hello everyone the algebra 2 regions is coming up and we hero preps want to make sure that you are prepared to succeed we're going to take a quick look at the test format there are four major clusters or units that will be on the exam number and quantity which is worth 5 to 12% algebra 35 to 44% functions 30 to 40 and statistics and probability which is 14 to 21 we'll go more in depth in our inunit reviews you'll have 3 hours to complete this 37 question exam which has four parts first we have 24 multiple choice questions each worth two points for a total of 48 then there are three short answer portions part two is eight short answers also with two points each for a total of 16 part three is are wor more and give be multi-art they're worth Four Points each but there's only four of them so part three is equal in weight to part two finally there's one last question were six points this one typically involves some sort of graph drawing but it's pretty easy so it should be a free six points you'll notice that this adds up to 86 points which is kind of a weird number you'll have to remember that this Region's exam is positively scaled so unless you get 100 you'll always get a higher scaled score than your raw score you need a 77% raw or a 66 out of 86 to get in level five you'll need a 47 out of 86 for a level four and a 20 out of 86 or 30% to get a three and pass the exam these numbers can change slightly from exam to exam but it generally holds that you need around onethird of the points on the exam to pass that's right just onethird to get a 65 which is a passing score all right let's get to some real material with unit one of Algebra 2 number and quantity so the first thing that we've got to know is the number system we have a real number system below and the two main categories we need to know are the rational and irrational numbers inside rational numbers we see here we have integers and inside integers we have whole numbers inside whole numbers we have natural or counting numbers rational numbers are essentially anything that has a decimal that terminates or repeats while integers require that the numbers beyond the decimal point are zeros for example -3.0 behind the decimal point there's just absolutely nothing whole numbers have to be equal to zero or greater so no negative numbers and natural numbers are equal to one and greater so numbers you could physically count on your fingers we see here this is kind of overlapping these circles here in the rational numbers are overlapping what that means is that any natural number is a whole number is it also an integer is also a rational number or any whole number is an integer and a rational number but not a natural number we can use this diagram to show what is what and to classify the different categories now over here a different Circle we have the irrational numbers irrational numbers are simply numbers that whose decimals do not repeat or terminate like Pi or E and next up we have some basic number properties all right I'm not going to read through all of these so you have to pause the video and write them down if you're taking notes which is something we recommend doing if you take notes your brain will recall these Concepts better on the exam and you'll have a higher score than if you just passively watching anyway you'll need to know these properties of radicals rational exponents logs and exponents for your exam now that we've got this set let's take a look at some polom division polinomial division is just like regular long division except you use two polinomial as your divisor and your dividend and you'll get a polinomial as your quotient and it's easy to explain with an example let's say we're dividing over here 2x^2 + 7 x + 6 and we're dividing that by x + 2 first we're going to see what value we can multiply by this by X+ 2 to cancel out one of these terms and usually you want to cancel out the first term so what could we multiply this by to cancel out 2x^ 2ar so we'd want to have a negative - 2x^2 right so we're going to multiply x + 2 by 2X and we'll be able to cancel this out because when we do long division we want this to cancel out by subtracting all right so we're going to take this we're going to do 2x we're going to multiply this by 2X and we're going to get 2x2 + 4x we're going to subtract this and we're going to get 3x + 6 as our new polinomial we're going to do this one more time we're going to multiply X+ 2 by 3 to get rid of this 3x and it just so happens that in this case we got rid of the six as well meaning there's no remainder and in this case a theorem called the factor theorem is going to apply right here if there is no remainder it means that x + 2 is a factor of the polinomial and in this case it's really written as x - A so it's x - -2 not X+ 2 but obviously that's the same thing anyway the remainder theorem which is another theorem over here will show what happens when we do have a remainder then the remainder will be F of a or whatever we had left over divided by the polinomial dividing so let's say this was really like a like an eight or five or whatever right it would be like say 3 or 2 over x + 2 would be our remainder another note on this topic you do not need to do synthetic division on the algebra 2 Regions exam and if you don't know what this is don't worry about it and if you do you can use it and you are allowed to but you do not have to by any means so next thing we're going to talk about is rational functions rational functions basically look like fractions and operate like fractions but with polinomial as their numerator our numerator and their denominator so as we see here fraction rules are going to apply and if you don't know your fraction rules you should pause the video to take a look at these the multiplication division or you can do a quick Google Search now on the topic of rational functions and fractions we're going to jump over to rationalizing denominators this is an extremely important topic that's probably going to make up like a question or two on your exam and you're going to need this skill if you have a fraction with a square root on the bottom see we have a square root right here so what we're going to do in rational to rationalize the denominator is we're basically going to take the square root we're going to get rid of it and we're going to put it in the numerator all right how we're going to do that is we're going to multiply both the top and the Bottom by the conjugate of the denominator now what does this mean a conjugate conjugate basically just switches the sign so if we have B minus < TK C we're going to multiply by B+ squ < TK C and again we're going to do this on the top and the bottom because by fraction rules if we multiply the same numerator and denominator by the same value we're going to get an equivalent fraction so in this case if this was really B+ < TK C we would be multiplying by B minus < TK C and when we do this we're going to eventually end up with something that looks like this and as we can see the square root is on the top and there is no square root in the bottom now we're going to quickly jump into factoring and then we're going to move over into our last topic which is complex