 Hello, this is a video about rational functions their asymptotes and intercepts So what exactly is a vertical asymptote well a vertical asymptote abbreviated VA is a vertical line Meaning it's of the form x equals some number that a graph never crosses So the graph will never cross a vertical asymptote to find a vertical asymptote You always want to factor and simplify your rational function first if possible and then set the denominator equal to zero That's the easiest way to find a vertical asymptote a word of caution for you If you don't factor and simplify first if the factor which causes the denominator to equal zero Is also in the numerator Then the x value is a hole. It's just an open circle in the graph somewhere not a vertical asymptote Now it is a horizontal asymptote. Well a horizontal asymptote abbreviated ha is a horizontal line It's of the form y equals some number that a graph sometimes Intersects sometimes To find the horizontal asymptote you will want to compare the degrees of the numerator and denominator So the top and bottom of the fraction and consider the following three cases Case one if the degree of the numerator is smaller or less than the degree of the denominator The horizontal asymptote is y equals zero. That means on top of the x-axis If the degree of the numerator is equal to the degree of the denominator then the degree is Or then the horizontal asymptote is y equals It's a ratio of the leading coefficients you take the leading coefficient of the numerator and put it on top here Over the leading coefficient of the denominator that goes on the bottom If the degree of the numerator is more than or greater than the degree of the denominator then there is no horizontal asymptote Instead there could be what is known as a slant or oblique asymptote for the sake of this course We'll simply say there is no horizontal asymptote and go no further To find intercepts just like you learned for many other types of functions to find the y-intercept you set x equal to zero And then to find x-intercepts you would set y equal to zero In other words, this means because remember y is the same thing as f of x or g of x or whatever your function is This means you set the function equal to zero But we got a good shortcut for you here since rational functions are fractions a Shortcut instead of setting y equal to zero instead of setting the function equal to zero instead of setting the whole fraction The whole rational function equal to zero the shortcut is to set the numerator equal to zero So now we have a few examples We're going to find the asymptotes and intercepts of each rational function. We will factor and simplify first if possible 4x plus 3 over x minus 1 is not a rational function. We can factor and simplify So immediately we can jump in and start finding our information Beginning with the vertical asymptote. So remember the process for finding a vertical asymptote. We will set the denominator x minus 1 Equal to zero It's all for x So x equals 1 the vertical line x equals 1 will be a vertical asymptote Next horizontal asymptote. We have to compare the degrees and The degree of DOT will represent the degree of top and DOB will represent the degree of bottom I did that so that way we can save a little bit of writing time So the degree of the top or degree of the numerator is well the highest power on the variable and the numerator is a one And the highest power on the variable and the denominator Is also one So the two degrees are equal So you have to figure out which case is the case that talks about what to do when the degrees are equal That would be case two So the horizontal asymptote is y equals the leading coefficient of the top over the leading coefficient of the bottom This means y equals four Over one the term that tells you the degree is the term that contains the leading coefficient so we have four over one or Simply put Y equals four That's a horizontal line that goes through four on the y-axis Now the y-intercept remember you will let x equals zero so that means Your y-intercept will be y or h of x or h of zero equals four times zero plus three or Or over zero minus one So that's three over negative one or y equals negative three Which has an ordered pair of zero Negative three so we cross the y-axis at negative three Now for the x-intercept remember the shortcut we let the numerator equal zero So that means we have to set four x Plus three equal to zero and solve for x Take away three from both sides and Lastly we will divide both sides by four. So x is going to be Negative three fourths. That's the x-intercept negative three over four comma zero So we found all the information we need In part b we have x squared minus one over x plus one Please make sure you factor and simplify first So I have x plus one x minus one Over x plus one So yes the x plus ones do cancel out. So you're left with x minus one over one I'm keeping it over one just so that way it keeps its fraction form so from now on We're allowed to consider the rational function in simplified form x minus one over one What this means when a factor of Factor containing x cancels out in both the top and bottom when a factor containing x is in both the top and bottom It means there's a hole At Wherever that factor is equal to zero So it means there's a hole at an x value of negative one So that's just a little bit of bonus information for us. There will be a hole in open circle at x equals negative one in the graph For the vertical asymptote we set the denominator equal to zero Well, guess what you look at this simplified format You set one equal to zero Yeah, that's pointless. You can't do anything with that. So there's no vertical asymptote in this case Horizontal asymptote. So let's find the degree at the top and let's find the degree at the bottom It's best practice to use nothing but the simplified version Of the fraction So degree at the top highest power on x is one Degree at the bottom you have a constant term which is a degree zero term So the degree of the top Is going to be Greater than the degree of the bottom So you have to think about what case this would be and this would be case three This is actually the case where there's no horizontal asymptote Let's find those intercepts there. So remember the y-intercept process is to let x equals zero You can use the simplified version of the function. That's a okay. That's the best practice So y equals zero minus one over one Or negative one over one Okay, so y equals negative one. That's your y-intercept zero negative one Is the ordered pair Remember the x-intercept shortcut is to set the numerator equal to zero You use the simplified version of the fraction of the function x minus one equals zero And that means my x-intercept will be x equals one Or one zero So that's the given input or the information they want us to find for big f of x that rational function So what about part c the first thing you want to do in part c is See if you can factor and simplify That's first So that means on top i'm going to have x plus three And on the bottom i'm going to have x minus seven x plus three The first step is to always factor and simplify So notice that the x plus three's cancel out on both the top and the bottom which leaves you with one over x minus seven This is the simplified function you will use for the rest of the question So notice that this means now that there is a whole At wherever x plus three the cancelled factor is equal to zero This means there's a whole At x equals negative three So We have that little piece of information that means there will be a open circle At x equals negative three on the graph of the rational function In the vertical asymptote it's found by setting the denominator of the simplified function equal to zero Giving you a vertical asymptote value of x equals negative seven. It's a vertical line that goes through seven on the x axis Now the degrees the degree of the top Is going to be remember look at the simplified function one over x minus seven the top has a constant So the degree of the top is zero and the degree of the bottom is going to be just one so this puts us in the Situation where the degree of the top is less than the degree of the bottom So this puts us in case number one Case one says that the horizontal asymptote will be y equals zero That means it will be The x axis that's what y equals zero is this is line right on top of the x axis so now We find the y intercept and the x intercept the y intercept remember we find it by letting x equals zero Use the simplified rational function one over x minus seven one over zero minus seven So you get negative one seventh as an ordered pair that zero Negative one seventh. So you crossed a y axis at a value of negative one seventh Now the x intercept We set the numerator equal to zero used a simplified form of the rational function One is equal to zero Which you can't do which means there is no x intercept in this case So that's all I have for you for tent for now. Thank you for watching