hello and welcome to unit 2 in this unit we cover two more functions exponential and logarithmic which makes it very long and it's worth a lot on your AP exam seeing as how you guys really like these unit review videos a lot I'm going to do a quick Shameless plug for 5 seconds ready I just uploaded probably my greatest video ever on my non-educational Channel please go watch it at the top Link in the description and subscribe to that channel and this Channel and also follow all my social medias okay I'm done what else do I need to say oh yeah this video is simply a clip together video of all my topic review videos have the AP style questions at the end of each video if you want to try your hand at some AP style questions associated with each topic the unit 2 playlist is in the description anyway let's get in to unit 2 Welcome to the common sense topic why do I call it the common sense topic just wait a sequence is really just a list of numbers sequences are split into two groups arithmetic and geometric an arithmetic sequence is really just a linear function the numbers have a common rate of change with a common difference which in this case is two a geometric sequence increases more and more as the sequence continues because of a common proportional change and therefore there is one number multiplying by each number which in this case is three so if you wanted to find the seventh number in the sequence you can just multiply time three and then times three and it's pretty easy and in the arithmetic sequence if you wanted to find the 20th term you just keep adding two until you get the 20th term pretty easy right I took this exam this most recent year only knowing that and I got a five so this video should be over right uh the course requires I teach four complex equations to go with sequences of course I'm not telling you you don't need to know these but you know anyway arithmetic sequences have two equations first is a sub Nal a Sub 0 plus DN where a sub n is the value of the term you are finding out a sub Z is the first term in the sequence D is the common difference and N is the position number that you trying to find but what if you don't know the first term in the sequence well that's why we have another equation a sub Nal a sub K plus d MTI n minus K where a sub K is the number of the already known term and K is the position number of the known term or you could just add the common difference to get the answer but you what do I geometric sequences have two equations first is G sub Nal G Sub 0 * R to the N power where G subn is the term you are finding and G sub Z is the first term of the sequence R is the common ratio or proportional change and N is the position number of the term you are trying to find but again if you don't know the first term we have another equation G sub Nal G sub K * R to the power of n minus K where J sub K is the term you know and K is the position number of the termy no okay topic 2.2 is a kind of clarification on sequences and an introduction to exponential functions so sequence equations we talked about last video you could probably tell I don't really like them because of my tone talking about them last video however before we get into this video I need to clarify something on particularly these two equations I said last video that the zero term was the first term of the sequence what would have been more appropriate would have been to say that it is the initial term because it's technically the term before the first term if you're still confused reference the answers to the questions at the end of my last video and they will clarify it so anyway though I don't really like these equations they do tell us some stuff and a massive surprise to everyone I'm sure an arithmetic sequence is really just a linear function and we see that in its equations as the form a sub Nal a Sub 0 plus DN is really just the equation y b plus MX which is the linear function equation and the other equation a sub n a sub K plus d MTI nus K can be expressed as another way of writing linear equations being f ofx = y i + m * x - x i where you include the point x i comma Yi in the equation geometric sequences embody a new type of function we'll be talking about this unit exponential functions the exponential function skeleton equation is f ofx = a to^ of X and you can see the similarity to the geometric sequence equation and funny enough the second geometric sequence equation can be transformed into an exponential function written as f ofx = y i * R to ^ of x - x i where it includes the point XI comma Yi in its equation so what do we take away from all of this well here it is linear functions have output values changing at a constant rate based on addition and exponential functions have output values changing at a proportional rate based on multiplication which is why if you see a graph table or even just two points you should be able to tell me whether it is exponential or linear exponential functions themselves are a next video topic 2.3 is all about exponential functions and their properties lots of students seem to struggle with exponential functions in math so to those kids I say they really aren't that bad just wait till we get to log functions but anyway we already know the skeleton equation to an exponential function is f ofx = AB to the^ of X where a is the initial value and B is the base and take a look at the graph it's a curve that keeps curving there are three simple rules of an exponential function first is that a can never equal zero the second is that b must always be positive and finally B can never be one other than those three things you can do really whatever you'd like with the function an exponential growth happens when a is greater than zero and B is greater than one it looks like this on a graph you can see it growing hence it being called a growth an exponential decay happens when a is greater than zero and B is less than one but greater than zero it looks like this on a graph and you can see it decreasing hence it being called a Decay I don't want an exponential function graph to trick you though despite it looking like it's curving up it is still going to positive Infinity on the x-axis this means all exponential functions will have a domain of all real numbers some other fun things an exponential function is either always increasing or decreasing meaning it is only concave up or down and never changes this also means that exponential functions have no points of inflection exponential functions are a little weird though because they don't have an ironclad parent function