 In this video, we're going to graph the function g of x equals 3 over x minus 2 plus 1 and you can see the picture of the graph illustrated here to write in blue. To get started with this type of situation, if you want to graph the function, you have to know the basic graph. What is the graph without any transformations to it whatsoever? And so we kind of get rid of stuff like let's give it this plus 1, this minus 2. You might look like something like y equals 3 over x and admittedly your basic graph could start anywhere if you know how to graph that thing. But most likely you're not going to have this three right here either. And so your basic graph when we're done with the day would look more like y equals 1 over x. This is the basic graph I would suggest starting off with because it's going to be one of the graphs we kind of memorize. If we switch over to this graph over here, we can use this to help us out here. When it comes to decimals, you can interactions just by hitting the slash button. I actually recommend you hit the slash button before you get started so that it gives you a clear numerator and denominator and we're going to hit 1 over x. You can use the arrow keys to navigate between them. And we see these pictures right here. We don't need these points illustrated. I'm going to take them off and their labels. So we see this graph right here. This is a rational function that actually does have a vertical and horizontal asymptote, which we can see illustrated in the graph right here. We'll talk more about those in the future. We then want to identify which transformations have we done here. Remember, as we go through this, we first can think of reflections. So reflections, we're looking for negative signs. Negative signs in either in front of the x, nope, x is good, or in front of the function. Well, since the function is a fraction, a negative sign you'd expect to see in front of the fraction bar, but it could also be hidden inside of the numerator itself, right? If I write negative 1, if I take negative 1x, 1 over x, that's the same thing as negative 1 over x, right? So a negative sign could be hidden, but in this situation, we don't see any negative signs here or here. So we're clear. There's no, there's no reflections in this example here. The next thing to look for are going to be stretches and compressions, right? What type of stretches do we see here? Well, so to find a stretch, we're looking for coefficients in front of the x-coordinate because our horizontal zone for this rational function is the denominator. And I don't see any coefficients in front of the x here. So that means there's no horizontal stretching, but if I'm looking for a vertical stretch or compression, I'm looking for some a in front of 1 over x, which again, as this is a fraction, this could be written as a over x. So rational functions are kind of nice because the denominator is a horizontal zone, but the numerator is our vertical zone. And so looking here, we see there is a 3 in the numerator. And so this 3 tells us we're going to have a vertical stretch, a vertical stretch by a factor of 3. This is going to get vertically stretch our graph here. And now the last thing to do is to consider shifts and moves of any kind in the horizontal zone. Like we said is the denominator, we're going to see a negative 2 right here. So this x-2 in the denominator represents you're going to get a horizontal shift to the right by a factor of 2. And then the plus 1, notice the plus 1 is outside of the fraction. The fraction was the basic function. The plus 1 here tells us that we're going to have some type of vertical shift. And this is going to be a shift up by a factor of 1. And so when we look at these together, there's three transformations in play here. We're going to vertically stretch the graph by a factor of 3. We're going to shift it to the right by 2, and we're going to shift it up by 1. And so we're going to shift over back to Desmos here. So we're going to stretch it by a factor of 3. So our A chord is going to go up by 3. I'm going to turn those points back on. These points right here, the labels, well, we don't need the labels. So these, look at that, so you can look at orange and pink point as a reference here. So as we stretch this to 3, this is going to get bigger, bigger, bigger, bigger. So this point, the pink point is now above where it used to be by a factor of 3. Next, we want to shift it to the right by 2. So we just increase the K from 0 to 2. You see that? And then lastly, we wanted to shift it up by 1. And that's our K value, so we shift that up by 1. And so we see the following picture right here, and I should turn off that orange dot. It's actually part of the graph. So we've now changed the picture. So we shifted into the appropriate stretch and come back here. That's exactly the picture we see in the graph right here. In this situation, I also did include those vertical asymptotes. This function, the basic function y equals 1 over x, it avoids the x-axis and the y-axis. What happens to the x-axis as we do this situation? If you start off with the x-axis, if you stretch the graph vertically, that means you're going to multiply the y-coordinate by 3. Well, if your y-coordinate is 0, a.k.a. the x-axis, stretching it doesn't do anything. If you were to horizontally shift a line on the horizontal line to the right by 2, it doesn't change anything. The x-axis was left unaffected by the horizontal shift, but you do shift it up and you're going to go to this new asymptote, which is y equals 1. That's what you see here illustrated here in green. Y equals 1. And then also, we do the same thing to the vertical asymptote, which is the y-axis. The y-axis, when you stretch everything by factor 3, that doesn't change the vertical line. When you shift everything over by 2, that does move it over here. So we get the vertical asymptote x equals 2, and then shifting everything up by 1 doesn't change the horizontal line as well. So when it comes to transforming the graph, if it has some type of asymptotic behavior like these vertical and horizontal asymptotes, we want to move those things around as well as we do these transformations. And don't worry too much about these asymptotes. We'll talk some more about them in the future. But in the meanwhile, if you do graph something like y equals 1 over x or y equals 1 over x squared, these graphs that do have horizontal and vertical asymptotes move them around. Those vertical and horizontal asymptotes will only be affected by shifting, stretching and reflecting will never change a vertical or horizontal line.