Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College Transformations. Y equal F of X + K is a vertical shift. If K is greater than 0, we're going to go up. If K is less than 0, we're going to go down. So we're taking your original function and adding some constant to it. The next one, Y equal F of the quantity X + H. That's a horizontal shift. If H is greater than 0, it's going to go left. If H is less than 0, it's going to go right. The way I have students think of it sometimes is to think about this inside parenthesis and setting it equal to 0. So if X + H equaled 0, then X would equal negative H, and that would tell us how we're shifting left or right. Y equal the opposite of F of X. That's a reflection about the X axis. So whatever the original Y value is, we're now taking the opposite value. Hence the reflection Y equal F of negative X. That's going to be a reflection about the Y axis. If you have Y equaling a * F of X, that's a vertical stretcher shrink. Where a is greater than 1 is a stretch, 0 less than a less than 1 is a shrink. Y equal F of BX is a horizontal stretch or shrink. If zero is less than B is less than one, it's a stretch. If B is greater than one, it's a shrink. One method of graphing these the order to do would be reflections. First stretches and shrinks and then shifts. Now we can do it in lots of different order, but that's usually the easiest to do. Reflections, stretches and shrinks and then shifts. Asymptotes. If as X approaches Infinity, or as X approaches negative Infinity, and the function of X approaches some L that's a constant, then the line Y equal L is a horizontal asymptote. So when we're getting way out to Infinity or negative Infinity, we get closer and closer and closer to some constant value. That constant value would be called a horizontal asymptote. A graph may cross a horizontal asymptote as the definition only looks at what happens as X2X as X approaches Infinity way out here or negative Infinity way back here. A line X = C is a vertical asymptote. If as X approaches C then the absolute value of the function of X is going to Infinity. A graph never intersects a vertical asymptote. So what this is saying is we have some value C here. As X is getting closer and closer and closer to the C, then our absolute value of F of X is going out to Infinity. It has to be true from both sides. So as X is getting closer and closer and closer to C, our absolute value of the function is going out to Infinity. So this piece here is going to Infinity. This piece here is going to Infinity. That makes it a vertical asymptote. We can never intersect a vertical asymptote. Thank you and have a wonderful day.