the parent function of an exponential function is said to be B to the X where B is greater than 1 for a grow or between 0 and 1 for a Decay so that means that any value you plug into B that is greater than one or between 1 and zero falls into the group of being a parent function for the exponential function and for these parent functions we can see some fun things first is that no matter what there will always be a point at 0 comma 1 because anything to the power of 0 is one the second thing we see is that there is a horizontal ASM toote at y equals 0 and this would mean the limit as X approaches negative Infinity of any growth parent function would be zero and of course as X approaches positive Infinity all growth parent functions would approach positive Infinity the end behavior of any parent exponential decay functions would simply be swapped from the growths but where the horizontal ASM tootes are and what the end behavior is are all impacted by the transformations of a function which is an issue for next topic 2.4 is all about different rules to do with exponents and how those rules impact exponential functions so for this video we will learn four properties we'll go in order and start with the first one the product property which says that b to the m mtip b to the N is equal to B to the m + n so what to take from this is that if you multiply two values with an exponent that have the same BAS or B then you are really just adding their exponents to explain this one on a graph we'll start with the parent function y = 2^ x if we change this to Y = 2 ^ x - 1 we see the whole graph shift to the right by one making adding or subtracting anything from the X a horizontal translation of the graph the second property is the power property which says that b to the m to the^ of n is equal to B to the MN meaning if you have double exponents you are simply multiplying the exponents together what this does to a graph is it functions as being the stretch or Shrink value to the graph meaning it is the horizontal dilation students sometimes are confused how to sketch a graph that has dilations my advice is to Simply make a table and graph that information to make it easiest but anyway the next two properties move away from graphs the first one is the negative exponent property which states that b to the N power = 1/ B the N this means that if you ever have a negative exponent it is simply equal to 1 over the original term removing the negative sign on the exponent finally we have the last property which doesn't really have a name so for for our cases we're just going to call it the exponent root property it states the following B to 1 / K is equal to the K root of B this means that if you ever have a power that is one over something it's really asking you to do what the denominator root of the function is so for example this equation is really asking what the third root of 8 is which is two and that is really all for this 2.5 is all about building exponential functions from scenarios now listen this isn't complex you can tell an exponential function is a best fit for a scenario if you see it based on multiplication and you should already know how to build an exponential function from my topic 2.2 3 and four videos and you should know how to model from scenarios from my topic 1.13 video so I'm not going to Simply repeat everything I said there is however a lot of other things from this topic we need to talk about understand that you only need to be given two points or Dare IA input output value pairs to derive an exponential function to fit the model you do this by solving a system of equations which you should have been taught how to do back in algebra but if you still need help with it here is two examples of me solving systems of equations with exponential functions a use the Transformations discussed last video to mess around or tweak exponential functions to what the questions are asking also understand that exponential functions are used and written in different ways to represent interest and compound interest in real life with b as the growth factor and though you don't need to memorize these interest equations for the AP exam they are often required for schools throughout America to teach which is why they're on the screen right now this all leads into e e is a massively long number that we're going to round to 2.718 e is the base of a natural exponential function that is used to to model continuous growth or decay in real life scenarios like continuously compounded interest unlike regular growth factors e is special because it allows us to model processes that change at a rate proportional to their current value e is a complex topic and takes a while to understand but it's not that crucial to understand for the AP exam one thing that is crucial is how to model exponential functions on calculators just as you have all the polinomial regressions you also have the EXP regression you can run to see if your data is exponential so keep that in mind top 2.6 is half about function modeling and half about residuals so really quickly we're going to do an overview of making equations for linear quadratic and exponential function data sets if the data is linear you have the skeleton equation Y = MX plus b you plug in whatever y equals when xal 0 for B and the slope calculated by this formula for M if the data is quadratic you have the skeleton equation Y = ATI x - b^2 + C where you plug in the vertical shift for C the horizontal translation for B and then you plug in a point into the equation and use algebra to solve for a if the data is exponential you have the skeleton equation yal a to the X where you plug in whatever Y is when x equals 0 for a and the rate at which the data is being multiplied by for B and that's it on the calculator portion of the exam you'll likely be given a data set and asked to find the equation that best fits it which of course you would find with a regression once you've determined the regression equation you might be given residuals a residual is just the difference between the actual data point and the value predicted by your model so if your model says the point should be there but the actual data is a little higher or lower the residual is the vertical distance between those two points a model is considered appropriate if the residual plot which is a graph of all the residuals appears without a pattern in other words if the errors are random then your model is a good fit for your data if the residual plot shows a pattern it means your model isn't fully capturing the behavior of the