 Many real-world phenomena display periodic behavior. The rising and setting of the sun, the daytime temperatures over a year, the water level in a wave. Since trigonometric functions like sine and cosine also exhibit periodic behavior, we can use them to model these phenomena. So we want to find A, B, C, and D for a trigonometric function of the form. Now, remember that we rename certain graph transformations K horizontal shift is a phase shift and a vertical stretch by a factor of K is the amplitude. There are formulas for converting the phase shift and amplitude into the coefficients of our trigonometric function. But remember, the fast way to failure is memorizing formulas. If you only memorize the formula, you will almost always answer these questions incorrectly. Remember, understand concepts. And so we present a better approach, box and transform. For example, suppose we want to model the level of water in a lake using a sine function, where we know the lowest and highest water levels. Now, since we want to obtain a function of T, we'll let T be the horizontal coordinate and so D will be the vertical coordinate. We also know the lowest point and the highest point. So the secret to graphing is graph first, then label. So we know where the highest and lowest points have to be, and so we'll mark those points. And then our water level has to go from the lowest to the highest, so maybe our graph looks like this. Now, let's compare this bit to the graph of y equals sine of x. The given information corresponds to part of this graph, and if we look closely, we see that this looks most like the part from here, x equals negative pi halves to here, x equals pi halves, and from y equals negative 1 to 1. So let's box our two graphs, the original graph and the graph of the water level. And so the important question is, what transformations can we apply to turn the graph of sine x into the graph of the water level? While we can do our transformations in any order, some orders are easier than others, and we'll always use the following sequence. First, we'll stretch vertically, then shift vertically, then stretch horizontally, and shift horizontally. So let's try and figure out what transformations we need. So note that our original box has a bottom of y equals negative 1 and a top at y equals 1 for a height of 2. The given data, meanwhile, fits into a box with a bottom at y equals 30 and a top at y equals 75 for a height of 45. So we need to stretch the graph of y equals sine x vertically by a factor of 45 halves, or 22.5, and this gives us the graph of y equals 22.5 sine x. Now having stretched vertically by a factor of 45 halves our bottom, which used to be at y equals negative 1, is now going to be at negative 1 times 45 halves, negative 22.5. And we want the bottom to be at d equals 30. And so we need to shift it vertically by 22.5 plus 30, 52.5. So this gives us the graph of y equals 22.5 sine x plus 52.5. Now remember our sine box begins at x equals negative pi halves and ends at pi halves for a length of pi. Meanwhile the given data fits into a box starting at t equals 35 and ending at t equals 218 for a length of 183. So we want a horizontal stretch by a factor of 183rd over pi and this will give us the graph of 22.5 sine pi 183rd's t plus 52.5. And stretching horizontally by a factor of 183 over pi puts the left side at negative pi halves times 183 over pi or negative 183 halves. We want it to be at t equals 35 so we need to shift it to the right by 183 halves and 35 more or 126.5. And this gives us the graph of where we switch to the variable we want to use, t as our independent variable and this is the function that models the water level.