[Music] hello and welcome to the eath instruction YouTube live Algebra 1 region review my name is wer and tonight we're going to be reviewing hopefully everything that you need to know for your Regions exam in Algebra 1 that's happening tomorrow June 4th 2024 all right and we're going to do a couple things you know one of the things you may have noticed is that your Regions exam is happening awfully early and that's because it is the first year for a new set of Standards called the Next Generation mathematics learning standards it's a mouthful but it's the first test in Algebra 1 being given according to those standards that makes it a little bit tricky because over the years what I've always done when I review for this exam is I simply take last year's Regent exam June 2023 Algebra 1 and I go through it and I'm going to do that again tonight in fact you're probably going to want to have the algebra 1 June 2023 region sitting in front of you so that you can work through it with me I'm hoping that most of you that are watching actually have already tried most of the questions that's when it's going to be most helpful to you that being said right this is still a test that follows what are what are called The Common Core State Standards now the Next Generation standards and the common core standards are extremely extremely similar so there's going to be one question on this test that I'm not going to do because they eliminated that when they went to the Next Generation standards shout out to Brett Whitman for helping me uh think about that question and why I should eliminate it uh we had a nice discussion last week so if Brett's watching thank you so much for helping me remember to take that question off this test all right but as well as going through this test I also have a supplementary set of problems that you can find on the emath instruction website if you go into our blog and you look at the second article from the top it'll be all about these regions reviews and I published this little set of additional problems it's going to look something a little bit like this supplementary problems for June 2024 Algebra 1 region review and these are sort of topics that AR aren't on the tests that we're going to go through but I think are very likely to come up on your exam tomorrow they're just they just weren't part of the common core standards for Algebra 1 but now are part of the Next Generation standards all right now normally at this point what I would do is I would just immediately launch into the algebra one regions and we're going to get to it in just a moment but before we do that when you sit down to take that exam tomorrow at the end of your exam is going to be be a formula sheet and I thought we would take a look at what that formula sheet looks like right now so it's What's called the reference sheet for algebra one then it says ngls here for Next Generation learning standards I don't know if that'll be on the test tomorrow but we'll see now at the top of the formula sheet you've got some really handy uh conversion factors so if you know you're doing some kind of a problem and the answer you know the units are in miles but for some reason you have to work it in feet you've got your conversion 1 mile equal 5,28 ft and you can kind of take a look at all of these I'm hoping that your math teacher has sort of gotten you used to seeing this formula sheet if not for the entire year then hopefully at least for the last few weeks as you've prepped for this exam but let's take a look at the real important stuff on this formula sheet which is the formulas right so they have a quadratic equation right and all that is is the equation of a quadratic function one that graphs in a parabola I don't even know why they put that on this formula sheet but then again something things are a mystery to me in life all right the second formula down is something that's very important the quadratic formula right that solves for the zeros or the X intercepts of a quadratic equation and you certainly used that quite a bit this year we'll use it when we go through that practice test in just a bit they also have the equation for the axis of symmetry specifically that's the axis of symmetry of a parabola that's that vertical line that cuts it right down the middle right it's also the x coordinate of the the turning Point all right they very nicely put the slope formula on for on here for you the change in y / by the change in X then they have the equation of a line y = mx + b in what's called slope intercept form and they also have the equation of a line in in a in a form that you may not have gone over this year or maybe you did called the point slope form and that's going to be on that supplementary set of problems that we're going to go through because that's a new topic in the Next Generation standards even though it's not actually in the Next Generation standards they put it on the formula sheet so we have to assume it's going to be on the test they they give you an example of a generic exponential function y = a * b x they also have the formula for what's called compound interest all right then they have what's called an arithmetic sequence formula that's going to predict any term of an arithmetic sequence if you know the first term the common difference and how many terms there are that's n they've got the same thing for a geometric sequence and we're going to see at least this one on that practice test I don't think that we end up seeing this one although there is sort of a question about exponential growth and finally they have a formula for the inter quartile range which you've been looking at since I think sixth grade which makes you wonder why they gave you a for a formula for it but based on that formula they also then have two formulas that tell you how to identify whether or not data points are outliers in a data set again knowing these two particular formulas and imply buying them is nowhere really in the Next Generation standards but given that they're on the formula sheet I created a problem that will hopefully give you an idea of how to figure out whether or not a particular data value in a data set is an outlier or whether it's a a normal data point better yet a not a non-outlier all that being said that's our formula sheet right that's what we're going to be working off of I'm not going to come back to this formula sheet until I hit some of those supplementary problems and really spefic specifically when we take a look at this for now let's jump into the June 2023 Algebra 1 Regent exam here we go all right our region exam now of course the algebra 1 region exam hopefully you've done some practice uh examples of these have