suppose we want to look at why equals to co sign of X. So we're taking into account the vertical stretch. So now, instead of going up to a height of one and down to a height of negative one, it goes up to a height of two and down to a height of negative, too. Still has a period of two pi. So that's going to look like this same thing on the other side. Okay, Now let's take into account the horizontal shrink, so that's going to make it narrower and change the period. Gonna squeeze that one and down here. So instead of having a width of two pi for one complete cycle, we have a width of 1/3 of that. So two pi over three gives us one complete cycle, and I'm just gonna put the two pi over three right here. You would have three complete cycles in a width of two pi instead of one complete cycle in a width of two pi still going up to two and down to negative two. Okay, so one complete cycle in a width of two pi over three is going to look like that. And if you want to show that you have three times as many in a width of two pi, you can just do a whole bunch of those and you could do that on the other side as well

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Johns Hopkins University

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Oregon State University

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University of Michigan - Ann Arbor

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University of Nottingham

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