 So, in a kind and gentle universe, you'll only ever have to do one transformation. But you don't live in that universe. Very often you have to do several transformations, one after the other. And we can perform several transformations, one after the other, but it's important to perform the transformations in the specified order, since we can end with very different results if we change the order. So, for example, if I shift something three units to the right, then reflect across the y-axis, I may end up with something very different than if I reflect across the y-axis first, then shift three units to the right. So, let's see what happens. Suppose the graph of y equals 3x plus 5 is shifted two units to the right, then reflected across the y-axis. Find the equation of the new graph. Also, let's find the equation of the graph produced by performing the transformations in the opposite order. So, again, we have our formulas for the transformations, but we don't really need them. If we shift the graph two units to the right, our original points x, y will go to the new points capital X, capital Y, where our new x and y coordinates are, which we can solve for the original x and y coordinates. We'll replace these in our original equation, do a little algebra, and if it's not written down, it didn't happen. The equation of the graph produced by shifting two units to the right will be y equals 3x minus 1. Well, that's one transformation. We still need to reflect the graph across the y-axis. So reflecting the graph across the y-axis takes our original coordinates x, y, and sends them to a new set of points, capital X, y, where solving for our original lowercase x and y, and we'll replace and simplify to get our new equation. And so, when we do that second transformation, reflecting this across the y-axis gives a graph with the equation y equals negative 3x minus 1. This happens if we do the transformations in the opposite order. So first we'll reflect the graph across the y-axis, so our new x and y coordinates will be, we'll solve for the original x and y coordinates, and replace, do a little algebra. So if we reflect the graph across the y-axis, the equation of the new graph will be y equals negative 3x plus 5. The other transformation requires us to shift two units to the right, so our new x and y coordinates will be, and again we'll solve for the original x and y coordinates, and replace and simplify, and summarize if we shift the graph to units to the right, the equation will be y equals negative 3x plus 11. And so we can compare if we reflect first, then translate, we get one equation. If we translate first, then reflect, we get a different equation, and the order of our transformations is important because we'll end up with very different graphs.