numbers our first example is when we have a clear greatest common factor we're just going to basically factor that out by taking the greatest common factor as we see here 5x and 3x^2 have a common factor of X so we're going to take that out we're going to get x * 5 - 3x this is pretty much a straightforward form of factoring you can get now in example two we have a square number minus another square number and we can rewrite this as the square root of the first plus Square Ro of the second * s < TK of the 1us < TK of the second if that sounded a little complicated let's take a look at this example our example shows that x^2 - 9 which is equal to 3^ 2 is going to be equal to x + 3 * x - 3 and you you can do the math to your by uh you can do the math on your own to make sure this all checks out now lastly if neither of these are possible or if you still need another example you'll need to use the poly Roots calculator function which may be different depending on what calculator you're using but you'll need to do that or figure it out yourself to find if two numbers that multiply to the C in this form X ax2 Plus BX plus C and add to this B make sure you factored out any great greatest common factors before doing this step as we saw in example one so here's an example we're going to need to find two numbers that add to -4 and multiply to three now let's take a look at the list of things that multiply to three well three is a prime number so that's going to be one and three and negative 1 andg -3 all right now we see that when we add one and three we get positive four so that's not our answer but when we um add netive - 1 andg -3 we're going to get ne4 so those are our factors and this is the factored form of the expression we we can check our work here by doing a little bit of box multiplication and if you want to take a look at this or do it yourself we should get the original value x^2 - 4x + 3 and you may also need to factor by grouping which is a new type of factoring that was probably not introduced to you in Algebra 1 this is most helpful when you have a polinomial or function with a degree of three or greater all right you're going to need to make two groups with a greatest common factor then you're going to have to factor out your greatest common factor and then you're going to have to simplify and here we have x cubed and 2x^2 now we notice that here we have a greatest common factor of x^2 so we're going to factor that out and we get x + 2 as basically what we just factored out and then we're going to do this again with the greatest common factor ofg -3 for these two terms and we're going to get -3 and X+ 2 now if it is possible to factor by grouping the things that are in these parentheses must be the same and when you take this you're going to rewrite this you're going to put it over here and then you're going to basically add these two things together so x^2 + -3 is X2 minus 3 make those into a factor and then multiply by these two by each other and that is essentially how you would simplify this first equation here to a factored form now our last topic is complex numbers all right we're going to go over here complex numbers the imaginary I is the basis of the complex number system and it is equal to the square root of -1 what that means is that I S when we Square both sides of this equation i^ s is equal to just -1 and if you want to learn more about that try looking at your curriculum again or Googling a little bit more of an in-depth summary again this is just a quick unit review so you should have already heard of this concept anyway we have complex numbers which come in the form a plus bi I where A and B are both real numbers and B is the coefficient of a set number of I rationalizing a complex number is basically just the same as a normal one you're going to multiply the top and the Bottom by the conjugate and we see here we have C plus di so our conjugate is going to be C minus Di and again we have to make sure that if we have a basically rational function in complex form we are going to have to multiply both the top and the Bottom by that conjugate and now we're moving on to the first part of algebra 2's second unit that is functions this one's extremely General but here's a topic by topic list of what this video will go over we've got domain and range composition functions inverse functions one one and on two functions end behavior multiplicity Transformations applications logarithms and regression functions first we're going have to Define what a function is function should pass the vertical line test meaning if you had a vertical line going through every single x value of the function left to right the function should only hit the line once for given x value now in this second example right here we see that a vertical line is hitting the equation twice at a given x value so it is not a function function should only have one y value for each x value now let's talk a little bit about a function's main qualities and characteristics firstly domain is the set of X values for which the function produces some sort of output or the X values for which it exists rational functions domains specifically could be found algebraic their domains are everything but where their denominators are equal to zero which is when they're undefined now this fancy notation right here with this r with two lines just means a set of real numbers and this line in between basically means like another stipulation to the rule we'll talk a little bit more about that when we get to probability so in this case if we have the function FX = x - 4x +2 obviously this is going to be equal to zero when x equal -2 so our second stipulation here is that X is not equal to -2 so the domain is true the domain is true when we have the real number set so we're in the real number set the set of real numbers and when X is not equal to -2 so domain is all real numbers if X is not NE -2 all right now let's take a look at radical functions other rules for radical functions domains are functions like a square cube root function these are all real numbers except when the radicant is less than zero all right so here if we have the square root of x - 5 this is equal to xus this is equal to Z when x equal 5 and less than Z for all X is less than that therefore our domain is all real numbers greater than or equal to 5 now let's talk about range range does not need to be found algebraically for the purposes of the Region's exam but it's just the set of all yv values that the function is defined for now let's talk about one: one and onto functions one: one functions have no repeating X or Y Valu so something like a sinus weal function like s of X is not 1: one since its y values as we can see here clearly are repeating you could adapt the vertical line test to say that one to one functions must also pass a horizontal line test where no point would touch a horizontal line twice an onto function over here is pretty simple just means that all X values have a defined yv value so just most normal functions all right now composition functions may be new to you they're basically just a function inside of a function put together and here's how we write them this dot by the way