hey there i know the title says full review but we aren't going to include unit 4 only because it's not required on the AP exam this video clips together all of my topic review videos for this course not including the AP style questions at the end of each video if you'd like to try your hand at some AP style questions associated with each topic of this course my full AP pre-calculus review playlist will be in the description let's go topic 1.1 is titled change in tandem it deals with input and output values a function is a mathematical relation where there is a set of input x's and output yv values such that each input value is mapped to one exact output value input values are the domain of a function and the independent variable output values are the range and the dependent variable a function is increasing if as the input values increase the output values always increase a function is decreasing if as the input values increase the output values always decrease moving over to graphs the graph of a function displays a set of input output pairs that shows how the values of the function's input and output values vary rate of change is a graph's slope a graph is concave up when the rate of change is increasing or when it looks like a u a graph is concave down when the rate of change is decreasing and it looks like an upside down u whenever a graph intersects the x-axis the output value or y value is zero meaning the corresponding x's or input values are said to be the zeros of the function the equation to know here is y = mx plus b where x is the input y is the output m is the slope or rate of change and b is the y intercept or whenever the graph crosses the y-axis topic 1.2 is all about rate of changes first how to find them and then how to describe them rate of change is simply a graph's slope the common slope equation you are given back in algebra will suffice to find it y2 - y1 over x2 - x1 if you are given two points on a graph and are asked to find the average rate of change between them you write the two points down label them accordingly so x1 y1 x2 y2 then you put them in the equation solve the equation and you get your average rate of change for those two points furthermore if you are asked to use that average rate of change just calculated to predict points on the graph simply multiply the value they give you in the word problem by the average rate of change and you got it and it's that easy remember that a positive rate of change means that as one quantity increases or decreases the other quantity does the same a negative rate of change means that as one quantity increases the other decreases topic 1.3 is coincidentally also about rate of change so essentially it's very similar to my last video this video is going to explain a few more things about rate of change and that'll be it so we've been twiddling with this idea of average rate of change over specific intervals in a linear function the rate of change will always be the same no matter what interval you assign to it this means that the average rate of change is changing at a rate of zero on the other side in quadratic functions the average rate of change is changing at a linear rate the thing with quadratic functions is that if you ever find the average rate of change over any interval you'll simply be calculating the slope of the seeant line created in that interval this means that the rate of change will never be accurate to the actual quadratic function if you were asked to calculate the average rate of change at a specific point like in this equation where it says to do it at x= 5 simply find what y equals at 5 and what y equals at a point right next to 5 so we'll go with 5.001 label them then plug them into the equation y2 - y1 over x2 - x1 and solve if you model this on a graph you would be finding the slope of the tangent line between these two points 01 apart topic 1.4 4 is all about what makes a polomial function along with some helpful terminology on the screen now is the technical way that a polomial function is written now though this may look intimidating understand that a subn is the coefficient and n is the degree and also notice that throughout the equations the degree keeps getting smaller just remember that polomials can't have a negative degree imaginary coefficients or any division within its equation you've been working with polomials and solving them throughout your entire schooling days the only polomials you'll see in this course will be the main four linear quadratic cubic and quartic in a later video we will go into more details on these now let's get into some terminology a local or relative maximum or minimum is a maximum or minimum of the function of all local maximums the greatest is called the global or absolute maximum likewise the least of all local minimum is called the global or absolute minimum and for those wondering if the function goes to infinity or negative infinity those don't count as maximums or minimums polinomials with an even degree will always have a global or absolute maximum or minimum between two zeros of a polomial function there will always be a local maximum or minimum points of inflection of a polomial function occur at input values where the rate of change of the function changes from increasing to decreasing or from decreasing to increasing or simply points of inflection are where the concavity of the graph changes topic 1.5 is all about polomials multiplicity degree and even/ odd this topic is actually very easy to understand first let's talk on zeros zeros of a function are whatever x equals when y equals 0 zeros can also be called x intercepts or roots the thing about zeros is they can be real or imaginary if a zero is actually shown on a graph you know it's real some zeros need to be solved with the imaginary number i which is equal to the square root of -1 any solution with i in it is imaginary another word for real zeros are linear factors and another word for imaginary zeros is complex zeros now is the time I tell you some helpful tricks the degree of a function is simply the highest value of exponent so for example the degree of this equation is five the way to calculate this from a table would be to calculate the successive differences from the x side until the numbers are all the same and then calculate how many times you had to do the differences now whatever degree the equation has will also be how many zeros either real or imaginary a function has so once again using this problem we would have five zeros another helpful thing to know is that if you have any solution in an equation that goes in the form a + b i then the conjugate a minus b i is also a zero moving on to graphs say you have an equation in intercept form here's another trick calculating the zeros is easy simply set each parenthesy equal to zero and solve each one but then if you graph the zeros on the graph how do you know how the line moves here's the trick I was talking about now we look at the exponent next to the parenthesis if the exponent is odd then that zero is said to have odd multiplicity and therefore the line on the graph would pass through it if the exponent is even the zero is said to have even multiplicity and therefore the line will bounce off of the zero instead of going directly through it okay final thing a function is even if it satisfies the property f ofx= f ofx or if reflected across