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AP Calc P.1

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hello everybody Welcome to our first AP calc video today we're going to be going over chapter P1 and our textbook that is going to be a review of precalculus topics now this video is going to move rather quickly through these objectives just because it is by definition a review and not necessarily A relearning and a note what I expect from you when you are falling along with me here I do expect that you are taking notes copiously and I do expect that when I grade your notes and take a look to see what you've done that they closely match what I've written down in this video if you would like to add extra things just to give yourself reminders that is more than okay I just would like to make it clear what the expectation is for success  so let's get started apps to let you know you do not have to write down these objectives that is the one exception of things you don't have to write down because our just a guide us let us know what we're going to be singing this video so we're going to go over sketch and graphs testing for symmetry finding x and y intercepts points of intersection tell me what that means and fitting models to data so let's get started first we're going to talk about sketching graphs now to sketch a graph you have a few options you can create a table and that's also has Point plotting or you can find intercepts or you can use parent functions and their transformations now what we're going to focus on this first example it's just finding intercepts let's refresh that pretty quick  so for example say you are given the task to find all intercepts for the function y equals X cubed minus 16x well you're going to be finding both your Y intercept and your X intercepts so remember just in general to find your Y intercept that's when X Goal Zero and find your x intercept that's when y equals is 0  so to find that X starset let's do that one second let start with r y intercept here to find our y-intercept going to draw a little arrow and stir turn over here for you we set y equal to zero and we get X cubed minus 16x now we're going to do some of that fact we learned back in Algebra 2 and we're going to take out an X and then we are going to be left with x squared minus 16 part of me just 16 inside the parentheses if you remember your perfect squares factoring then you'll remember that we can change this into x + 4 x - 4 lastly to get some value 4X we can use that zero product property set each of these things equal to zero and then we do that we'll find that x equals 0 ask what equal -4 an exit Eagle positive for if we solved all of those for zero awesome however I'd like to stop ass right here and let you know that this is not how we write our answer in AP we have to be very careful but how we report our answers it is sing us the question to find all intercepts so I do not report my values this way I need to report them as intercepts as coordinates so  I need to say this my X intercepts are zero zero -4 zero and at40 and now to deal with why are X intercepts oh I did that backwards that's okay every do my arrow hear you guys can see it in my work a little bit backwards there that's okay he's RX stop so just move that Aaron over by mistake first video I'm sure I'll get better working out all these Kinks later okay we found our rec center by plugging in 04 why now it's fine are y intercepts and we can just kind of do this one mental you guys check it out if we were to take her can you function and plug-in 0 4 x wow we would just get y equals 0 so right away just doing some Mental Math I can see that I have a wife intercept at 0 0 and that would be all  okay that's fine intercepts now what if we were asked a different what if we were asked not to just find our intercepts but to graph  using intercepts  and our function was x minus y I squared equal to 1 well we're going to start this off just the same we're going to find those intercepts first to begin let's start with our x intercept and remember as we just reviewed sat y equal to zero hey that's not so bad X would equal 1 sorry I would have a text intercept at10 awesome if I was going to find a y-intercept  then I would need to set that x equal to zero and I would be left with the equation can you buy squared equals want from here I can transfer that negative over so I have y squared equals negative one I can take this root of both sides and when I do that you may have recognized this as our I  plus or minus square root of -1 is not a real number that is an imaginary number and when you get a result like that remember that when we are graphing we are graphing in the real coordinate plane what this essentially means is that there are no why  intercepts okay so I know two things right now I know there are no y intercepts and I know I have a point at10 quite frankly I just give me a lot of information so what I'm going to do is I'm going to go back up to my toolbox here create a table find intercepts or use parent since I'm going to use that parent function to help me out here  in order to get that parent function I'm going to take my original and I'm going to solve it for why I can really see what I'm dealing with here  so if I were to first start solving for y I would move that X over and I'm getting is y squared equals negative X plus one  I would change the sign of everything or multiply everything by -1 forget y squared equals x minus one I would take the square roots as both sides y equals radical x -1 and I would I look at this I can clearly see a parent function here I can see the parents change of y equals the square root of x now I know that is just that sideways Parabola looking thing and I know that it has a train relation of moving to the right by 1 and that means it's going to Pastor that point 1, 0 great now I've all information I need to sketch have some going to do just that get my coordinate plane here going and I'm going to have that point at10  and I know the current function looks like that sideways parabola  so there we have it we have draft using intercepts okay moving on to our next objective looks like my computer Frozen on you but I think we're all good okay so our next active was testing for symmetry I need you ask right that right here real quick  testing for symmetry add to review their the three basic types there is symmetry the x-axis  there is symmetry about the y-axis and there