multiple choice problems and they have free response problems we're going to begin obviously with the multiple choice and there's no time like the present let's start with question number one the expression 9 m^2 - 100 is equivalent to which of the following all right so right this is clearly a question in factoring and it's an example of what we call factoring the difference of perfect squares now we know that if we have anything that has this kind of structure a^ 2 minus b^ 2 that can be factored into a minus B * A + B and that's exactly what we have have here now sure it's 9 m^2 - 100 so it's a bit more complicated looking than this but keep in mind right that 9 m^2 - 100 is the same as 3 m * itself - 10 times itself right so we have the structure of something squared minus something squared and therefore that's going to be equal to 3 m - 10 * 3 m + 10 nicely enough they started us off with Choice one all right let's take a look at question number two which expression represents an irrational number all right well I'd love to talk forever about rational versus irrational numbers but probably the thing that sticks in your head is irrational numbers are numbers that have decimal representations that never terminate and never repeat their pattern where rational numbers are numbers that have decimal represent representations that either terminate or repeat in some kind of a pattern for instance 1/4 is rational because its decimal is 0.25 1/3 is rational because its decimal is is 0.3 repeated but a number like Pi is irrational because it starts off as 3.1415 Etc and never repeats its pattern now one of the things that you should have learned in Algebra 1 is that square roots of numbers that aren't perfect squares will always be irrational and square roots of numbers that are perfect squares will be will will be rational and the other thing that you should have learned is the following whenever we take an irrational number and we add it to a rational number we always get an irrational number it's almost like multiplying a negative by a positive and getting a negative when we take an irrational number and we add it to a rational number it's always irrational and it kind of makes sense if you thought about adding like the decimal versions right then you know if I took like five and I add which is rational and I added it to Pi which is irrational you know I'd get like 81415 Etc right the decimal would go on forever without repeating Etc so one way to do this problem is to just start chucking these things into your calculator and seeing if their decimals terminate repeat or go on forever but that can become problematic especially if your calculator is set to sort of have the decimals truncate in other words only show me a certain number of decimals then you might think that the decimal representation just stopped so let's take a look at kind of what we have here Choice one we have the square < TK of 16 plus the squ < TK of 1 but that's just that's just 4 + 1 that's just five that's a rational number right likewise in Choice 2 < TK of 25 + 4 that's 5 + 2 that's 7 right choice 4 very similar the < TK of 49 + the square of 9 that's 7 + 3 that's 10 right all of those are rational so why is choice three irrational well choice three is the correct Choice it's irrational because the square < TK of 36 is 6 but the square < TK of 7 is an irrational number so here we have an example of a rational number the < TK of 36 plus an irrational number the square < TK of 7 and that results in an irrational number all right so here we have choice three all right let's take a look at question number three which linear equation represents a line that passes through the point -3 comma 8 great right so the question really is which of these lines contains this point and one of the things they really really want you to feel comfortable with in Algebra 1 and it's not just with lines it's with lines parabas inequalities and all sorts of things like that is that this point will fall on one of these lines if when you substitute it into the equation it makes it true or another way to do it is substitute -3 into each one of these equations for x and see if it gives you8 for y if it does then it passes through that point now one of the unfortunate things about this problem is that this problem really has to be done by sort of guessing and checking you have to be like all right let me put this thing into Choice One see what happens put it in Choice two see what happens that's really the only way to do it other than slapping all four of them into your graphing calculator and producing a table which would be wildly inefficient right now one of the things that is a little bit unfortunate about this problem is that the correct choice is Choice one so let's take a look at why that is literally if I take the equation Y = 2x- 2 right and I can either just substitute -3 in for x or I can substitute both of them in to do my check right what I'll see is 2 * -3 is -6 -6 - -2 is8 that checks right and so because when I substitute this point into Choice One it makes the equation true that point lies on that line or that line passes through that point if if I chose any one of these other ones for instance let me do choice three real quick right if I did choice three and I put 8 in and I put -3 in right I'd get 8 = -6 + 13 and that would be 8 is equal to 7 well that's not true right so it wouldn't lie on choice three all right the thing is once I know it lies on Choice one I'm pretty much good to go so let me get rid of that all right let's take a look at Choice four Choice four question four the expression 5x^2 - x + 4 - 3 * the expression x^2 - x - 2 is equivalent to which of the following all right and this might be a little bit hard for you to read because I've got that eight nicely sitting over the X squar so let me get rid of all of that let me move this up a little bit and let's take a look at this problem this is a pretty straightforward um combining like terms problem in that first expression there's really nothing going on right in fact those parentheses aren't doing anything at all because nothing is multiplying the 5x^2 - x + 4 so I'm just going to remove the parenthesis but in the second expression what I'm going to do is I'm going to multiply through each term by -3 so x^2 * -3 will be -3x 2x * -3 will be POS 3x remember a negative * a is a positive and -3 * -2 is pos6 and I get a red pen just showing up out of nowhere all right well if you've watched my videos before you know that the magical red pen will show up whenever it wants to all right so we've now got this with no parentheses we can now rearrange terms to kind of have like terms lining up right so I