the y-axis the graph looks the exact same a function is odd if it satisfies the property f ofx= f ofx or if rotated across the origin 180° it remains looking the exact same and today we have an easy one topic 1.6 is all about n behavior and polomial functions first the notation then how to find them so let's start with talking about what end behavior is end behavior is exactly what it sounds like it describes how the graph ends on both the left and right side the notation we use for this is called limit notation let's start with this graph first we need to see where x ends you see on the right x increases without bound and on the left x decreases without bound this means x will be going to positive and negative infinity we write that in limit notation like so then we need to examine what y is doing when x is doing these things so we need to write on both of them f ofx equals to symbolize that now what is f ofx or y doing let's start with when x approaches infinity you see that y is also increasing without bound therefore it is approaching positive infinity now what is f ofx or y doing when x approaches negative infinity you'll see y increases without bound toward positive infinity this means that our answers would be positive infinity and this right here is the end behavior for this graph and real quick before we move on to how to find this in equations just so you know if the end behavior has a minus in the exponent of it it means the end behavior as it comes in from the left of the graph and a plus means as it comes in from the right side of the graph now let me teach you a trick to find in behavior just from an equation all you need to find is the degree of the equation and whether or not the leading coefficient is positive or negative then just reference the table below the fact is you will always have at the start of your end behavior equations the limit as x approaches infinity and the limit as x approaches negative infinity this table shows what y would equal under the different scenarios for example in this equation the degree is odd but the leading coefficient is negative referencing the table I just plug in the y values from it so my answers would be the limit as x approaches positive infinity of f ofx would equal negative infinity and the limit as x approaches negative infinity of f ofx would equal positive infinity memorize this table and you will be all set to go now we transition to talking about rational functions so what are rational functions answer is that it is two polomials over one another this division causes new stuff to appear with the most notable being vertical and horizontal asmmptotes which are invisible lines that values of the graph will continue to get close to but will never touch this video will just simply be how to calculate end behavior in rational functions and a little bit about horizontal asmmptotes you already should know how to do this from a graph from my last video in rational functions you again only have to worry about what y does when x approaches positive and negative infinity because you already know how to do this from a graph let's talk about how to do this from a rational function equation here are your three rules if there is a higher degree in the denominator compared to the numerator the function is said to be bottom heavy the limit as x approaches negative or positive infinity of f ofx will be zero this is because there will be a horizontal asmtote at y equals 0 if between the numerator and the denominator they have the same degree it is said to be same heavy the limit as x approaches positive or negative infinity of f ofx will be the ratio of the leading coefficient over the leading coefficient this will also be the horizontal asmtote of the graph if there is a higher degree in the numerator the function is said to be topheavy the limit as x approaches positive infinity of f ofx will be positive infinity the limit as x approaches negative infinity of f ofx would be negative infinity or positive infinity if the degree is odd however if the leading coefficient is negative this rule would be swapped for topheavy functions there will be no horizontal asmmptotes rather slant or oblique asmmptotes to solve for these simply use polomial long division to divide these two functions and the quotient is the slant or oblique asmtote 1.8 is all about how to find real zeros of a rational function now you might look at a rational function and see the division and think this would be really hard but it really isn't to solve for the real zeros of a rational function simply set the numerator of the fraction equal to zero and solve the only new thing you need to worry about with rational functions is the denominator you must also set the denominator equal to zero and solve for any zeros that match between the numerator and denominator it is not a zero it is a whole any remaining zeros that don't match in the numerator are real zeros of the function now technically for this course I don't need to teach you how to solve zeros of polomials because you should already know how to do it on the screen now are completed problems solving polomial functions for zeros but that is all I am obligated to offer for this topic and this course in general topic 1.9 is about how to find vertical asmmptotes and rational functions and this is actually really simple to find vertical asmmptotes on a rational function set both the numerator and denominator equal to zero after you've solved cross out any zeros that match between the numerator and the denominator as those are holes then any zeros remaining left over are your vertical asmmptotes but better question what is an asmtote this is an invisible line that the graph will keep trying to approach but will never reach it all the way to positive or negative infinity you'll be able to tell where the vertical asmmptotes are because of what the lines on the graph seem to approach but will never reach and since nothing on the graph can be placed within the range of the vertical asmtote the limit notation approaching the vertical asmmptotes from the left or right in the parent rational function would be positive infinity and negative infinity you can also do this with horizontal asmmptotes now one today topic 1.10 is all about holes and rational functions a hole occurs at a graph when in the equation once you solve the numerator denominator you get a common factor between the two this hole on the graph is indicated with an open circle instead of a closed one a hole is just a point on the graph where nothing exists if we say that a hole exists at the point C comma L the limit as X approaches C of the function would equal L because nothing would exist in that hole this would also affect the domain and range of the function but anyway a video that is going to be hard to keep under 3 minutes there's so much random stuff packed into this topic taught at once and a lot of it is just repeated from other topics three things you need to learn in this video first is that sometimes you might be given the roots of a function and told to build a function all you need to do is put them in parenthesis and multiply each factor by one another okay that's number one number two is polinomial long division sometimes to shrink a polinomial down when it has too many values in its expression we use polinomial long division your goal in polinomial long division is to