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CCSS.Math:

let's do a couple more examples graphing rational functions so let's say I have Y is equal to 2x over X plus 1 so the first thing we might want to do is identify our horizontal asymptotes if there are any and I said before all you have to do is look at the highest degree term in the numerator and the denominator in the highest degree term here there's only one term it is 2x and the highest degree term here is X they're both first degree terms so you can say that as X as X approaches infinity Y is going to be as X gets super large values these two terms are going to dominate this isn't going to matter so much so then our expression then Y is going to be approximately equal to 2x over X which is just equal to 2 and that actually would also be true as X approaches negative infinity so as X gets really large or super negative this is going to approach 2 this term won't matter much so let's graph that horizontal asymptote so it's y is equal to 2 so let's graph it so this is our horizontal asymptote right there or is on ttle asymptote Y is equal to 2 that right there let me write it down horizontal asymptote asymptote that what is what our graph approaches but never quite touches as we get to larger and larger pop more and more positive values X or more and more negative values of X now do we have any vertical asymptotes here well sure we have when X is equal to negative 1 this equation or this function is undefined so we say Y undefined when X is equal to negative 1 that's definitely true because when X is equal to negative 1 the denominator becomes 0 we know what 1 over 0 is it's not defined and this is a vertical asymptote because the X doesn't cancel out the X plus 1 sorry doesn't cancel out with something else let me give you a quick example right here if I have us to have the equation y is equal to X plus 1 over X plus 1 in this circumstance you might say hey when X is equal to negative 1 my graph is undefined and you would be right because if you put a negative 1 here you get a 0 down here in fact you'll also get a 0 on top you'll get a 0 over 0 its undefined but as you can see if you assume that X does not equal negative 1 if you assume that this term and that term are not equal to 0 you can divide the numerator and the denominator by X plus 1 and this is going to be or you could say well that over that if it was anything else over itself it would be equal to 1 you would say this would be equal to 1 when when Y does not sorry when X when X does not equal negative 1 or when these terms don't equal 0 it equals 0 over 0 which we don't know what that is when X is equal to negative 1 so this in this situation you would not have a vertical asymptote so this graph right here no vertical no vertical asymptote and actually you're probably curious what does this graph look like and I'll take a little aside here to draw it for you this graph right here if I had to graph this right there what this would be is this would be Y is equal to 1 for all the values except for X is equal to negative 1 so in this situation the graph it would be y is equal to 1 everywhere Y is equal to 1 everywhere except for Y is equal to negative 1 and Y is equal to negative 1 it's undefined so we actually have a hole there we actually draw a little circle around there a little hollowed-out circle to show that we don't know what Y is when when X is equal to negative 1 so this looks like that right there it looks like that horizontal line no vertical asymptote and that's because this term and that term cancel out when they're not equal to zero when X is not equal to negative one so when you are identifying vertical asymptotes let me clear this out a little bit when you're identifying vertical asymptotes you want to be sure that this expression right here isn't cancelling out with something in the numerator and in this case it's not in this case it did so you don't have a vertical asymptote in this case you aren't cancelling out so this will define a vertical asymptote X is equal to negative 1 is a vertical asymptote for this graph right here so X is equal to negative 1 let me draw draw the vertical asymptote it will look like that it will look like that and then to figure out what the graph is doing we could try out a couple of values so what happens what happens when X is equal to 0 so when X is equal to 0 you have 2 times 0 which is 0 over 0 plus 1 so it's 0 over 1 which is 0 so the point 0 0 the point 0 0 is on our curve what happens when X is equal to 1 we have 2 times 1 which is 2 over 1 plus 1 so it's 2 over 2 so it's 1 comma 1 is also on our curve so that's on our curve right there so we could keep plotting points but the curve is going to look something like this it looks like it's going to approach it's going to approach negative infinity as it approaches as it approaches the vertical asymptote from the right so as you go this wiggle is negative infinity and then it will approach it'll approach our horizontal asymptote from the negative direction so it's going to look something like that and then let's see what happens when X is equal to what happens when let me do this in a darker color do it in this red color what happens when X is equal to negative 2 we have negative 2 times 2 is negative 4 negative 4 and then we have negative so it's going to be negative 4 over negative 2 plus 1 which is negative 1 which is just 4 so it's just equal to negative 2 4 so negative 2 1 2 3 four negative two four is on our line and what about what about well let's just do one more point what about the negative three so the point negative 3 on the numerator we're gonna get 2 times negative 3 is negative 6 over negative 3 plus 1 which is negative 2 negative 6 over negative 2 is positive 3 so negative 3 3 1 2 3 1 2 3 1 2 3 so that's also there so the graph is going to look something like that something like that so as we approach negative infinity we're going to we're going to approach our horizontal asymptote from above as we approach negative 1 as X equal to negative 1 we're going to pop up to positive infinity so let's verify that once again this is indeed the graph of our equation get our graphing calculator out and get the graphing calculator out we want to define Y is 2 X divided by 2 X divided by X plus 1 is equal to delete all of that out and then we want to graph it and there we go looks just like what I what we drew and we we don't that vertical asymptote it connected the dots but we know that it's not defined they're just trying to connect the this pot super positive value all the way down because this just trying out all the graphing calculator is actually just making a very detailed table of values and then just connecting all the dots so it doesn't know that this is an asymptote so it actually tried to connect the dots but there should be no connection right there hopefully you found this example useful