data meaning the goal is to see Randomness the difference between the predicted and actual values is the air in the model sometimes having an overestimate or underestimate might be appropriate for a data set but it all depends on the context toic 2.7 is meant to essentially prepare you for the SAT the reason I say this is because the only place you'll find the things in this topic is in the SAT so if you're given two functions and asked to find F of G ofx or something similar it's pretty simple you just substitute any instance of X within the F ofx function for the G ofx function and solve now for some reason AP precalculus made this topic very long however if you actually read the topic it's all really just common sense like let me give you an example of what it wants me to teach if you're asked to find F of G of X with the F function is f ofx = x then the answer will be G of X obviously like this topic has got to be an example of college board just trying to pad their course and make it look beefy but the thing is I am nice to you guys so there's only really one more thing you should know from this topic and we can get out of here say you had a random function like f ofx = the < TK of 1 + x^2 remember that you can break this down into two functions and the original function would now become the result of G of H of X so keep that in mind also just so you know this is the same as this just don't get confused on the congratulations this video right now symbolizes something special the official halfway point of the course content on the AP exam lucky you I want to take this time to congratulate you and ask you to subscribe to my other non-educational YouTube channel so inverse functions these babies are weird man an inverse is typically notated like instead of f ofx being F to the1 of x to understand inverse functions let's take the parent cubic function y = x cubed we know that the first three points on the graph would be 1 comma 1 2A 8 and 3A 27 the inverse of this function would have these points but simply swapping the X and Y to find the inverse function equation you would take the original function in this case y = x cub and you swap the X and the Y and solve for y and that's the inverse function for a function to have an inverse function it must be one: one meaning each output value is produced by exactly one input value and I don't think I can overstate the importance of that last line enough because on the AP exam you are guaranteed to be asked to explain why a function has or doesn't have an inverse and this definition will give you full points on a graph you know a function is 1: one if it passes the horizontal line test where you put a horizontal line on the graph and if it only intersects the graph once it's 1: one any more times than one and it's not one: one last thing about inverse functions the inverse function and the original function swap domain and ranges so the original's domain becomes the inverse is range and The Originals range becomes the inverses domain on all right it's time I'm not going to downplay this lots of people don't enjoy logs even teachers don't but I promise they aren't as bad as you think you just need to learn how they work if you had the expression 2 the x = 8 you can of course infer that xal 3 but to rearrange this into log form we say log base 2 of 8 the answer to this is three the way we write this with variables is log base B of C is equal to a where of course B to the^ of a is equal to C there are two rules of logarithmic Mythic logarithmic logarithmic logarith there are two rules of logarithmic Expressions they are as follows B has to be positive and B cannot be one and also if you ever see a log with no base or B it is known as a common log and the base is automatically 10 the reason for this is something to do with science or something you have to understand that logarithms bring a new scale to life for example on a standard scale the units might be 0 1 2 and so on while in a logarithmic scale using logarithm base 10 the units might be 10 to the 0 10 the 1 10 2 and so on but the fun thing about logarithms is what it allows you to do going over to a calculator the way to input this here is math Alpha a where you can input a base and the answer to find the power once you learn how to habitually use this it becomes very useful for solving for Powers which you sometimes have to do in exponential functions I mean heck look at the topic 2.5 video's answers and you'll see I used it there now there's also another way to do this with a change of Base formula using common logs but I absolutely despise that way so I will not be teaching it 2.10 introduces log functions now you might be confused because last video we introduced logs but in that video we introduced log Expressions not functions the log function looks like this where of course a cannot equal zero B has to be positive POS and cannot be one if you remember back to inverse functions the way they work is the coordinate points get swapped between the original and the inverse function while log functions are inverse to exponential functions meaning if an exponential function has a point at T comma s the inverse log function of it will have a point at s comma T and let's look at a log graph you'll notice it kind of looks like an exponential graph just flipped around and that's because it is the logarithmic graph is simply a reflection of the corresponding inverse exponential graph over the line Y = X isn't that fun and it also makes this very easy to teach because you should already know exponential functions well from my previous videos in this unit really if you didn't care to understand log graphs all you have to do is find the inverse exponential function and you could understand log graphs that way but don't let me sway your mind logs are pretty easy to understand after all but that's a next video problem okay so we had an introduction to log functions last video now we'll talk even more on them and fully explain them so you have to understand that log functions are not so different from exponential functions the reason for this is because they are inverses of one another this means the same rules of a parent function applied to a log graph where the parent function is log base B of X where B has to be between 0 and 1 or greater than one for all parent functions the domain will be any real number greater than zero because of a vertical ASM toote at x equals 0 the range would be all real numbers also understand a log function is either always increasing or decreasing meaning their graphs will be either