always make it so whatever you put on top multiplies by the divisor to get whatever the first degree of the dividend is then you subtract which cancels out the first degree and you rinse and repeat this until you're left with either zero or a remainder which you would add on to the end of the quotient over the divisor so your rule with polinomial long division is to multiply then subtract and rinse and repeat till you were done okay that is part two the last thing we need to talk about is binomial theorem using Pascal's triangle say you had an expression like x + 5 to the 5th now you could foil forever and figure out the answer to this but Pascal's triangle is a shortcut to figuring it out pascal's triangle is a diagram of numbers where each number is the sum of the numbers above it adding powers to Pascal's triangle you can see the pattern of exponents going up and down in this triangle consider the f term referring to the factor and whatever factors in the parentheses we plug into the f spot and of course the fifth row because it's the fifth power then solve the exponent and multiply each term and you get the answer this is the shortcut way using the binomial theorem and throughout this entire course you'll find that any function we talk about except for maybe polar functions these transformations throughout this video will remain transformations can split into two groups additive and multiplicative let's start with the additive transformations g ofx= f ofx plus k is known as a vertical translation and it will move the graph either up or down g ofx= f ofx + k is known as a horizontal translation that moves the graph of f by negative k units either left or right so if you see like x - 5 in the parenthesis you know the graph goes 5 units to the right because it's always the opposite now for the multiplicative transformations g ofx= a f ofx when a is not zero is a vertical dilation of the f graph the higher the number gets the more closed the graph will get the lower the number the more open it gets if A is negative it will result in a reflection of the entire graph over the Xaxis g ofX equals F ofXB when B doesn't equal zero is a horizontal dilation of F by a factor of 1 / B if B is negative it will result in a reflection over the Y-axis now of course all of these transformations are not just exclusively used by themselves many of them are combined and used in what equation this would of course have an impact on the domain and range as well you should practice noticing transformations both in equations and on graphs and for that I put some help in this 1.13 and 1.14 are practically the same 1.13 is about predicting function models by context clues while 1.14 tells you how to actually construct these functions so I should preface this all by saying as someone who took this exam this most recent year you will get a lot of questions asking you to fit a function to a set of data or a word problem the real best way to know what function to use is to use your calculator however let's say you were given this on a no calculator problem let's discuss some tips a linear function will model data sets or aspects of contextual scenarios that have a constant rate of change quadratic functions will always have the rate of change shifting but it's typically associated with a function that has one distinct minimum or maximum geometric contexts involving area or two dimensions can often be modeled by quadratic functions geometric contexts that involve volume or three dimensions can often be modeled with cubic functions a peace-wise function is really just multiple functions on one graph knowing this is useful for modeling a data set or contextual scenario that demonstrates different characteristics over different intervals but just know the best way to figure out what model matches a scenario is a graph graph the information they give you and see what function it models moving on a problem might have underlying assumptions on how quantities change together or maybe what is consistent in the model what I'm essentially saying here is to read the entire question and make sure you understand each part of what it is asking and saying also remember to model each answer to the real world scenario given in the problem like for example you can't have.5 people in a room or.7 printers because of this some domain and range restrictions might exist in an equation 1.14 serves as a kind of part two to topic 1.13 so please watch that video first if you haven't already so you get the idea how to predict function models from scenarios now let's understand the ironclad way to do this and that's with the calculator so on your calculator you'll have your stat button and then you can press edit now what comes up is tables essentially so what you do in here is you input a set of data so for our purposes we'll just input a linear set of data so and then we head over to L2 by pressing to the right now you see we have a linear set of data here now of course we know it's linear but what if we didn't know what it was and we had to use the calculator to find out what the data is we go to stat then calc you'll see that we have a bunch of regressions that we can run here the only ones you need to know for this unit are linear quadratic cubic and quartic so essentially you run these regressions by just pressing enter and pressing enter on this specific calculator program that I'm using it doesn't provide an RV value but essentially there will be an R value right about here and what it will say is that like R will equal a certain number and on all regressions you want to run to make sure that R value gets closest to one whichever regression has R value that's closest to one is the best fitting model for the set of data that you have then say you have the regression that you need we know it's a linear regression so we want to store this function so we go varss y bars function and then we can store it in any of these i'll just do y1 and then of course we see in y1 we now have the function stored so now we can see stuff like we can go to the graph see what the graph looks like and we can do like uh well we can see value like for example we want to know what y equals when x= 3 well y equals 9 so yeah and listen I'm glossing over a lot of calculator things and that is because your teacher should be teaching you how to use a calculator in this course but if you still need help I'll link videos in the description on different calculator mechanics you need to know for this course moving on rational functions can also be modeled with calculators but only if you manually put in the equation yourself so if you click on your y equals you'll see you have the equation here you can just put in a rational function equation so we go y equals we have the fraction that comes up and then we can just do let's do the parent function of a rational function y overx or sorry 1 /x graph that is a rational function and the same thing you can do with rational functions like if you needed to find what x equals when y equals I don't know let's say or sorry what y equals when x equals let's say 3 then we know that y equals 1/3 and stuff like that in this course you won't need to really worry about putting rational functions in from scenarios but it will be nice to have it for the calculator portion of the multiple choice part of the test and as we shift from unit one to two I'd like