hello everyone and welcome to the first ever AP daily video for AP pre-calculus we're here to make some history today as you guys will be the first students ever to take the AP pre calcus exam my name is Jamal sadiki I'm coming to you from East Bridgewater Massachusetts and I teach at East Bridgewater Junior Senior High School hi and my name is Becky Barnes I'm a math teacher at West Springfield High School in Springfield Virginia and I'm pretty excited to be here with you guys today we're going to be focusing this first video on unit one reviews so let's go ahead and jump right into a multiple choice question okay we have the polinomial function p is given by P ofx equals a time xn power plus b where A and B are non-zero constants and N is a positive integer as the input values of P increase without bound the output values of P decrease without bound which of the following must be true if we look at our four multiple choice options we can see a combination of a either being positive or negative and if we look at the limit as X approaches Infinity of P of X we have answer choices of negative infinity or positive Infinity Jam why don't you take this one for us sure Becky what I want to do is I want to take a look at this question try to break down what it's really asked so I'm looking for some important terms or phrases and the first thing I see is input values now whenever I think about input I'm thinking about the independent variable for our function as we keep going it says that they're going to be increasing without bounds now when you see those words and you're going to see those words a lot throughout the AP pre-calculus what that means is we're going to Infinity so what these two phrases together really mean is that X is going to be going to Infinity as we continue to read we now see the output values that's going to be talking about the dependent variable and this time we're told that they're going to decrease without bound in this case we know that means we're going to negative Infinity so what we have here is that our output our P of x value is going to negative Infinity so when I look at these answering choices right away I think that we can get rid of answer C because that said the output was going to positive infinity and answer d as well let's take a look at our first rapid review a rapid review is when Becky and I want to review some material that you've had throughout the year but we're going to do it in a quick fashion because we know that you've already gone over we just want to bring it to the Forefront of your mind so in this one we're talking about polinomial function and behavior what we want to do is we want to look at the limit of f ofx as X goes to negative Infinity which again what that means is it's the output when we go all the way to the left of our graph and the limit of f ofx as X goes to positive Infinity which will be the output as we go all the way to the right of our graph there are a couple of different cases that are going be really important we think about polinomial in the first case we want to talk about when the degree is even and the leading coefficient is positive so whenever we have that I like to think of an example and the example that pops in my head right away is something like an x to the 4th so if we think about a class X 4th graph we might see something like this when we look at the end behavior the limit as X goes to negative Infinity of this function we're saying what is the output when we go all the way to the left and you can see it's going to be Infinity now when we go to the other side when we talk about the right end behavior we see that that's also going to Infinity so now let's take a look at what happens when n is still even in other words our degree is even but our leading coefficient is negative again I'm going to do an example and just kind of visualize this I want to think about something like Negative X 4th which you know that all that's going to do is just reflect My Graph over the x-axis so in this case when we talk about going all the way to the left we see that our function is beginning at negative Infinity so that limit as X goes to negative Infinity of FX is going to be negative infinity and you can see kind of the same behavior on the other side as we go to all the way to the right we're going all the way down on the graph so we're also going to have an output of negative Infinity now you might be wondering what about when the degree is odd well in this case we want to think about an odd degree with a positive leading coefficient so something like an xq graph in a cubic graph and again there's one that would pop into my mind when I think of that as we go all the way to the left in this situation we see that we're starting from negative Infinity so that first limit is going to be negative infinity and this time when we take the limit as X goes to Infinity of f ofx we're ending going up increasing without bound that's going to give us an output of infinity in the last situation we're going to have an odd degree but our leading coefficient is going to be negative remember that's going to reflect that graph over the xaxis so when I'm thinking about something like a Negative X cubed we might see something like this and this time the end behavior as we go to the left is going to be positive infinity and as we go to the right we're going to get Negative Infinity so jam I know we have a couple answer choices left that have to do with the sign of the a term what I'm noticing from your table is when the a value is positive regardless if I have an even or an odd degree polinomial on the right side my x value increases without