 So to find the transformations needed to produce a particular trigonometric graph it helps to think about fitting one wave into a box. One wave of y equals sine of x fits into a box that has length 2 pi, has height 2, and has base y equals negative 1 and top y equals 1. And the box starts at x equals 0. And however you transform the box will be how you transform the equation. Now it's helpful to remember it's easier to do stretches before translations. Now since sine and cosine are just horizontal translations of each other it doesn't really matter which one we start. So let's use sine waves. So first we'll identify what one sine wave looks like and that might be this. And now let's try to fit one sine wave of our graph into a box. And so that might look something like this. And so now let's compare our two boxes. So our sine wave box starts at x equals 0 and ends at x equals 2 pi for a length of 2 pi minus 0 or 2 pi. Meanwhile the box for our graph starts at x equals pi thirds and ends at x equals 7 pi thirds for a length of 7 pi thirds minus pi thirds or 2 pi. And so the boxes have the same length. Our sine wave box has bottom at y equals negative 1 and top at y equals 1 for a height of 1 minus negative 1 or 2. The box for our graph has bottom at y equals 0 and top at y equals 4 for a height of 4 minus 0 or 4. And so our box needs to be stretched vertically by a factor of 2. And so that means the first thing we need to do is stretch our graph of y equals sine of x vertically by a factor of 2. Our sine wave box begins at x equals 0 while the box for our graph begins at x equals pi thirds. And so the box needs to be shifted horizontally to the right by pi thirds. And we have to adjust our equation. And so now we have the graph of y equals 2 sine of x minus pi thirds. Now notice the horizontal stretch has made our sine wave box have bottom at y equals negative 2 while the box for our graph has bottom at y equals 0. And so the box needs to be shifted vertically upward two units. And what happens to the box also happens to our graph. So we need to shift the graph vertically by two units upward to get the graph of y equals 2 sine of x minus pi thirds plus 2. Or how about packaging this one up? So we'll find a single sine wave and a single sine wave of our graph. We see the box for our graph begins at x equals negative pi halves and ends at x equals pi halves for a width of pi. And so we need to compress our box by a factor of 2. And so compressing the graph of y equals sine of x horizontally by a factor of 2 gives us the graph of y equals sine of 2x. The box for our graph has a bottom at y equals negative 2 and a top at y equals 0 for a height of 2. So we don't need to adjust the height. We see the box for our graph begins at x equals negative pi halves. So we need to shift our box left by pi halves. Shifting the graph left by pi halves gives us the graph of, and we see the box for our graph has a bottom at y equals negative 2. So we need to shift our box down by one unit. And so shifting by one unit gives us the graph.