 So, how many mathematicians does it take to change a light bulb? One, they change it into an elephant and thereby reduce it to the previous joke. Now, I could take away two things from that joke. One, I should probably not give up my day job and go into a career as a stand-up comedian. But the other thing that's important to take away from that is it reflects a very common thing to do in mathematics, which is to take a problem and say, hey, this is really like another problem we've already solved. And one way we do that is through what's known as transformations. And this is really a geometric idea. Given a geometric object, we can apply one or more transformations to it. We can translate it by sliding every part of it the same distance in the same direction. We can rotate it by turning it around a fixed center of rotation, or we can reflect it across a line of reflection. There are other transformations we can do, but these are the three principle ones. So how does this relate to things we've done already? When we graph an equation, we find all points x and y that satisfy the equation. So what equation describes the transformation of a graph? So let's focus on vertical and horizontal translations first. It's easiest if we focus on one direction at a time. If we shift a graph vertically by k units, then whatever our original x and y coordinates were, our x coordinate will remain the same, because we haven't moved left or right. Our y coordinate, on the other hand, will change by k, so our y coordinate will now be y plus k. And we can express that as follows. Our original coordinates x, y will be transformed into the new coordinate x, y plus k. Similarly, if we shift a graph horizontally by h units, our y coordinate will remain the same because we haven't moved vertically. On the other hand, our x coordinate will change by h, because we have moved horizontally a distance of h units. And so we can express that as follows. Our original coordinates x, y will be transformed into the new coordinates x plus h, y. So how does this affect our equation? Well, suppose our coordinates x, y satisfy some equation. If we shift the graph horizontally by h units, then the points of the new graph are capital X, capital Y, where capital X is x plus h, and capital Y is y. Now, we know that lowercase x and y satisfy the original equation. So let's rewrite these to find lowercase x and y. Lowercase x is x minus h, lowercase y is still y. And here's the important thing. Since we already know an equation satisfied by lowercase x and y, equals means replaceable. Which means we could replace lowercase x and y with these new expressions, and we can obtain an equation that's satisfied by capital X and Y. Let's take a look at that. Let's try to write the equation corresponding to the graph of y equals 3x plus 5 that's been shifted to the right by 5 units. So again, if x, y are the coordinates of a point on the graph, we know that y equals 3x plus 5. Now, if we shift this point horizontally to the right by 5 units, then the new coordinates capital X and Y will be, where we've increased our original x coordinate by 5, but because this is a horizontal translation, we haven't changed our y coordinate. Now, we do have an equation that is in terms of lowercase x and y. So let's solve for lowercase x and y. And equals means replaceable. So we'll replace lowercase y with capital Y and lowercase x with capital X minus 5 to obtain, which is a new equation. And we'll take one last step, which is mostly cosmetic. Since capital X and Y are just coordinates of a point, there's no reason to retain the capitalization. So we can just write the equation using our lowercase letters. And so we get our equation. Now, this suggests the following. Remember, one of the worst ways of learning mathematics is to simply memorize theorems without understanding the underlying concepts. Here, the underlying concept is that this horizontal translation corresponds to an alteration of our x and y coordinate, followed by a replacement in the original equation. And if you understand this idea, you don't need the theorem. Converrywise, if you don't understand this idea, the theorem is a power tool that you're using without safety equipment. It's sort of like using a chainsaw while you're wearing a swimsuit. You can do it, but I wouldn't. But anyway, here's the theorem. Let a graph be shifted by h units horizontally, where h greater than 0 corresponds to a rightward shift and h less than 0 corresponds to a leftward shift. The equation of the new graph can be found by replacing x with x minus h. What if we have a vertical shift? So let's find the equation when the circle is shifted down by four units. So if x and y is shifted downward for units to capital X, capital Y, the new coordinates will be. And we can solve these coordinates for lowercase x and y. Now, we know an equation that's satisfied by lowercase x and y is this equation of a circle. Equals means replaceable. So we can replace lowercase x and y with an expression in capital X and capital Y. And we get a new equation. And again, capital X, capital Y are just coordinates. So we can write them as lowercase x and y to get a new equation for this object that has been shifted for units downward. Again, the important concepts here are that if we have a vertical shift, that's going to do something to our coordinates. And we can then replace them in our original equation. But if you want to grab that chainsaw while wearing your swimsuit, there's the theorem. Let the graph be shifted k units vertically, where k greater than 0 corresponds to an upward shift and k less than 0 corresponds to a downward shift. The equation of the new graph can be found by replacing y with y minus k. So let's take a look at combining our two transformations. So we have our two theorems. But you know what? I don't think I want to use them. Let's build this up from the basic concept of what a transformation means. So if we shift our graph three units to the left, then any point x, y will be moved to capital X, capital Y, where capital X will be x minus 3, because we've gone to the left. And because this is just a horizontal shift, our y-coordinate doesn't change. So I can solve these equations for lowercase x and y. I know that my original point x, y satisfies the equation. So I can replace lowercase y and lowercase x to give us a new equation. And again, capital X and y are just coordinates, so I can write them as lowercase x and y. Now the important thing to recognize here is that we've only shifted the graph three units to the left. We still have the other transformation to do. So let's take care of that. So if we shift the graph vertically upward by four units, then lowercase x and y will be moved to capital X and y, where our x-coordinate remains the same because we haven't moved left or right, and our y-coordinate will be the original y-coordinate plus 4. And that means we can solve for lowercase y. And again, we know that lowercase x and y satisfy one equation, so we'll replace giving us a new equation. And again, capital X and y are just coordinates, so we'll write down our final equation, which will be the equation of the graph that corresponds to this one, shifted three units to the left and four units upward.