 We can sketch the graph of y equals sine of x by plotting many points. At x equals zero, y is sine of zero, is zero, so zero zero is on the graph. As x increases to pi over two, y goes to one, so the graph rises to pi over two, one. As x increases to pi, y goes back to zero, so the graph falls to pi zero. As x increases to three pi over two, y goes to negative one, so the graph continues to fall until three pi over two, negative one. And as x increases to two pi, y goes to zero, and so the graph rises to two pi zero. And since sine is a periodic function, the graph repeats, and we end up with duplicates of this fundamental period. Now, since this is a graph, all of our graph transformations can still be applied, but we give some of them new names. So what we used to call a horizontal shift is also called a phase shift. And if we had a vertical stretch factor of k, we're going to call that the amplitude. So for example, let's try to sketch the graph of y equals sine of x with a phase shift of pi, and then write the equation. So remember, phase shift is just another word for horizontal translation. So as with horizontal translations, a positive number corresponds to a shift to the right. So the phase shift of pi, this is a horizontal shift of pi to the right. So we'll start with a graph of y equals sine of x, shift by pi units to the right, and our equation with a horizontal shift of pi will become, or maybe we want to have a vertical stretch. So let's have a vertical stretch by a factor of three and find the equation and the amplitude. So the amplitude is the same as the vertical stretch factor three. So we'll start with a graph of sine of x and stretch this vertically by a factor of three, and that vertical stretch has the same effect as it would on any other function. The new equation is y equals three sine of x. So one important transformation is the following. Let's take the graph of y equals sine of x and do a phase shift of minus pi over two. Since the phase shift, I mean, horizontal translation is negative, we're shifting to the left by pi over two units. And so our equation will be. Now notice that our equation gives us the sine of a sum. And if we use the sine of a sum identity, we can simplify. And remember cosine of pi over two and sine of pi over two have particularly nice values. So this simplifies further to cosine x. And this leads to a useful observation. The graph of y equals cosine x is the graph of y equals sine x shifted to the left by pi over two. So what if you want to determine the equation of a graph? It helps to consider a single sine wave. Now some of the key features of that sine wave, one wave, stretches from, well, say, zero to two pi. The low point is at y equals minus one and the high point is at y equals one. The wave rises from zero zero, reaches its top at pi over two one, and falls through pi zero until three pi over two negative one, then rises again until two pi zero. And so if you consider these to be the basic features of the sine wave, we'll try to match the features of the graph to the features of the sine wave. For example, let's say the graph shown is a horizontal translation of y equals sine of x. Let's find the equation and then find the corresponding equation using cosine of x. So remember that on the graph of y equals sine of x, the graph rises through the point zero zero. On the graph shown, the graph rises through the point pi fourth zero and this suggests we shifted the graph of y equals sine of x to the right by pi fourth units. So we'll make that our first transformation. If we shift the graph of y equals sine of x to the right by pi fourths, we get the graph of y equals sine of x minus pi fourths. What if we wanted to describe this as a transformation of the graph of cosine x? We might begin with the graph of y equals cosine x and notice the graph of y equals cosine x falls through the point pi halves zero. But on the graph shown, the graph falls through the point five pi fourth zero and this suggests we shifted the graph of y equals cosine of x to the right by three pi fourths. And so alternatively, we could shift the graph of y equals cosine x to the right by three pi fourths to get the graph of y equals cosine of x minus three pi fourths.