to say that if you've been helped by any of my videos this semester please consider pressing the join button and becoming a channel member all right exponential and log functions let's go 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 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 subn= a sub0 plus dn where a sub of n is the value of the term you are finding out a sub0 is the first term in the sequence d is the common difference and n is the position number that you are 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 subn equals a sub of k plus d multiplied by n minus k where a sub of 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 know what do I geometric sequences have two equations first is g subn equals g sub0 * r to the n power where g subn is the term you are finding and g sub0 is the first term of the sequence r is the common ratio of 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 subn equals g subk multiplied by r the power of n minus k where j ofk is the term you know and k is the position number of the term you know 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 subn= a 0 plus dn is really just the equation y b + mx which is the linear function equation and the other equation a subn= a of k + d multiplied by n minus k could be expressed as another way of writing linear equations being f(x)= y + multiplied x - x i where you include the point x i 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 the^ 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 ^ x - x i where it includes the point x i 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 problem project 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= a b ^ 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 one for a growth or between 0 and one for a decay so that means that any value you plug into b that is greater than one or between 1 and 0 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 1 the second thing we see is that there is a horizontal asmtote 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 asmmptotes 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 multiplied 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 base 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 ^ of 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 power 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 to 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 our cases we're just going to call it the exponent root property it states the following b to the 1 / k is equal to the kth root of b this means that if you ever have a power that is 1 over something it's really asking you to do what the denominator's root of the function is so for example this equation is really asking what the third root of 8 is which is 2 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 I say 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 and 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 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 polomial regressions you also have the exp regression you can run to see if your data is exponential so keep that in mind topic 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 x= 0 for b and the slope calculated by this formula for m if the data is quadratic you have the skeleton equation y= a multiplied 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 y= a b to the x where you plug in whatever y is when x equals z 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 topic 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 pre-calculus 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 ofx where 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 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 the -1 ofx 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 1a 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 cubed 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 to 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's range and the original's range becomes the inverse's domain honestly 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 2x= 8 you can of course infer that x= 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 power of a is equal to c there are two rules of logarithmic logarithmic logarithmic logarithm 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 light for example on a standard scale the units might be 0 1 2 and so on while on a logarithmic scale using logarithm base 10 the units might be 10 to the 0 10 the 1 10 the 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 videos answers and you'll see I use it there now there's also another way to do this with the 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 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 well 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 comm 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 equals 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 apply 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 asmtote at x= 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 asmtoic to xals 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 asmtote 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 2.12 this is where logs start getting a little hard you will learn three properties of this video and an intro to natural logs let's go in order and start with the product property this states that log base b of xy 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 base 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 the change of base formula using commonal 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-calc 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 and to 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 too 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 yaxis 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 polomial regressions and the exponential regression you have the ln regression which is just a log regression you can run to find log functions 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 multiplied by log base b of x just as how you need to remember the skeleton equation for an exponential function being f ofx= a b 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 yaxis 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 yaxis 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 and as we shift from unit two to three please consider following my Instagram now where you can find just the greatest content all right trigonometric and polar functions let's go so unit 3 is here and with it brings a fun spectacle of severely long topics of great difficulty i mean that isn't going to stop me from making each topic in under 3 minutes but still this topic is a nice and easy start to this unit talking all about periods and math so for a graph to be periodic it has to have a continuous cycle of the same pattern happen over and over over equal length intervals this unit staple is s and cosine graphs which are by all accounts periodic graphs because they continue going up and down all the way to positive and negative infinity this means the graphs are continuously repeating a cycle of the same pattern a period is how long it takes a graph to complete one cycle the way to calculate this on a graph is to find the horizontal distance between two maximum points next to one another or calculate the horizontal distance between two minimum