only concave up or down and never shift or have any points of inflection with their limited domain logarithmic functions in general form are vertically ASM totic to x equal 0 this means that the end behavior will finally change from what we're used to instead of X now going to positive and negative infinity x will go to positive infinity and zero or whatever the vertical ASM toote is and of course you would need to fill in the Y part too oh and one last thing all the Transformations talked about in the 1.12 video back in unit 1 will always apply to log functions as well so just keep that in mind okay topic 2.2 this is where logs start getting a little hard you learn three properties of this video and an show to natural logs let's go in order and start with the product property this states that log Bas B of X Y is equal to log base B of X Plus log base B of Y meaning if you have two logs of the same base added together it is equal to the log keeping the base of the two terms multiplied by one another this means that the opposite of subtracting two Logs with the same base will equal the log base of the term over one another on the log graph this functions as being the horizontal dilation and if you need help graphing it uh just roll the dang clip students sometimes are confused how to sketch a graph that has dilations my advice is to Simply make a table and graph that information to make it easiest the power property states that log based B of x to the K is equal to K * log base B of X this means that if you ever have a log to the power of something you can just move that power onto the front of the log on the graph this functions as being the vertical dilation and you know what roll the clip again students sometimes are confused how to sketch a graph that has dilations my advice is to Simply make a table and graph that information to make it easiest now you might remember what I said back in time now there's also another way to do this with a change of Base formula using common logs but I absolutely despise that way so I will not be teaching it well here I am being forced to teach it so here it is if you want to take calculus you should definitely know this but for the AP pre-cal exam I promise it is not a big deal if you forget it and finally a little note log base e of X is notated as Ln of X and it is known as a natural log I promise you will see this quite a bit so remember this anyway okay so this is the topic that people start hating logs over I will try my best to be civilized and present this as easily as I can everything you have learned from this unit exponential functions different properties to do with exponents inverse relationships and finally logarithmic function properties all of that culminates into one thing a very intimidating looking question to solve this one specifically was taken from the 2024 AP pre-calculus exam so let's go over it together first I'm going to use the product property on the first two terms then I will use the quotient property which is just the opposite of the product property on the other term now I'll multiply the numerator using the product property then I'll divide using the quotient property which is just the opposite of the product and I'll get my fully simplified answer I guarantee you you will get a question like this on the AP exam remember your properties and you'll be fine I promise and remember the way to solve for inverse functions even when talking about logs is to swap the X and the Y into solve for y also if you ever have two solutions to a log function you need to plug them both back into the original problem because often times you'll find one would be impossible because you can't take a log of a negative number don't overthink This I Promise it's not that bad topic 2.14 is all about modeling logarithmic functions from data so understand this you really only need one or two input output value pairs to derive a log function you do this by entering the points into the skeleton log function and rearranging it and using algebra to solve for b but more than likely you'll have an A to solve for to which in this case you would just have to solve a system of equations like you did back with exponential functions for data you'll know it is modeled by a log function when the X's go up at a proportional rate based on multiplication this might sound familiar because exponential functions have the Y AIS going up proportionally based on multiplication and of course you can use transformations to tweak log functions to conform to real life scenarios given in problems but I promise you won't be given much of that the calculator regression however is always important just as you have the polinomial regressions and the exponential regression you have the Ln regression which is just a log regression you can run to find log function and always remember to save these regressions no matter what they are on the exam to use them to solve and predict later values on the graph and also I feel like since I've been talking about logs I haven't said this enough so remember the skeleton equation for a log function is f ofx = a mtip log base B of X just as how you need to remember the skeleton equation for an exponential function being F ofx = A to the X and that's oh my goodness oh my goodness we are at the end of unit 2 and I hope you are as happy as me because this unit is a real pain luckily we have an easy one to end it on so you've been dealing with a certain coordinate plane your entire life known as the cartisian plane the thing is that there is a lot of other planes in math such as the polar plane which you and me should both be afraid of coming up at the end of this course of course throughout all of these videos we've examined the log graphs on a cartisian plane but there is another plane that is used called the semi log plot this is a plot that either has the x or y AIS logarithmically scaled while the other axis remains linearly scaled if the x-axis is logarithmically scaled you'll notice if you graph a log function it will look linear if the Y AIS is logarithmically scaled you'll notice if we graph an exponential function it will suddenly look linear this type of plot can be used to confirm if a function is exponential or logarithmic so definitely remember which axes correspond to each function so I hope you're now prepared because next unit is a total shift on everything you once knew on math get ready thank you so much for watching Remember the Salt Lake City video in the description it's so tempting anyway I'll see you in unit 3 bye-bye [Music] [Music]