is symmetry about the origin now  give you a graphic example of what that means something that's symmetric about the x-axis is kind of like weed saw something that for every x value it has a positive and is y value and something that is symmetric about the y-axis  grass are you because think of that plain old Parabola write the classic one were used to staying where for all of our y-values there is a positive and a negative next value and you can see that they reflect each other reflection over the y-axis  about the origin is it just a little bit it's sort of like we combine both of those Concepts  and something that would be symmetric about the origin could look like a cubic function something that looks like this  where if you were to take any point from this side and 180°  you would get a point over here and you can also see that relationship of a positive XY  negative XY  now what you can do in order to find it function is symmetric about any of these things you can just plug in different values into the function  in order to test it out so say that every positive why you would substitute in a negative way and if it's the same then it is symmetric  ran over here if you reply list every every positive acts with a Negative X and you get the same I should say function  same function then yes it's symmetric about the y axis  over here if you were do both things if you replaced all the positive axes with negative x's and all the positive wise with negative wives and you result in the same function then it is also symmetric about the origin say we were given the function y equals x squared over x squared plus once let's see if this is symmetric about the x-axis let's replace this y with a negative Y and see if we get the same function we would get Negative y equals x squared over x squared plus one now it's just a simple matter of asking yourself is this the same as this well no so it is not symmetric about the XX next not symmetric about the next  axis okay now we're going to replace every acts with a Negative X to see if it's symmetric about the wax enemy do that it's going to look like this and now we are pretty comfortable with the rules of exponents hear any negative times itself is just going to be a positive again so what we have here here are essentially the same function so this is symmetric missing an app about the y-axis lastly see if it's symmetric about the origin by replacing y with negative and acts with negative X  negative y equals  negative x squared over negative squared plus 1 well again we run into that same problem where we end up getting negative y x squared x squared and this is not the same as what we started with right here's our original  and this is only equivalent here not here not here so we know it is not Patrick  about the origin as well  and if we really want to get in the habit of answering the way that we are expected to on the AP exam and then we would go a step further here and we would fully state that  the function and you state your function y equals x squared over x squared plus one  is Shrek about the Y okay and that is just a quick review of symmetry next we're going to talk about those points of intersection poems  intersection now you may remember points of resection also as systems of equations write the same as solving systems  of equations because that's all a point of intersection is correct at point of intersection is play a solution to an equation and so  are hard me it's the point where two equations enter that resolve both of them we will find the point where they cross each other there's a really a lot of ways you can do this your book gives you really good example of substitution if you would like to see an example of that again I'm going to go through the process of elimination so say we are given system  we are asked to Pitbull example here find the points of intersection for the system and then we are given two equations x squared minus one equals 3 and x minus y equals 1  now the elimination strategy tells you to eliminate one of your variables by making them opposites and it looks like we've got this single variable why here that would be pretty easy to if we set one of these multiplied one of these by -1 Griff wine  then will you get our system would now look like this if it's in parentheses here  we'd still have squared minus y equals 3 because we didn't do anything to that and on the bottom we would have Negative X Plus y equals negative 1 now we can just do some basic algebra here we can see that are positive and a negative wise with cancel out or x squared play sax really can't be simplified and 3-1 get us that too and now we can see pretty clearly by this we're going to be dealing with a quadratic function and so we can start to move our to over set an equal to zero do that lovely factoring that we got so used to in Algebra 2  where we and we have minus two and a half plus one and again if we do the zero Mattox property then we know that we have two answers for x we've got x equals 2 + x equals negative one awesome I've some values for X however we are asked to solve a system and we're trying to find points of intersection right now we've only sounds the X this is what we truly found we found two points  Army solder x coordinates for those two points we are tell missing the Y  add a fill in these blank spots here to find the Y values we got one more thing to do we have to take these X values and plug them back into one of our two Originals I don't know about you but this bottom one looks a lot easier to plug into so I'm going to do that now what we're going to have is we're going to have for one case 2  minus y equals one and for another we're going -1  minus y equals 1  and if you're just to do some Mental Math hear you we can see the we would subtract 2 on both sides we would get negative one we would get Negative y equals negative one which would just end up getting a positive one  now what x equals negative wanted to start solving it we would add one to both sides you get Negative equals to we would move that negative over and we would get -2 when x equals negative 1 and those would be our two points of intersection  amateur does a, between those points or space because they are not multiplying moves used together they are just coordinates  now  let's move on to our last objective fine as fitting you model to data let's move on down here now when you are asked to fit a model to data you're typically given some