points next to each other so for example on the standard sign graph the period would be 2 pi this is because in the length of 2 pi no matter where you look on the graph it is completing one full cycle of its pattern but don't be fooled graphs don't look all perfect and nice like this all the time some period graphs can look like this it's still a periodic graph though because though the pattern might be strange it still repeats the same pattern over equal length intervals and if you were given a period length of a periodic graph and asked to continue drawing the graph you can do this because you already know the pattern that needs to be repeated just remember that a period is a full cycle of a graph's pattern and you'll be fine so trigonometry here we go trig kind of has to start set up here with a circle putting a graph into the circle we can start at the angle 0° then 90 180 270 and 360 in this course we will be working with a new angle measure known as radians the entire circle is equal to 2 pi radians meaning 180° is pi 90° is p /2 and 270° is 3 p /2 this would mean the conversion of degrees to radians would be this now that we have the start setup the trig idea goes something like this if you make an angle in the circle you are creating a terminal side equal to the radius of the circle or r you have the initial side which is just the x-axis and an angle called theta the terminal ray will keep going forever but where it intersects the circle it makes a point we call that point point P the coordinates of point P are X comma Y a unit circle is not complex it simply means the same exact circle when the radius is equal to 1 now if we introduce s cosine and tangent we see they also function differently on the circle s deals with a vertical displacement which is y / the radius r in the unit circle the radius is one making s the y-coordinate cosine deals with the horizontal displacement which is x / the radius r again in the unit circle since the radius is one it would make cosine the x coordinate tangent deals with the slope meaning rise over run or y /x in the unit circle it would mean tangent is equal to s over cosine quadrantal angles means one of the big four angles or any angle that has a multiple of 90 it is pretty simple to solve the sign cosine and tangent of these quadrantal angles first you make coordinates for each one since the radius is 1 we simply make coordinates for each one pretending like it's a graph that only stretches out by one so 1 comma 0 comma 1 comma 0 and 0 comma 1 remember these coordinates are x comma y or cosine comma s meaning if you were asked to find s of let's say 180° or p radians it would simply be zero or let's say cosine of 270° or 3 p over2 radians it would be zero if you were asked to find tangent of a quadrantal angle so let's say 90° or p over2 radians we can do this because we know tangent is just s x which would be at p /2 radians 1 / 0 which is undefined because you can't divide anything by 0 of course if the radius was something else like three you would just simply change the quadrantal coordinate points to be 3 instead of 1 okay so the trig circle it's cool and all but let's make it disappear think of it this way of course remember the angle measures but just think of a graph we already know how to find s cosine and tangent of quadrantal angles but what if we have something like s of 30° let's draw a line draw the 30° angle but this time we draw a line to make a triangle just so you know no matter where you draw a line it needs to go to the x-axis then since we know one angle is 30 and the other is 90 this would make a 30 60 90 triangle so filling in the sides knowing the 30 609 triangle we know sign is opposite over hypotenuse meaning 1 /2 meaning s of 30° is 1 /2 and if we had something like cosine of 45° then we can construct the 45 459 triangle and we know cosine is adjacent over hypotenuse which in this case is 1 over 2 thing is you can't have a square root in the denominator so you must rationalize it by multiplying both the numerator and denominator by 2 and solving this gets us 2 /2 as our answer to cosine of 45° or p 4 radians and if you wanted to find say tangent of 60° or p over 3 radians you construct your 30 60 90 triangle we know tangent is opposite over adjacent which in this case is roo3 over 1 or roo3 so to spare this video from being over three minutes if we were to do this anywhere on the unit circle where any special right triangles are possible we can get the parentheses answers here that show the answers in the format cosine comma s to each of the angles and of course if you wanted to find tangent it's just s cosine you need to memorize the circle to make this easiest for you all right a handful of other things we need to talk about if you were given an inverse sign cosine or tangent equation it's not that difficult so for example in this one where it's inverse sign of roo3 over2 all it's asking is what angle will have roo3 over2 as an answer for the sign of it so looking at our unit circle we see it in two places thing is dealing with inverse functions certain inverse functions only exist in certain quadrants of the graph because of domain restrictions since inverse sign can only exist in quadrants 1 and 4 even if there is an answer of roo3 over2 at 120° it can't be an answer because inverse sign can't exist there meaning our only answer would be 60° or p over 3 another way to notate inverse function is using arc notation just know it means the same exact thing angles can also be negative by going clockwise instead of counterclockwise so 30° is equal to -3° you would still solve the s cosine and tangent of them the same way these equations know them here's why if you were given a question like this you were asked to find the x and y you could just plug in r and theta to get these answers or in this question where you're asked to find r and theta you find r by using this formula and theta through algebra okay we figured out how to solve all these sign and cosine values on the unit circle from the last few videos let's start by taking all the sign values if we were to plot all of these on a graph it would look like this you'll see the sign graph is a curve that keeps going up and down all the way to positive and negative infinity and since it does go to positive and negative infinity it makes all sign functions have a domain of all real numbers now if there are no transformations on a sign graph it will typically go on the y-axis from 1 to 1 this means the range of any sign graph would be whatever the minimum to maximum is put into brackets now let's graph cosine and all of its values you see that the graph looks practically the same to the sign graph that is because it basically is the cosine graph is a sign graph with a horizontal translation of minus p /2 so that means that all the rules of a sign graph apply to a cosine graph the domain is all real numbers and the range is minimum to maximum in brackets the period of a s and cosine graph would be 2 pi and to write out the basic s and cosine equation you would say something like f of theta is equal to s of theta or f of theta is equal to cosine of theta theta in this case would just be an angle measure so it would function the same as something like say X or Z sinosodal functions congrats this is the second to last function you need to know for this course a syosoidal function by