is a hollow dot it does not mean multiplication rather it means F of G of X all right and I'll show you what I mean with this example if we have AG of X is equal to F of G of X and G of xal 3x2 and F ofx equal < TK of 9x that means that H ofx equal of 9 * 3x^2 all right let's take a look at that again all right so we see how the 3x^2 essentially just takes the place of the 9x inside of the F ofx that's kind of what we're looking for here and here's just some work that would simplify that equation now we're going to shift over here we're going to take a look at inverse functions all right so when we've got an inverse function they're pretty simple they're basically what you get when you reflect any one: one function over the line yal X so let's just pretend that I'm a perfect drawer in this line is yals x when we reflect this thing here y does y is equal to 3 the X then we're going to get y = log base 3 of X you can also solve to find inverse function by swapping the Y and the X values of the original function all right let's take a look at this let's say we have y = 3x2 + 5 we're going to swap the Y and the X we're going to get x = 3 y^2 + 5 do a little algebra and we get the < TK of x - 5 over 3 is equal to y we use this notation here it kind of looks like f to the^ of1 this just means inverse functions now we're going to switch gears and we're going to talk about n Behavior multiplicity n behavior is pretty simple and on the test if you don't know what the N behavior of a polinomial is you can just graph it you'll see at the end of the functions will either go to positive or negative infinity and that's pretty simple but here's a quick trick to know if the leading degree of your function is odd like X cubed or X the 5th the NS will point in different directions if the leading degree is even like X 4th or X to 6 they'll point in the same Direction and if the coefficient of the leading degree is positive like 4X Cub for odd functions it'll look like this right here where we see it kind of looks like a just like you know yals X where we'll have as we approach uh as X approach is positive Infinity y approach is positive Infinity as X approach is negative Infinity y approaches negative Infinity 2 all right but if it's negative like negative 3x to the 5th for odd function it'll look something like this other one right here kind of in Reverse like YX even functions will have both ends going up for a positive leading coefficient and both ends going down down for a negative one now here's another thing multiplicity is basically just how many times something bounces and we can see here as we go from one goes to three it kind of stretches a little bit goes to five it stretches a little more we go to two it's kind of normal four it stretches six it stretches that's basically all we need to know about that all right now let's go to a big topic here Transformations it's usually one or two questions on the regions and if your teacher didn't get through them through the year they're pretty simple but the best way to understand them was some practice questions however here's a brief review and some really helpful acronyms that I used when I took Algebra 2 to remember function Transformations firstly there's six different types of Transformations that the r wants you to know that is horizontal translations otherwise known as horizontal shifts horizontal dilations AKA um horizontal shifting or scaling screen or sorry shrinking or scaling reflections over the y- axis and over the x-axis vertical dilations and vertical translations let's start with the horizontal dilations um translations and why Reflections first and you'll see why in a second firstly right here the equation for a horizontal translation you're going to take the equation F ofx and you're going to basically make an F parentheses x plus or minus a again parenthesis is important we're going to see why in a second but let's say we had x s then a horizontal translation would be we see we put the four inside of the X inside of the parentheses right all right and now this would be a right shift right a right shift because we subtracted that number a each point is shifted by some value a as we shift see in this graph here so this point kind of shifts here this point shifts here all right now again let's take a look at that formula right here the change is happening within the parentheses within the parentheses take note of that and also realize that adding an a results in a leftward shift while adding while subtracting an a is a rightward shift now let's take a look at dilations dilations shrink or stretch functions and if we multiply them by some integer K it'll get skinnier and if we M and if we uh multiply them by some one over K it'll get small R again this occurs within the parentheses as we see by our rule here F of K of X or F of 1 K * X finally our y-axis reflection is pretty self-explanatory we take the red function flip it over the y- axis and you should have this background from geometry also if you weren't already aware the red function is the original while the orange or blue one are images anyhow the equation for that is f ofx becomes F of negx again inside the parentheses is where our change is occurring now we're going to take a look at some vertical stuff it's all the same idea for a vertical translation except adding a moves it up and subtracting and a moves it down all right so it'll look something like this now this is different from what we had before it is outside the parenthesis see how the x is unchanged inside that parenthesis all right it's not going to become like x - 4^ s it'll be x^2-4 all right now vertical dilations are essentially the same except again the rule changes it's outside the parentheses so it's K * F ofx and 1/ K * f ofx X all right xais refle reflection is also similar F ofx becomes negative f ofx again outside of the parenthesis and we're going to flip over the x-axis now after all this you might be wondering how on Earth am I going to memorize all this is just too much for me well I got some simple acms for you hi ya and vxv what do these stand for we'll start with Haya it means horizontal inside the Y AIS again this doesn't really make sense but when you think about it any sort of transformation involving the y- axis like reflection and horizontal dilation and translations involves changing the rule on the inside of the parentheses all three of those I just went over were inside the parentheses likewise vxv is vertical xaxis vertical outside all these rules are going to happen outside of the parenthesis and you'll have to remember that translations always involves adding or subtracting an a that dilations involves multiplying by K or one over some K and that Reflections involves adding in a negative but that made a lot for me I remember VX VI by thinking of the Music Company Vivo now one last thing about transformation will finish up with odd and even functions um regression logarithms the order matters you always must perform your transformations in the following order hdrv you can see what those stand for on my screen horizontal translations dilations Reflections and vertical translations if you don't do this it'll mess up how your