 Another type of geometric transformation we can apply is a reflection. We can reflect a geometric object across a line or a point. We'll focus on reflecting across the x-axis, the y-axis, or the origin. If we reflect across the x-axis, a point whose coordinates x, y becomes a point whose coordinates are x negative y. If we reflect across the y-axis, a point whose coordinates are x, y becomes a point whose coordinates are negative x, y. And if we reflect across the origin, a point whose coordinates are x, y becomes a point whose coordinates are negative x, negative y. So we can find the equation corresponding to the graph of y equals 3x plus 5 reflected across the x-axis, the y-axis, and the origin. So again, let's suppose x, y is a point on the graph of y equals 3x plus 5. If we reflect this graph across the x-axis, then the point's capital X, capital Y on the new graph will be capital X is the same as the original x-coordinate. Capital Y is the negative of the original y-coordinate. Or that also means that lowercase y is negative y. Since we already know an equation lowercase x and y satisfies, we can replace lowercase x with capital X and lowercase y with negative capital Y to get an equation that capital X and y satisfy. And since capital X and y are just coordinates, we'll write them as lowercase x and y. And finally, as a matter of style, since we got the equation in y equals stuff form, we'll give our answer in that form as well and solve for lowercase y. If we reflect the line across the y-axis, then the points capital X and y on the new graph will satisfy. And we'll solve these equations for lowercase x and y. We already know an equation that lowercase x and y satisfies, so we'll replace. Again, capital X and y are just coordinates, so we'll write them as lowercase x and y. And since our equation is already in the form y equals stuff, we can leave it. Finally, if we reflect a line across the origin, the new coordinates will be solving for x and y, substituting, replacing. And again, we'd like to write the answer in the same form the question was given, so we'll solve for lowercase y. These equations suggest a few theorems. Again, the worst way to learn mathematics is to simply memorize theorems. It's more important to understand the concept, so when we reflected a point across the x-axis, we got new coordinates, we solved for the new variables, and we replaced them in the original equation. But if we put all these things together, this suggests the following theorem. Let a graph be reflected across the x-axis. The equation of the new graph can be found by replacing y with negative y. When we found the reflection across the y-axis, we found the new coordinates and solved. We replaced them in the original equation and got our new equation. And if we put all these steps together, this suggests the following. Let a graph be reflected across the y-axis. The equation of the new graph can be found by replacing x with minus x. And again, when we reflected the line across the origin, we found the new coordinates, solved for the original variables, and substituted to get our new equation. And again, if we put all of these things together, this suggests let a graph be reflected across the origin. The equation of the new graph can be found by replacing x with negative x and y with negative y. Again, the theorems are useful, but they're not actually important if you understand the basic concept of a reflection. If we reflect the graph across the x-axis, then our original coordinates x and y become x negative y, giving us, and we'll do a little bit of algebra because we can, which gives us our new equation. If we reflect the graph across the y-axis, then our original coordinates x, y become new coordinates minus x, y, and this gives us a new equation. And again, we'll do some algebra because we can. And this is not the same as our original equation. Our graph will appear different. And if we reflect the graph across the origin, then our original coordinates x, y will become minus x, minus y, giving us the new equation. And why not? Let's do the algebra.