definition is a periodic function that continuously oscillates between a set minimum and maximum point the syosoidal functions in this course will just be the basic cosine and s functions the actual details of the equation will be covered next video so let's talk about the specificities for the graph now first of all as a review from my last video cosine of theta is equal to s of theta plus p / 2 this is [Music] because of course you know how to calculate periods from the 3.1 video and in the parent s and cosine functions it is 2 pi the frequency is simply whatever the period is with a one on top of it a middle line of a sinosoidal function is the invisible line that's halfway between the minimum and maximum points the middle line of the parent sign and cosine graphs is y equals z the amplitude of a sinosoidal function is the vertical distance from the middle line to the maximum point on the graph that means in the parent sign and cosine graphs the amplitude is one you'll see that on a s and cosine graph the concavity changes at wherever the line passes the midline and as a reminder on concavity just memorize this image here okay one more thing to talk about and to understand it let's go back to my unit one topic 5 video where I sounded truly dead inside a function is even if it satisfies the property f ofx= f ofx or if reflected across the y-axis the graph looks the exact same a function is odd if it satisfies the property f ofx= f ofx or if rotated across the origin 180° it remains looking the exact same if we were to apply these properties to the s and cosine functions or try the rotational and reflective symmetry on the graphs you will see that y= s of theta is an odd function and y= cosine of theta is an even function okay manipulation a tool that can get you really anything you want in life and of course I'm talking about this in the context of sinosoidal functions i mean what else could it mean this video serves as a continuation of my topic 3.5 video but instead of asking you to rewatch it I'll just reuse some clips from the video in this one and since I said it here you're not allowed to get mad at me take a look at this equation this is the skeleton sinoso equation i want you to know this same skeleton equation is used for s and cosine functions let's examine it from left to right a is the amplitude of the function and roll the clip the amplitude of a sinosoidal function is the vertical distance from the middle line to the maximum point on the graph that means in the parent sign and cosine graphs the amplitude is 1 b is a little strange b is equal to 2 pi over the period and remember whatever the period is will show how long it takes for the graph to complete one cycle of its pattern c is the horizontal shift or as it's known in the trig world as the phase shift just like any horizontal translation it moves the graph by negative C units left or right so negative p /2 would move the graph p /2 units to the right because it's always the opposite and finally D is your vertical shift but you know it better as the middle line and if you want to know what that is roll the clip a middle line of a sinosoidal function is the invisible line that's halfway between the minimum and maximum points the middle line of the parent sign and cosine graphs is y equals z i should also mention d can be on the front of the equation as well and mean the exact same thing okay so let's practice really quick take a look at this graph to construct an equation from it first we find the midline which is three then find the amplitude by finding the distance from the midline to the maximum point and it's two then we find the period which is pi and put it under 2 pi which is reduced to two then we take either the sign or cosine parent graph and see what the phase shift would be to follow the pattern we see for cosine it wouldn't have a phase shift but for sign the phase shift would be plus pi over 4 this topic is very important in other topics of this course we might say oh yeah this will probably be on your AP exam no this topic is literally guaranteed to be on your exam it is written in the exam description that this is a literal requirement so listen up for this topic we're going to take an actual question from the 2024 AP exam and I urge you to pause the video now and try it on your own if you don't want to try it at least pause to read it because I'm not going to read it to you all right all good let's go so the way I like doing this is to make a graph first and then the equation when you make a graph from one of these scenarios you will always want to make five points and I'll show you why so when the tire touches the ground it will be our start point at 1 /2 then it touches the ground again at 5 over2 so let's write down the three points between those two to get up to our fivepoint total good now let's plot the y-axis the maximum is 18 making the midline 9 at 1 /2 the tire is on the ground or at zero and it is also at zero at 5 /2 so we just fill in the extra points from there the way these points are modeled imagines the tire spinning in a circle great so they give us the sign skeleton equation in the problem so let's make it the amplitude or A is 9 the period is two so we can put that under 2 pi to get pi as our B the midline or D is 9 and finally C here's a trick for C cosine graphs start at the maximum and go down to the midline sign graphs start at the midline and go up to the maximum so where do we see on the middle line it going up to our maximum that's right at 2 over two or one meaning C would be minus one because it's always the opposite and this is our fully simplified answer all right one more thing to talk about calculators if you look through your regressions page you will find that you also have a sign regression which is cool but here's the most important part your calculator functions on modes particularly radians and degrees if I were to do say sine of 90 when I was in degrees mode it would be doing s of 90° but if I were in radians mode I would need to change it to be s of p /2 radians since you only will be dealing with radians in this course you should always have your calculator be in radians mode wow that was a lot of info let's take a quick break if you're not following my TikTok yet you seriously are missing out on absolute masterpieces such as my AP pre-calculus unit 1 video has more views than Sabrina Carpenters's video oh my goodness this is very fair don't judge me just truly marvelous now that we're refreshed let's jump into the second half so you might be wondering why did we start this unit solving s cosine and tangent but never talk about the tangent graph well my friends it's because tangent is a little strange of a graph let's look at the unit circle but only with tangent solutions we get these solutions off of the original unit circle by knowing tangent is equal to s over cosine you notice tangent is not defined at p / 2 and 3 p /2 that simply means tangent cannot exist at those two points which would make vertical asmmptotes on the graph if you were to graph all of the tangent points on the graph it would look like this look at all those vertical asmmptotes you notice that the graph goes to positive and negative infinity and looks real fun this is the parent tangent graph f of theta is equal to tangent of theta the period is not 2 pi but rather now pi the way you find it is to find