image is constructed I remember this by thinking helicopters do rise vertically and you can make up your own little thing if you want but that's just how I remember it all right even and odd functions a function is basically just called even if it is symmetric about the Y AIS so that's something like the cosine function or the function X squ a function is called odd if it has 180 degree rotational symmetry about the origin so F of negx is equal to negative f ofx for all x's in the domain right so that's something like X cubed or S of X and you see this little fancy e just means like in so all X is in the real number domain all right two topics left we'll go through these real quickly regression this is different for everyone since we all have different calculators if you have a TI Inspire I remember I recommend looking at our all you need to know for our Algebra 1 calculator guides the same idea same U ression function if not just know how to recognize all these models these six models here very important except for power functions don't really show up very much but those other five definitely important um and be able to use your calculator to find regression um equations when you're given those data points and applications of logs is our final topic so we all know this is the basic way to write a logarithmic equation right here but here's some ways that that's actually useful and you know applicable in real life and these may show up in word problems especially on those fourpoint or twoo questions on your regions the first one is that we are going to show exponential growth in Decay so if we have this equation a of tal P * 1 plus or minus r to the power of T we can draw exponential growth through Decay by showing that P is our initial Val our start value one is just a number kind of to represent like 100% plus or minus so minus is going to be a negative rate plus is going to be a positive rate um and then our rate is always expressed as a decimal so if we have something like 3% interest right that's going to be 03 and T is of course time now compound interest can also be shown using this equation the numbers mean the same thing so for n right here which means the number of compounds per year all right so common compounding periods are monthly quarterly and daily or yearly it's how many it's how many times per year of 12 months in a year four quarters and 365 days and of course one year in a year so that would just be one now the last thing we have today is continuous compounding which requires the use of the number e and yes it's the number e it's a constant that's approximately equal to 2.72 and is the limit if you know what that means the function 1 + 1/ n to the n as n approaches Infinity in other words as this equation right here gets has n getting infinitely bigger that's what that's equal to and you don't need to know that you just need to know how to use this equation right here when given a situation with continuous compounding now that we've discuss most of the major functions on the exam it's time to take a look at the final function family as part of Algebra 2 that is trigonometric functions we're going to go over radians and degrees trig functions the unit circle special triangles identities inverses graphs and parts of trig functions first let's start off by identifying the way that we measure angles that is radians and degrees these equations are both in your reference table so you don't need to memorize them but to go from degrees to radians simply multiply by pi over 180 and to go from radians to degrees multiply by 180 over Pi now let's get into some real trig Algebra 2 involves the three primary trig functions that is s cosine and tangent as well as the reciprocals that is cosecant secant and Co cotangent all right these are a little bit less common but you still need to memorize all these functions again most of this trig unit is memorization now since sin Theta is defined as y r cos Theta is X over R and tan Theta is y over R they can also be called opposite over hypotenuse adjacent over hypotenuse and opposite over adjacent respectively this can be memorized as soaa you may have heard this acronym before the reciprocals are of course the exact same thing but reciprocal r y RX and X over y now that sounded confusing let's look at an example triangle right here here let's focus on this orange angle right here first this here is the side adjacent to the angle that is not the hypotenuse again the longest side is of course always the hypotenuse the hypotenuse is always the longest side and again this also has to be a right triangle triangle we are looking at right angle um trigonometry only in out number two now we have the hypotenuse we have the adjacent that means that the last side must be the opposite angle all right it's also opposite from the angle we see it's not touching these two sides are touching the angle this side is not touching the angle now if we're to focus on this blue angle here that means that this side becomes the opposite because this side is opposite from the blue angle while this one becomes adjacent because this one is next the adjacent and the hypotenuse of course stays the same now let's see how our trig functions like this can relate when we have some real numbers involved that is with special triangles the two special triangles that is triangles with common angles like 30° 60° and 40 5° are the 2 > 31 and the < TK 211 triangle let's look at the first one right here all right if we wanted to find s of 30° which is the same thing as saying s of pi over 6 right we're going to go to the 30° angle which we have right here and then after we do that we're going to find the opposite side of the hypotenuse all right which one's the hypotenuse it's obviously going to be this one which one's the opposite it's going to be this one right we know the HP side is the longest of course and the opposite one's just opposite the 30° angle so since we know that s is opposite over hypotenuse we're going to have our opposite which is one over our hypotenuse which is two that means it's 1/2 if we do this for the other triangle let's say we want to find find S of 45° we're going to focus on our 45 degree angle we have our opposite we have our hypotenuse so it's 1 over < TK of two if we want to radicalize this denominator which we learned in our previous unit we simply multiply both the top and the bottom of the frac by < tk2 obviously * < tk2 is simply two and the top becomes < tk2 all right we're going to into what these y's x's and rs that I mentioned up here mean in a moment but for now make sure that you have these triangles memorized these special triangles have to be memorized so that you know what those values are and you can use your calculator to find the S cosine tangent cosecant see or cotangent of any angle but it'll spit out a decimal and sometimes the question will ask for the answer in exact form like this over here this Square 2 2 over two which your calculator will not give you it'll just give you again another decimal and these angles 30 60 and 45 are the only ones you're ever going to need to know exact values for you know with other angles you can use a calculator but if you're asked