bound and when my a value is negative on the right I decrease without bound I think that might help us finish up this problem what do you think Jam Becky I I think you really hit the nail on the head on this one because on this one we've got X going to infinity and that means that P of X is going to be going to negative Infinity so we need that that right end behavior to be negative INF and the only way that can happen like you said is when that leading coefficient is negative so we're going to eliminate answer B because we know we're going all the way to the right and all the way down and that means our final answer here is going to be answer a awesome now janal on the previous page you mentioned this term dominate in terms of the terms of my polinomial can you dig into that a little deeper for us yeah I'd love to so when we think about a polinomial function we're talking about X radius to a power so what I've done here is I made a table where we've got X the 1 x the 2 all the way up to X the 5th and we want to think about what happens when we input certain values of X so for the first group of data here we've got X values starting at one and going up to five and what you might notice is that when we're at one it doesn't really matter which power we're using we're getting one for the output for all of them but as the inputs start to increase okay you can see that the X to the 5th column is really really increasing far faster than any of the other columns and as we get bigger and bigger that X the fifth column is going to become more and more significant and leave the other colums as less significant and we've only gone to five here so imagine what's going to happen when we go up even higher up to 100 up to a th as we increase without bound you can see that every time we move over the column before it becomes less and less important matter fact by the time we get to x to the 5th I had to look this up Becky I think that's actually one quadrillion that we have there so that clearly would dominate the other degree terms so as we're thinking about polinomial it's really that highest degree term that plays the biggest role as we talk about their behavior you know I think it's time for our first spin-off uh a spin-off is g to be a question that's related to the question we just talked about but picks in a little bit of a different direction so what we want to do is we want to ask you what's going to be a very common question we feel on the AP pre-calculus exam describe the end behavior so here we've got a rational function f ofx is = to x^2 + 3x - 2/ x cub - 5x^2 + 4x - 9 and we want to not only get the end behavior correctly but we want to use that limit notation correctly so when we think about this one we want to think about what's going on on the left hand side and again that means X decreasing without bounds so the limit as X goes to negative Infinity of our rational expression now if you're following along with what we just said this really comes down to the degree the degree of the polinomial in the numerator is going to dominate that numerator and the same thing in the denominator so I want to locate those two terms that are going to be the most important so for this one the numerator is going to be X2 the denominator will be X cubed so what I'm saying is to find the limit of this expression let's actually do a different limit that will have the same answer let's take the limit of the ratio of those two dominant terms so here we're going to take the limit as X goes to Infinity of x^2 over X cubed and we're going to use some some algebra skills we're going to take that down to 1/x and what we're going to find out is that this answer actually turns out to be zero so jam I'm noticing that this term dominates coming up in more than one way when we look at the numerator and the denominator we're taking that leading or that dominating term to make a simpler limit expression but then we're also looking at the degree of the numerator denominator to figure out if the numerator or the denominator dominates in this case because the denominator dominated we went to a value zero that's a great observation Becky I'm glad you brought that up because we you're right we are looking about this in two different ways so in this next one we've got the right end behavior so we're laying X increase without bound so the limit says X goes to Infinity of our same function and again it's about finding out that dominant term in the top and the bottom so we're going to do the limit that's going to give us the same answer but this time again we're using X squ in the top X cub in the bottom and that's going to take us to the same limit we kind of saw before but notice this time we're going to Infinity not negative Infinity but as X goes to Infinity we still have one overx and like you said Becky it's the bottom that's dominating that's going to pull this answer down to zero so we want to make sure we answer this question correctly in describing the end behavior we have found basically a horizontal ASM toote here so we're going to say that this results in a horizontal ASM toote going to both the left and the right at the Y value zero so J I noticed in this question our leading coefficients were one in the numerator and the denominator how might this change the problem if our leading coefficients aren't one that's a good question Becky wait Jam how did you know I was gonna ask this question well you know it's not like we've been practicing this for like the last four weeks you know so let's take this on let's go describe the end behavior of f ofx equals our rational expression 9 x^2 - 2xb - x + 5 all over 7 x CU + 3x^2 - 4x + 8 using correct limit notation we're going to mirror the steps that we just saw in the last example so for this particular one we're going to take that limit as X goes to negative Infinity of our expression and again we're looking for that highest degree term but be careful because in this one there's a little bit of a Twist to it they didn't write our polinomial in the numerator in descending order so don't fall for the Trap make sure you actually locate that highest degree in this case it's -2X cub and when we go to the denominator it's that leading term 7x Cub so instead of taking the limit you see right now we're going to make this much simpler we're going to take the limit as X goes to negative Infinity of the ratio of those two dominant terms so when we look at that we see -2X Cub over 7x Cub we'll use a little bit of algebra and this is going to come down to the Limit as X goes to negative Infinity of -27 now that might look a little strange to some of you because there's no variable in this expression but but that's actually makes it an easier problem because -27 is a constant and by definition that constant is not going to change so as we take this limit as X go to netive Infinity of the constant -27 we're going to end up with the value -27 so Becky why don't you take this next one I would be happy to jam did all the hard work for me let's go ahead and take a look we're gonna do the limit as X approaches positive Infinity we're going to look for that leading term in the numerator which is that -2X cub and that leading term in the denominator which is 7 x cubed and simplify our limit expression to the Limit as X approaches Infinity of -2X Cub over 7x Cub we're going to go ahe and simplify that down to the limit as X approaches Infinity of -27 and just like jam said a moment ago that's going to lead us to a final answer of -27 because we're approaching a value of -27 on the left and the right we're going to result in a horizontal ASM toote of y equals -27 when you're asked to find the end behavior any rational function remember that just means we've got a polinomial function the numerator a polinomial function the denominator so in this case f ofx over G ofx and we're letting F ofx be defined as a polinomial of degree M so the leading term is ax to the m and we've got all the trailing terms but remember when we're talking about n Behavior it's really that leading term we're talking about in the denominator we're going to have a polinomial and the leading term is CX to the N with the trailing terms but the important thing to remember is the degree of the numerator here is M and the degree of the denominator is n so let's summarize kind of what we just talked about what you want to do is you want to compare the degree of that numerator to the degree of the denominator in case one if your degrees are the same and you just saw that in the last example where the degree of the numerator and the denominator both three in that problem what you're going to do is as X goes to negative infinity and as X goes to Infinity you're going to end up with the ratio of the coefficients and that's always going to lead you to a horizontal ASM toote at y equals that ratio awesome so if we look at that next row it says m is less than n so the degree of the numerator is less than the degree of the denominator well in this case if we simplify out we'll see that our our denominator or bottom heavy that's what I like to call it in my class as our fraction is bottom heavy as X increases without balance the fraction is going to approach a value of zero so my n behavior on the left and the right will be zero I'm approaching an ASM toote of y equals z now in this last case the degree of the numerator is actually greater than the degree of the denominator so you know Becky referred to that last one as a bottom heavy case this is actually I I I do the same thing I call it a topheavy case and when you've got a topheavy case it really means that numerator is really dictating what's going on so when you think about this what's going to happen is this rational expression for its end behavior is going to act like the polinomial where you've got a first term that will have X raised to the m minus n because you're going to have to subtract those exponents from the top and the Bottom now in terms of getting an answer on your pre-cal exam the one case that's really really very specific is if you have a difference of one if the degree of your numerator is one higher than the degree of your denominator that's going to mean that you've got a slant ASM toote you're going to want to use some um long division polom long division and find the equation of that slant ASM toote when asked to do so we don't have time to actually go into that but I'm sure you've covered this throughout the year so hopefully you'll be prepared when you see this type of example on the exam hey Jimmy I think it's time for another multiple choice question why don't I give us a read for us says the function f is given by F ofx = x^2 - 9 over the quantity x + 3 * the quantity x^2 + 2x - 15 which of the following is true about the graph of f in the XY plane Jim when I look at these answer choices I see that we're looking at these input values of xal -3 3 and5 and trying to determine if we have vertical ASM tootes are holes at each of these places yeah Becky it's true the big question always seems to be when you've got the these zeros in the rational expression are they going to be a whole are they going to be a vertical ASM toote are they going to be a zero and I think the best thing to to give you for advice is Factor start these problems by