information you're given either a table or a grass or a set of points and then you were asked to find a model and remember the word model you guys is just an equation  that sets given data and the example that we're going to talk about we're given data El pardon each we're going to find a model but then it's also important that you understand we don't just stop there we're not just done we find her a model Wakely you're asked to make a prediction and that makes sense because it doesn't really matter that we found a model or an equation unless we use it somehow so for this given example our data is going to be dealing in the number of cell phone  site n  the US  and we are given the open table in terms of years a number of sites and let's get all the spaces that we need here we have six of them in 2003 there 163 sites 2005 184 2205  2007 there were 213 2009 there were 247 and 2011 there were 283 and 2013 there were 304  now we're going to need to find a model and this particular question zest to find a model of the Forum ax squared plus BX plus c part of me this is T not X  there we go we're dealing in terms of time T okay you can see then that the type of model we're going to be finding is not linear type of model or going to be finding and quadratic and there's another one piece of information that they tell us they tell us also that tea is equal to 3 in 2003 that's very important what they're telling us to do there is to change our table we are no longer dealing in terms of two how is things were going to deal in terms of  much smaller numbers so instead of running out 2003 I'm going to manipulate the table  and I'm going to change it according to what they gave me that looks like they're dealing in number of years since 2000 so we just fill in or numbers accordingly remove all of our values 40 and all of our dependent values on the bottom so what we are going to be doing next is or going to be reviewing a bit here on a calculator keep this as clear as I can okay your talk later is going to be really big help here if I go through the instructions pretty quickly don't will be giving you guys a hand out in class on to guide you on your own calculators so  first of all what we're going to need do you are some memory clearances organ press II  and then let me just let me zoom out a little bit more here are we going better  we're going to press seconds and then memory down here and we're going to go down to where it tells us to clear all press enter okay now Oliver lists are cleared out wonderful next we're going to set up our step plots we're going to press Next Step we're going to make sure this first one is turned on so we're going to press enter make sure it says on press enter and here's telling us telling us that we're going to be plotting a scatter plot of points not a line or bar graph add a telling us that are independent variables are going to be in our list one  I'm not a little bit more  hang put some technical difficulties here there we go okay then married Arkansas a little better so we have our L1 or L2 for wife or dependent and our marks are going to be is open access Nets going to be in the color blue in class might not do all these colors but the purpose will be the same  okay now that leaves that's enough we can press the second quit and we're going to perform our regression we know that they wanted us to perform a quadratic regression but before we do that we can do to put in our data we're going to press stat we're going to press edit and we're going to put in our data so for our exes are Independence that's going to be our to Artie's or years so we going to do 3579 11 + 11 and 13 now for our wise we're going have 163 184 - 13 - 47 - 83 + 3 Austin everything is where it needs to be and so now we go back to stat and we're going to press help and underneath caliper going to go to where it says quadreg and quadreg as you might have guessed just means that quadratic regression and that's what we saints were going to  press enter an RX list that's what we need our exes are at ones or what rl2 and then we're here where it says story regression equation you're going to want to do that so you're going to want to store it in your burst in your y1 and you can do that pretty quickly by pressing Alpha trace and where it says why one price and now we're going to press calculate  alright so now he's got some good information here turn the form ax squared plus BX plus C we know that our equations going to look alike going to be 0.125 x squared + 12.8 x + 1 your 20.475 and there was also an important number there they gave us our r squared which is 0.9 9396  and remember if we take the square root of that that is how we get our correlation and basically that's just going to tell us how closely our model is to her actual data enemy take the square root of that we get 0.9 9698 efficient and you can see this pretty closer original data if you think about multiplying something by one by itself you get exact the same thing so if we're multiplying are add a dive this by Point 99698 that's pretty close to 1 so it's pretty close to what our function actually is pretty close to our dad and this is great we've found our model we have found a how close it is to our fictional data but then we're going to make that prediction like we talked about before we're going to predict the number of cell phone sites in 2025 and check it out 2025 we're not interested in 2025 we're manipulating our variable correct like they told us too so we're not interested in this value were interested when t is 25 right  so what we can do is we can go back to our calculators  and we can grass this and then calculate R-value so first we're going to press grass and there you can see that these original boxes are our points table and this line is that model this line is that quadratic model that we make awesome so now what we need to do is find that value when T equals 25 so we can press seconds Trace so we get the calc function repress M14 are you do you want to know when x equals 25  and it tells us right there in the middle of it so you can see a little bit better a small at us when x equals 25 y equals 518.6 awesome to win a riding down or answer please say 518.6 again we were in a very careful that how we report our information Siri say 518.6 cell phone sites  and that is just a review of those topics thanks for watching our first video and see you guys in class

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