for an exact value you have to provide it like this with a square root in there and your calculator will not do that for you now let's take a look at some identities but these are basically just facts about Trigg equations first we have that s is 1 over cosecant and this makes sense since they're reciprocals therefore these relationship carry on for the other five trick functions as we see here now tangent cotangent has some other special properties tan Theta is equal to sin thet over cos Theta and cotangent Theta is equal to cosine Theta over sin Theta I have a little proof below to show each one you can just take a look at that and pause the video if you need to we also have the Pythagorean identities you may remember Pythagoras Pythagoras for the from the Pythagorean theorem but these are his identities now you just need to memorize that these are true right you need to memorize that these are true and they may pop up in the exam they probably will not but for example a question might ask you say What's the value of cosine squ Theta and it'll give you the value of sin Square Theta well you just plug this value in here you use this and then you find this algebraically very simply all right and now you still may be wondering where did all that X Y and R stuff come from let's regroup and take a look at that again all right we're going to look at a circle which may be weird since you thought we were talking about triangles but specifically we're going to be talking about a unit circle and what that means is that the radius of the circle is equal to one unit one unit all right Let's do let's do a little bit of a deep dive into what that's going to mean for us if we draw a radius or several as I have here all of these lines here they're all radi right if we draw one of these radiuses what we're going to see is that we have a triangle that's made between the y axis the x-axis let me draw little stter the y- axis the x-axis and a unit circle all right and this is radius now the radius is of course going to be the hypotenuse of this triangle right it's the longest side of course okay but how do we know where to draw these triangles and how does this really help us well here I've marked a bunch of different angles in both radians and degrees if I wanted to find s of 30° or pi over 6 it would be pretty simple all right we're just going to find the degree so this Theta right so let's say this angle here is our 30° right that's 30° now when we have that 30 ° we're just going to use the Y value of this point where we intersect and the x value so this is some point right this is some point x y right and what we're going to use is that this is y and this is X right and then this is one now we know that if we wanted to find the sign of 30 we're just going to do the opposite which in this case is y over the hypotenuse which is r or 1 this is where y over R comes from but we know what this Y is right we know this from our spectral triangles that this Y is just going to be 1 and that would be two of course we have a radius as a h is one here so this would really be a half so we should get then would be um in this case we would get 0.5 over 1 which is equal to 1 over 2 now let's think about what we would H what would happen if we were given the sign of some value but you did not know the angle in that case we would use the inverse trigonometry functions which are arc sign which is also written in this type of notation AR cosine and arc tangent let's see what I mean here say we're given that s of X is equal to 12 we're going to use our inverse trig function see that we had the little negative 1 kind of thing up there and we're going to basically swap it so we put the 1/2 here we put the X here all right now we're going to have AR sign of 1/2 is equal to X and we saw before this was 30 but if you put this into your calculator it would give you this value this to be 30 if you were in degree mode if you were in radian mode it would give you a different answer give you pi over 6 I also included some graphs of the functions these all have restricted domains and ranges because as we mentioned in a previous unit inverse functions must be one to one functions normally trig functions are not one: one as we saw earlier but they are one to one if we only draw them for a certain domain and decides up a little bit here's how we might draw an angle this is the really the proper way we have the initial side here we have the reference angle here and we have the terminal side here remember that if you're trying to find the arc sign or Arc cotangent or tangent or whatever of another thing you may need to add or play with the angle to figure out exactly what quadrant you're supposed to be in or what quadrant you're finding in because these are restricted to different quadrants right you may need to find the angle in one quadrant say this quadrant here and then add 90° right because if we find the angle here we find the angle here these are the same angle but we would need to sub ract this from 180 to get what this angle is now if that sounded a little bit confusing make sure to go back to look at your notes again this is a unit review video so you probably should already know how to do this if you don't make sure to check out our topic videos or go back and look at your notes now let's move on to what an actual trig equation would look like the three common ones that you'll see come in this format here we have yal a sin or cosine or tangent or whatever of B2 - * x - C + D all right now A is the amplitude which is 1/2 times um the max value yv value minus the minimum yv value absolute value of that B1 is the trig function use B2 is the frequency and note that also the period is 2 pi over the frequency C is the horizontal shift with a positive C being a right shift and a negative C being a left shift and D is the vertical shift with positive being up and negative being down all right now to end the video let's look at this function up top to see if we can determine what its equation is and first we clearly see if this is a sign function even if we didn't see this sign function down here we should be able to recognize what a sign function looks like remember sign starts at zero and it kind of goes up and then down and then it goes again whereas cosine will start at one and then go down tangent looks completely different and that should should be pretty easy to recognize anyhow we can see in our unit circle let's relate some Concepts here we can see in our unit circle that these functions are going to match up with the unit circle so let's say at zero right cosine is or X is at its maximum value it's at one over one right because there's no triangle formed it's just a line it's at one over one and that's at zero or 2 pi and we see over here in this graph that at zero and at 2 pi cosine is at its maximum value likewise s of X is is maximum value at Pi / 2 and if we go back and we look at our unit circle it's at its maximum value at pi over 2 where y over R is just 1 over one right anyway we know that it's sinx now our amplitude is going to be 1/2 times the absolute value of the max minus the Min value the thing between those two is going to be the midline so this