 Another type of transformation can be viewed as stretching and squashing a geometric object. Translations and rotations are called isometries. Lengths are unchanged by a translation or a rotation. Other types of transformations alter lengths. A scaling expands or contracts the lengths in one or more directions. For example, if we expand in all directions by the same amount, we have a dilatation, usually referred to as a dilation. Otherwise, we just have a scaling, a stretching along the direction. Again, a key goal is to try and describe these geometric transformations using the language of algebra. So, suppose we stretch a figure in the horizontal direction by a factor of k. What this means is that any point on the graph with coordinates x, y will be moved to a point with coordinates k, x, y. So, we could write this down as x, y goes to k, x, y. Likewise, if we stretch a figure in the vertical direction by a factor of k, a point with coordinates x, y gets moved to a point with coordinates x, k, y. So, for example, let's find the equation for the graph produced by stretching the graph of y equals x plus 3 by a factor of 2 in the horizontal direction. So again, if x, y is a point on the graph, then a new point will be capital X, capital Y, where our new x-coordinate is twice what the old x-coordinate was, and our y-coordinate remains the same. So with all of the transformations, we'll solve in terms of the original variables x and y. We know an equation that the original x and y satisfies, so we can replace lowercase x with capital X over 2 and lowercase y with y to get an equation that capital X and y satisfies. And since capital X and y are just coordinates, we can write them using their lowercase forms, x and y, and getting our new equation. And again, the important idea here is this notion of rewriting our new coordinates, but we can translate this into a theorem if we want to. Let a graph be stretched horizontally by a factor of k. The equation for the new graph can be found by replacing x with x over k. What if we stretch our graph vertically by a factor of 5? So again, if lowercase x and y is a point on the graph, then a new point will be capital X and y, where our new x is the same as the old, and our new y is 5 times the old y. Solving for our original variables, and again, since we know an equation lowercase x and y satisfies, we can replace lowercase x with capital X and lowercase y with capital Y over 5 to get an equation x, y satisfies. And once again, capital X and y are just coordinates, so we can write them as lowercase x and y to get our equation. And again, the important idea here is that our new x and y coordinates will satisfy an equation. But again, we can reduce this to a theorem. Let a graph be stretched vertically by a factor of k. The equation for the new graph can be found by replacing y with y over k. So for example, let the graph of y squared equals x cubed plus x plus 8 be stretched horizontally by a factor of one-half. Let's write the equation of the new graph. So that's a horizontal stretch by a factor of one-half. We'll replace x with x over one-half, also known as 2x. And we'll do some algebra because we can. If I want to do a vertical stretch by a factor of 3, so now I'm going to replace y with y over 3. And while we could leave this as an equation for the new graph, remember it's good form to write the answer in the same dialect the question was asked in. We're originally given the graph equation in the form y equals stuff. So we should write our answer in the form y equals stuff. So we can multiply both sides by 3 and do a little algebra.