the horizontal distance from x intercept to x intercept the skeleton equation is basically the same from sinosoidal functions d is your midline you'll notice the graph changes from concave down to concave up once it passes the midline c is your phase shift as for B since the parent function has a period now that is pi B is now equal to pi over the period for only tangent functions and now that gets us to A you'll notice on the tangent graph that it goes to positive and negative infinity since the amplitude is the distance from the midline to the maximum what the heck is the amplitude now well here it is on the horizontal axis look between two X intercepts then divide that quadrant by four and find what the point is here it is one the amplitude is whatever the distance between your midline and the Y point here is so in this example it is 1 i hope that makes sense look from x intercept to x intercept divide by 4 then find the y-coordinate of that point find the distance between the middle line and that y point and that's your amplitude and though I of course have no knowledge on what might be on your exam this year when I took the AP pre-calculus exam there was not a single question on the tangent graph so you know it might not be that important to know everyone I honestly forgot this topic existed the reason I say this is because I thought I needed to teach this earlier so I taught it in my 3.3 video i do still have some more things to say on inverse functions but for now let's get a reminder on what I said back in my 3.3 video if you are given an inverse sign cosine or tangent equation it's not that difficult so for example in this one where it's inverse sign of roo3 over2 all it's asking is what angle will have roo3 over2 as an answer for the sign of it so looking at our unit circle we see it in two places thing is dealing with inverse functions certain inverse functions only exist in certain quadrants of the graph because of domain restrictions since inverse sign can only exist in quadrants 1 and 4 even if there is an answer of roo3 over2 at 120° it can't be an answer because inverse sign can't exist there meaning our only answer would be 60° or pi over 3 another way to notate inverse function is using arc notation just know it means the same exact thing so what the heck do I mean when I say certain inverse functions only exist in certain quadrants of the graph essentially because the corresponding trigonometric functions are periodic they are only invertible if they have restricted domains the restricted domain of s is /2 to p /2 or the way I like looking at it quadrant 1 and 4 the restricted domain of cosine is 0 to pi or the way I like looking at it quadrant 1 and 2 and the restricted domain of tangent is over2 to p /2 or once again quadrants 1 and 4 now I don't want to confuse you if you see an inverse function you can assume it will always be solved within their restricted domains but if you see a trigonometric equation like this you no longer need to worry about restricted domain you can solve it and put both of the solutions p over 3 and 5 over 3 but since these functions are periodic you need to model all the periodic solutions to the trigonometric equation as well so that means when you solve trigonometric equations that don't have any domain you must slap a plus 2 pi k on the end of each solution to showcase all possible co-terminal angles you will notice that I actually already did this in my topic 3.3 video so your takeaway is to always use restricted domain if you are solving inverse functions but find every possible solution with trig equations like this okay okay hey there before you watch this video you need to make sure that you've watched my topic 3.3 and 3.9 video and as always my unit 3 playlist is in the description below the link for you to subscribe to my second channel okay take a look at this problem 2 multiplied by cosine of x + 1 is equal to 0 here's my advice get rid of the cosine and add it back later so now it's just 2x + 1= 0 so 2x=1 and x =1 /2 great now that we've solved for x we can add back the cosine so cosine of x is equal to -1 /2 that would be the same thing as saying r cosine of 1 over2 which looking at our unit circle it's at 2 over 3 and 4 over 3 and since we don't have any domain for the problem we need to slap the plus 2 pi k on it and then this is our fully simplified answer let's try this problem it's a calculator problem tangent of x - tangent of x - 6 is equal to 0 let's remove the tangent making it x^2 - x - 6 = 0 factoring it we get x - 3 and x + 2 so we get x = 3 and x= -2 so then we add the tangent back and we get these arc equations these aren't traditional values on the unit circle so this is where we plug it in on our calculator in radian mode and we get x= 1.249 and x=1.107 and then we slap the plus 2 pi k on it and we are done hey wo wo wo it's not 2 pi k tangent has a period of pi so for this it would actually be plus pi k it really isn't that bad for these problems i promise let me introduce you to an evil word a word that should always be hated when in the context of math identity an identity is something that equals something else next video is all about proving identities and it is so long so I need to catch a head start in this video so let's talk about the start of these being the reciprocal identities this video introduces three functions cosecant seeant and cootangent cosecant is equal to 1 / s seeant is 1 / cosine and cotangent is equal to 1 / tangent this also is true for the opposite so 1 / cosecant is equal to s 1 / secant is equal to cosine and 1 / cent is equal to tangent oh and another thing since tangent is equal to s over cosine is equal to cosine over s so if you were asked to find the secant cosecant or cangent of an angle you just find the reciprocal's answer to that angle and then put a one on top of it and solve now let's talk about the graphs for these functions understand that for things like exponential graphs log graphs and s and cosine graphs you need to understand kind of every little thing about it but the tangent graph and the graphs in this video are just kind of something you need to get a vague idea of what they look like okay okay let's start with seeant the graph looks weird it's a bunch of U shapes but watch this if we overlay the reciprocal cosine graph it suddenly makes sense each extremum from the cosine graph matches with the seeant graph now for cosecant it follows the same pattern if we overlay the sign graph we see that the cosecant U shapes match with the sign minimum/maximum points as for co-angent co-angent is actually really easy remember this is what a tangent graph looks like the co-angent graph looks like this mhm yeah it's just the tangent graph but the lines go the other way and the asmtotes shift just a little just kind of understand the vague idea of what these graphs look like and you'll be fine my friends I'd like to preface this video by already apologizing for the content in this video but I just need to get going so we already know the reciprocal identities from last video now we need to add another set of identities known as the Pythagorean identities all of these identities can be used to prove an identity true when you see an identity like this and it asks you to rewrite it in this case in terms of tangent let me show you how you do it also this question was taken from the 2024 AP exam