right here is our line we're going to call it the midline between the two all right because this is our in this is our Max and we can see what is our amplitude well we're going to go 1 2 3 up so our amplitude is three and again another way you could calculate that is 12 times the absolute value of 8 minus 2 of course is 12 * 6 which is three all right now the next thing we have here is the frequency in these two pies we are completing it looks like two Cycles so that means our frequency is going to be two now the horizontal shift is in it's very interesting it looks like we are shift pi over 2 to the right because when we are at zero here we're going down instead of up where we should be is right here if there was no horizontal shift that means we went pi over two right we went pi over two to the right therefore right because we went this way so that's our um that is our horizontal shift and finally our vertical shift it looks we five above the xais that means our vertical shift is five and given this equation that means we can write our amplitude is three function is s right time 2 which is our frequency time x minus our horizontal shift which is Pi / 2 plus 5 which is our vertical shift 35 to 44% of your region examination will be on this unit that is algebra we're going to go over linear equations quadratics and sequences SL series there's also a super important topic focus in direct trics that will definitely be on your R exam and has been for years now we'll look at it but firstly let's look at some linear stuff now linear functions or linear equations are just functions with a degree of one and are always straight lines when drawn in a graph and they always have a clear slope too right here we see some examples of a positive slope negative slope zero slope and undefined slope slope which is also known as the average rate of change or a change of change in y over change in X rise over run it can also be found by taking the yv value at B at some x value B minus the yvalue a some x value of a over the B minus a value this didn't make sense think of it this way b and a are different X values and we plug in those X values to an equation F ofx we're going to get some sort of yvalue so this is really just saying change in y over change in X in a different way now let's take this example here we have two points that is 36 and 04 and our change in y here we see is positive2 our change in X is pos3 so our slope is 23 or 2/3 because we have that change in y over change in X now when we see a linear equation in the form y = mx + b right m is the slope and B is the Y intercept or it might be the starting value in a word problem now point slope form as seen in the second thing right here use the same M for slope but includes some sort of random point on the line now let's say here we could take the that point we had earlier 36 to take the places of y1 and X1 so we would get y - 6 = 2/3 * xus 3 now this should all be reviewed from Algebra 1 but what's new in Algebra 2 is three variable linear systems if these show up in a multiple choice question what you should be doing is just using the Lin solve function or calculator we have another video to that be sure to check it out as a calculator guide to Algebra 1 and Algebra 2 reges examinations on short answer questions though you'll be required to show your work so just doing a bit of a calculator hack is not going to help you here's a six step plan that you could do that first thing you're going to do is group your first and second equation and then you're also your second and third equation then you're going to eliminate the same variable from both systems consolidate both systems to one equation use those new equations you just got to make a new system then eliminate another variable and then once you find that and solve for that variable you're going to go back and find the other two and that sounded pretty complicated so let's go over it with a little bit of an example let's say I have this three variable system x + y + z = 1 2x + 4 y + 6 Z = 2x + 3 y - 5 Z = 11 so we're going to split it off again we have the first group here we have G1 our first group is the first two equations we have G2 is our second two equations and if you do not know how to solve a two variable system of equations then you should probably go back and review your notes or take a look at your algebra one stuff because this should all be reviewed from Algebra 1 now once we take this first group here we're going to multiply that top equation by -2 so we can cancel out the X's we're left with 2 y + 4 Z = 0 likewise we're going to multiply this one by two so we can get rid of those X's we're going to get 10 y - 4al 24 and we see we've completed this step right here we have eliminated the same variable from both systems that we we have eliminated the X variable now we're going to use these two new equations that we just got here we're going to make a new system of equations that is 2 y + 4 Z = 0 and 10 y - 4 Z = 24 we're going to solve for one of the equations we eliminated this four right here just by adding them together we're going to get that yal 2 and now that we have that we can go back and use this equation that we just had here to solve for Z we get Z equal Nega 1 then we're going to go back and take this equation all the way back up here that had the three variables pretty plainly and we're going to insert the things that we know that are now equal to Y and the thing that we know is now equal to Z and we're going to get that X is equal to zero so our final solution here is that Y = 2 zal1 and xal 0 all righty next up is quadratic so we are going to start with quadratic right over here quadratics are pols with a degree of two and have a U or a horseshoe shape their standard form for an equation is right here we have y = ax2 + BX + C they are also symmetrical and their standard and their exis of symmetry is right here where we have B over 2 a b and a are the same values that we see here in our standard form equation right if your quadratic is not in the form ax S Plus BX X plus C where a B and C are all integers you need to take a look and re-evaluate and then manipulate your form or your equation algebraically so you can get into standard form now the sum of the roots and the product of the roots are illustrated by these two equations again using the same b and a roots of course are just your X intercepts or your zeros let's also take a look at the quadratic formula this is a Formula that is given to you on your reference table but it is very helpful to have memorized not too hard to memorize if you haven't found the quadratic formula song you might as well give that Google um anyhow the discriminant is the most important part of this formula since we've already gone over what this means again this is the axis of symmetry this discriminant is basically just this little bit here b^2 - 4 a c what this does is it determines the number of real roots and the nature of those roots so if you plug in those B A and C from before and it is negative that means there are no X intercepts no real X intercepts no real Roots so you have complex Roots but there are no X intercepts no zeros this also means that basically your