taking that rational expression just factoring all the the numerator and the denominator out completely so in this particular example that numerator is going to factor to X+ 3 and xus 3 and the denominator is going to factor to x + 3 which we had from the beginning and then factoring that quadratic Factor we're going to get an x - 3 and X+ 5 I think the safe place to start is to take a look at your numerator and just find the zeros of your numerator in this case we've got3 and three and then likewise find the zeros in your denominator because these are going to be the important values that we're going to have to analyze what we want to do now is we want to go and kind of compare the zeros in the numerator the denominator side is this going to be a vertical ASP or is it going to be a ho so our first result we've got this xal ne5 value that is only a zero in the denominator if you're a z in the denominator with no corresponding zero in the numerator you are always going to be a vertical ASM toote now when we look at the other two values we see kind of as x equals -3 for both the numerator and the denominator and one factor of each that's going to result as a whole and that's going to be true for both cases since they have this one factor of each for both of these problems so going back and looking at our answers the answer that would make sense is answer B the graph of f has a vertical asut xal 5 only and there are exactly two holes in this graph of f one at x = -3 and one at X = pos3 now one thing I was talking about when I went through that problem is the number of factors or the number of times these zeros show up and that's actually called the multiplicity the multiplicity is the number of repeated factors each zero of a polinomial has when you looked in this problem all these zeros came from just a single Factor so the multiplicity for all these zeros is one now you might be wondering what happens if the multiplicity is greater than one how do we handle a situation like that and we've got a rapid review to kind of go over that to make sure you guys are going to be comfortable so if m is the multiplicity of the zero x equals a in the numerator and N is the multiplicity of that same zero xals a in the denominator the following will hold true if the multiplicity of the numerator is greater than or equal to the multiplicity denominator we're going to see a whole at that value of a if the multiplicity in the numerator is less than that of the denominator that's always going to result as a vertical ASM toote at xal a so let's see if we can put this into practice with a spin-off problem here we're given the function f ofx you can see there are far more factors and many of them are being repeated so we're going to see how the multiplicity plays a role in this problem so in the numerator we have x * the quantity x - 1^ 2 x - 5 4 the denominator we have the factors x -1 Cub * x - 5 4 and x + 7^ 2 so when we look look at this we want to find out for each of these zeros classify them as a whole a zero or vertical asut for this function so what we want to do is we really want to just solve each of these factors equal zero and compare their multiplicity so in that numerator our zeros are given by zero one and five in the denominator we're going to have zeros of -7 one and five and now before we can classify we're going to want to see the multiplicity of each so x equals z there's only one factor that leads that that's got a multiplicity of one for xal 1 you can see that's coming from a squared Factor so the multiplicity is two and for five we're going to have a multiplicity of four in the denominator you can see the multiplicities listed here we find them the same way by counting the number of repeated factors so when we look at all these zeros and we try to find the result for each for xals -7 it's only showing up in the denominator that means we have a vertical ASM toote for xals 0 only showing up in the numerator we're going to have a zero in that case for xal 1 we see more a higher multiplicity in the denominator than the numerator so that would be a vertical ASM toote and we go to the xals 5 we see an equal multiplicity for the numerator denominator that's going to lead us to a hole you know Becky I think we're just about done this first review I'd like to end this with some important words to remember that are going to show up on the exam over and over again the first one is whenever you guys hear that phrase increase without bound it always means that you're going to be going to Infinity likewise if you're decreasing with that bound your your value is going to be going to negative Infinity one of the most popular things we talked about today is end behavior whenever you're asked to find end behavior make sure you use your limit notation and take that limit of your function as X approaches negative infinity and the limit as X approaches Infinity multiplicity comes into play when classifying those zeros the multiplicity is the number of repeated factors if zero of a polinomial will have and then our last two is comparing the whole to the vertical asop a whole is always going to be when the multiplicity of the factor in the numerator is greater or equal the multiplicity of the factor in the denominator and likewise a vertical ASM toote is going to be when the multiplicity of the factor of the numerator is less than the multiplicity of the factor of the denominator Becky I I really appreciate you being a part of this with me today and going through this first one I can't wait to see what you've got in store for us for our next one for unit two have a great day everybody we'll see you next time