so the truth is there is no right answer on what to do you could do things completely different for me and get the same exact answer i'll just show you how I'd do it so first I want to try and simplify each term in the numerator we have 1 - sin^ square of x but here's the thing we know sin^ square of x + cosine^ square of x is equal to 1 so if we rearrange this we know 1 - sin^ square of x is equal to cosine^ square of x from the reciprocal identities we know is equal to 1 / cosine now we can multiply the two fractions together we are able to cross out the cosine on the bottom and the square root on the top and that leaves us with cosine of x over sin of x which is equal to x but we need to simplify it in terms of tangent as the question asked so we know cotangent of x is equal to 1 / tangent of x so this is our final answer a trig identity like this is guaranteed to show up on your exam so at the end of this video there's a handful of them for you to try but wait there are more identities to talk about my friends we have something called sum and difference identities there are also quite a bit of them but for this course you only need to know the sum identity for cosine and s and none of the other ones you also need to memorize these double angle identities so that means in total you need to memorize this screen of identities for this exam and if you're wondering this is why people say pre-calculus is harder than calcul but I know you guys are lazy so I'll just say as someone who took this exam these ones don't really matter but definitely know these really quick let me show you how to use the sum identities if you had an expression like s of 75° we can rewrite this as s of 45° plus 30° and using the identity we can plug it into it then simplify and we get our fully simplified answer and as for the double angle identities they really aren't that important here are some questions to try if you really care about them so much and here are the answers but I wouldn't really care too much for them all right my friends here's your full identity screen again memorize it good good oh my goodness it's so close the end is near just a few more videos and we can finally be over with this course welcome to the last function you need to learn in this course polar functions here's how I want you to think of polar functions think of the unit circle earlier in this unit where we had angles on a graph that's essentially a polar graph now imagine a bunch of invisible circles on the graph those are your points polar functions don't use the xycoordinates they instead use r comma theta coordinates so if we needed to have a point at 2 p over4 we go to where p over4 is on the graph then we go out to the second circle and plot our point isn't that fun if we had like 3a 4 p over 3 we go to 4 p over 3 and then three circles out and plot our point but what if we had something like -2a 5 over4 well here we use something called crossing the pole where we flip the negative point across the origin and we go out by two it's really quite a simple coordinate plane to understand so here's a fun question if we have a sinosoidal point in the format x comma y like this one roo3a 1 how do we turn it to polar coordinates in the format r comma theta well you could take the equations from my 3.3 video and use them but I'll just lay out the basic way here no matter what r is equal to the of x^2 + y^2 and if x is positive theta is equal to r tangent of y /x and if x is negative theta is equal to arc tangent of y /x + in this case x is positive being3 so let's plug it into the first equation and solve then we plug each value into the r formula and solve and that my friends is how you get polar coordinates from cartisian coordinates so if you were given a polar equation like r is equal to 3 + 2 sin of theta as an example we make a table of values for sign and then plot those points on the polar graph or you could make a sign cartisian graph and use those points to plot on the polar graph honestly you do you but graphs are a next video so polar graphs they're kind of fun and cool they make fun drawings but really they are kind of boring because good luck finding a real life scenario that models this so let's just get through this as fast as we can polar graphs come in the form r= f of theta polar coordinates work in pairs where the input value is the angle and the output value is the radius so that means changes in input values correspond to changes in angle measure from the positive x-axis and changes in output values correspond to changes in the distance from the origin you can restrict the domain of polar functions by limiting the range of theta values which effectively cuts off part of the graph but let's be real we only really care about what the main shapes of the graphs are polar functions typically come in four forms circles cardioids limosons and roses to understand these let's model the skeleton equation r= A + B sine of theta or it can be cosine it doesn't really matter a circle happens when you don't have an A it looks like a circle on a graph a cardioid happens when A is equal to B it looks like this on a graph and I've heard it being compared to a heart before a limosan happens when B is greater than A it looks like this on a graph so kind of like a heart shape but with a loop and finally a rose happens when you have something multiplying by theta it can look like a lot of things but this is kind of generally what it looks like all right this is it the last video of required course content for the AP exam thank you all for sticking with me through this journey if you want to stay in the loop on what I do next follow my Instagram also if you have found this content to help you at all this year and would like to support me in a more involved way consider becoming a channel member by pressing the join button but for now for the last time let's get into it we've talked about how polar functions work how to convert between coordinate systems and how to graph different polar shapes but what does it actually mean for a polar function to increase or decrease think of the polar function r= f of theta like a moving point on a graph if r is positive and increasing the point moves away from the origin if r is negative and decreasing the point also moves away because it's flipping direction if r is positive and decreasing the point moves toward the origin if r is negative and increasing the point also moves toward the origin if r changes from increasing to decreasing or vice versa that means the function reaches an extremum which is a maximum or minimum the average rate of change of r with respect to theta is just how fast the radius is changing per unit of theta if the rate is positive the radius is increasing if the rate is negative the radius is decreasing think of it like driving a car in a circular track how fast you're moving outward or inward depends on the rate of change of r you can use the average rate of change to estimate values of r within an interval it's just like finding the slope of a normal function except now it tells you how fast the radius is changing per radian and that's it the last piece of AP pre-calculus content thank you so much for watching now if you do me a favor take a break and watch this video it's a very cool video I made going to Salt Lake City on my other channel and I would love it if you could watch it [Music]