function never hits the x-axis so it has a y intercept that is not zero all right if it's zero then we have one x intercept if it's two we have two x intercepts and if that number this number here ends up being a square number then when we apply the square root function it just become a regular integer which means that we would have a rational root but if it wasn't then we would have a square root up here still and it would be an irrational roote and now you've of course been waiting for the torture of focus and directrix this is on every Regions exam and is usually a two or four point question every year it's on there they are removing this from the curriculum for the 2025 to 2026 school year but until then we are stuck with focus and directrix now the focus of a quadratic is essentially a point on the line of symmetry that is the same distance so this distance here is equal to this distance from the line known as the directrix let's list this line right here these are both perpendicular to the line of symmetry and this distance right here from here to here and from here to Here is known as P so from here all the way down to here that would be 2p all right and this occurs at the same x value as the turning point of the minor Max now if we have a quadratic we can write it in this thing called vertex form which is right here all right you just have to do some more algebraic manipulation to switch from standard form to vertex form or whatever form you would have and what we do is we write the vertex in the form h k with h being the x value of the turning point right here and K being v y value right the equation to find the focus is H comma k+ P since the point the directrix is yal K minus p and the vertex form is y = x - h^2 over 4 p + K all right this all applies for only a vertical Parabola so when the things are going vertical they're going up or down or whatever now if your arrows are going left to right then we have an horizontal Parabola and these are the rules we're basically just going to switch the H's and the K's the x's and the Y's because we are going in the other direction you can take a look pause the video and write some notes if that is what you need to do but we are going to move on to a practice problem I usually don't include these in unit summary videos but this is important to consolidate our knowledge this is from the June 2023 practice region um it was there was a similar problem on the January 2024 regions but I like this one better so our answer is two I'm just going to give that to you straight away but if you want to pause the video which I highly recommend doing and check your work see what you get all right so the DirectX is yal 4 and we know that this means it's a vertical Parabola because only a vertical Parabola has Y in it right a horizontal one would have X all right so we know this is a vertical Parabola we know that yal K minus P from our work over here that means that therefore 4 = K minus p and k = p + 4 if we do a little algebraic manipulation likewise if we use our Point here from the focus and we have H um comma K plus P we know that H is zero right because it just has to be zero and we know that K plus P equal 6 which means that K is also equal to 6 - P now these are both equal to K so we can set them equal to each other and we get that P equals 1 after little algebra which also means that when we plug P back into here we get 6 - 1al K so 5al K now we have that final equation we have our H value we have our K value we have our P value H is zero p is 1 K is 5 we can plug that into this equation here for vertex form we get this equation we do a little algebraic manipulation that looks a lot like our answer right here all right final thing we've got sequences and series sequences are list formed with terms and series are basically just the sum of those terms and there are two types of sequences arithmetic and geometric arithmetic sequences are like linear equations you just add or subtract to get to the next term you might have heard these as now or next equations geometrics are like exponential you multiply or divide by some sort of ratio or number to get to the next term and unlike equations these are only defined for the integers they are not defined for decimals or anything in between so while a linear equation like yal X is defined for something like xal 1.5 it would not be defined like that for an arithmetic for an arithmetic or geometric sequence all right there's two different types of equations for sequences we have explicit types and recursive types so the explicit one is used to find the nth term while the recursive is used to find the next term all right here are the equations to those Below in a little bit of a table all right note that D is the common difference between terms which is basically the amount you need to get from one term to the next well R is the common ratio which is the amount you need to multiply or divide by to get to the next term pause the video and take a little note on this if you want to write these down these are not on your reference table now finally we have Sigma notation Sigma notation is pretty simple as well we're going to use the sum symbol to add up terms in a sequence and you can use your calculator Sigma notation to easily find the sum of a sequence and here's an example if I were to do it by hand we see here that this four means that our last term is four we saw that from this right here all right we see that two is our first term we see this right here and we see this is the equation that we need to use to plug this into so we're going to have to plug in two first then we're going to plug in three and then we're going to stop at four so we put this all together we put it in we put in the equation then we get that our final answer is 21 your calculator can do this for you now geometric series formula this is pretty easy just find on your reference table this is the only thing that's on your reference table that's super helpful um just plug in the same variables as we had down below and it will give you your answer this unit was definitely a lot but the main things you got to remember is you have to know your algebra one you have to know things like completing the square I did not go over stuff like that factoring Etc we talked about that a little bit in our first unit but make sure that you know these Concepts and have them fundamentally down so you can learn things that take it a step further like our three variable linear systems and our focus in dirc we've arrived at the final unit for Algebra 2 statistics and probability this makes up 14 to 21% of your exams so usually you see one or two multiple choice and two or three short answer problems we're first going to talk about the types of studies we have sample surveys observational studies and controlled experiments in this course sample surveys are like online polls experimenters take a randomly selected sample of answers from a larger survey and analyze them and they cannot draw inferences but they can draw generalizations from their results in observational studies we have a lot less work on the part of the experimenter and they're often a lot more coste effective they simply look at something that's already happening and analyze the results of whatever that treatment might be as such this is a non-random treatment and selection from this one could find causation but not a greater generalization because the results of this treatment only represent what is true in an observed population not for the greater world population lastly there's controlled experiments which involve active experimenters who randomly assign different groups of people to treatment groups participants may or may not be randomly selected to participate for example brain MRI studies often involve some form of payment to the surveyed peoples because of a risk of radiation poisoning from the studies a generalization can be made if the participants were randomly selected otherwise only an inference can be concluded because of potential bias studies can also be used single blind or double blind in a single blind experiment the participants don't know which side of the experiment they on for example in a vaccine study participants might not know if they got for example a placebo or a real thing this eliminates some bias and mistaken results double blind experiments have this too but the experimenters themselves don't know which group is which until after they've analyzed the data using the previous example they would analyze the data without knowing which group got the placebo and which group got the vaccine until after they had collected and fully analyzed the data this would eliminate the most bias now good experiments always involve three things the random assignment of treatment such as in a single or double bind experiment have a a lot of participants and include a control group let's dive in a little more complicated statistics firstly you should know terms such as standard deviation mean mode and median from Algebra 1 in this case we're going to Define mean and standard deviation using different variables depending on whether we look at a whole population or just a sample we're going to use an X with a line over it for a sample mean and mu a Greek letter for the population mean standard deviation is an S for sample and a lowercase Sigma for population we'll apply these definitions through the next couple of minutes the first thing we're going to need to learn is what a normal distribution curve looks like a normal distribution is symmetrical and 68% of the data as we see here is within one standard deviation plus or minus one standard deviation from the mean 95% is within two and 99.7 is within three if you're given a specific individual and need to find the percentile that it represents within this distribution you'll want to use the norm CDF function on your calculator because every calculator is a little bit different be sure to look up how to use the norm CDF function on your own personal calculator one last thing to not about this diagram is that is a perfectly distributed curve with the mean mode and median representing the same value the midline of the curve we have a few more equations to go over before we move on to probability first thing is confidence intervals a confidence interval is a measure of how good your data is and it can be used be calculated using this equation CI is equal to the mean plus or minus Z times lowercase Sigma over the square root of n where n is your sample population and Z changes based on what your confidence level is if you want to find a confidence level of 90 you use 1.645 95% 1.96 and 99% 2.57 five these three values must be memorized for your regs exam and there is no way around it now let's say you want to calculate the margin of error a margin of error is greater When there's less people this makes sense just think about it if you have a survey with a lot more people it's going to be more accurate than a survey with less people the margin of error can be calculated by multiplying Z * P * theare otk of P * 1 - P / n this Z is the exact same Z that we used right here and we would use these exact same values a different Z is found in zc scores which are just used to find how many standard deviations of value is from the mean we're going to take those differing values from earlier and create two equations that essentially mean the same thing this first one is for samples and this first one's for population but again these variables essentially have the same meaning just for different Sample versus a different population all right Z is equal to the selected value minus the mean over the standard deviation value same thing here and again you can find standard deviation values using your calculator from Algebra 1 this just about finishes statistics if this seems a little bit abstract don't worry because on your algebra 2 exam statistics should be pretty clear and they shouldn't involve too too much guessing they should be pretty concise as well and you shouldn't need to show too much work now on probability which at this stage is just common sense and again memorization the first expression here is set as the probability of a or b and this is just a notation we use the U right here means Union it's the same deal here this means the probability of A and B occurring and given that A and B are different events probability should always be written and calculated using fractions as well this by the way this upside down u means intersection let's finish this up we've got the probability of a complement which is the probability that an event say let's call it event e does not happen this is written as P of event or E Prime is equal to 1 minus P of e so basically 100% probability minus the probability the event does happen is of course the probability that it doesn't happen and this just really makes sense again going back to the idea that probability in Algebra 2 is really just about using common sense now mutually exclusive events have nothing in common and don't affect each other this means that the probability of A and B separately is essentially the same as the probability if as if both of them or either of them happen and independent events means that the outcome of the first event let's call it event a does not affect the outcome of event B right there are three ways that we can prove this as we see here using conditional probability is probably easiest and we'll learn that in a second but that's just basically where we have this P of a a little line B and in parenthesis now a dependent event is obviously just the opposite of an independent event and it means that the outcome of the first event will affect the second event this is how we calculate that again let's look at conditional probability this is written like this it is the probability of a happening given that B has already happened and we know the outcome of B and we can write it like this at the end of the day these are really just formulas that you can memorize or you can just think about it on the exam I mean using common sense you can usually derive them yourself and just think about again this example of compliments the probability of one is just 100% right so subtracting the probability that something will happen we just give the probability that it doesn't we here our preper wish you good luck and a high score on the Regent exam