can have 5x 2 - 3x^ 2 - x + 3x + 4 + 6 I can combine these two quadratic terms that'll be 2x^2 I can combine these two linear terms that'll be + 2x I can combine the four and the six that'll be + 10 and I get Choice two all right that's it let's take a look at question number five the 24th term of the sequence -5a -11 comma -17 comma -23 is is which of the following all right so first things first we have to really understand what kind of sequence this is now there's two types of sequences really that you study in Algebra 1 you study what are called arithmetic sequences and we saw these on the formula sheet and you study geometric sequences all right arithmetic sequences you go from one term to the next term by adding or subtracting something called a common difference now mathematicians always like to think about it as adding and not subtracting um you know but you would be adding a negative number and geometric you're going to get by either multiplying or dividing each term by the same number to get to the next term so let's take a look and see if we can understand just exactly what kind of sequence this is to go from -5 to -1 right I would need to subtract 6 to go from -11 to -17 I'd have to subtract six to go from -17 to -23 I'd have to subtract six so the question is if I keep subtracting six over and over and over again what will be my 24th term all right so this is actually where we could conceivably come over to our formula sheet and use this particular formula I'm looking for the I think it was the 24th term yep the 24th term so n is 24 right I know my A1 my A1 was -5 I know my D that's my common difference and that's -4 and then I multiply it by n minus one all right let's go back to the test and see if we can really understand what's going on here right in any arithmetic sequence if I want a particular term I always start with the first term term and then I add to it my common difference times 1 less than the number of terms right now why do I do one less than the number of terms well notice to get to the third term which is -17 I would subtract six twice to get to the fourth term I would subtract six three times the reason it's always one less is that you don't have to subtract or add anything to Simply get to the first term right you don't have to do any anything to get to the first term so it's always one less in other words right to get to the 24th term I'm going to take -5 and I'm going to subtract six from it 23 times right so now you just take your calculator out you literally type in Nega I might as well open up my calculator right now I forgot to do that up front um I'm going to be using the ti inspired today all right just for the record but you know what we're now looking to do is we're now looking to simply subtract 6 from -5 or add a -6 to -5 23 times right and when I do that I get - 143 and that is our answer Choice two all right great we're done with the first page we're making progress all right let's take a look at question number six when completing the square for x^2 - 18x + 77 equal 0 which equation is a correct step in this process awesome let's go through how we use completing the square to solve a quadratic equation now you've learned how to solve quadratic equations in at least three if not four distinct ways in this course one of them is by factoring one is by using the quadratic formula and probably your least favorite completing the square although you never know there's some people that love it now the good news is we don't have to go all the way there in this problem and we'll see why in just a moment so first thing I'm going to do is I'm going to write the problem down hopefully there we go and I'm going to take a constant if there is one on this side and I'm going to move it to the other side it doesn't have anything to do really with completing the square now recall what I will always do do as long as the leading coefficient is one as long as the number multiplying x^2 is 1 I will take the coefficient that's multiplying X I will find half of it and I will Square it so half of-8 is -9 and when I Square it I get 81 so what I will now do is I will add 81 whoops to both sides of this equation 77 and that's 77 not seven I don't know why that happened thank you Joey shout out to Joey chevel the man behind the curtain he's my producer um so right now we've got x^2 - 18x + 81 and we can now Factor this trinomial as x - 9^ 2ar and over here -77 + 81 is 4 so I've got x - 9^ 2 is equal to 4 at this point I would take the square root of both sides I'd get plus or minus two then I would add nine to both of the plus two in the minus 2 but that doesn't matter because I see that choice one is exactly where I'm sitting right now so often when they test uh using completing the square in a multiple choice format they only have you go halfway through because they can't be like solve this equation by using completing the square because they don't know whether you used completing the square or your calculator or the quadratic formula or factoring or whatever right they don't know so you know they got to like just be like well which step is part of it kind of thing right and this is one step in getting to the solution of that equation using completing the square all right let's take a look at number seven which function will have the greatest value when X is greater than one now this is a fascinating problem and it's trying to test a principle that basically says look exponential functions always that are increasing exponential functions that are increasing will always outpace any other type of function eventually all right but you don't really even need to know this this question says which of the following functions will have the greatest value when X is greater than one basically meaning if you take any x value greater than one any x value greater than one and you substitute it into each one of these four formulas whichever one is the biggest is the correct answer now what number should you put in well you could put in two you could put in 1.1 you could put in 100 I would suggest two so like literally because they say x is greater than one I would just take x = 2 and if I substitute it into this equation what I would get is I would get 2 * 5^ 2ar which would be 2 * 25 and that would be 50 if I put two in here I would get 2 * 2 which is 4 + 5 which is 9 if I put 2 in here I would get 2 * 2^ 2 + + 5 that would be 2 * 4 which is 8 + 5 which is 13 and here if I put 2 in I'd get 2 * 2 cubed 2 cubed is 8 * 2 is 16 + 5 is 21 so 21 13 9 and 50 hm which one's got the greatest value looks like it's the exponential function Choice one now another way of doing this is to take all four of the curves put them in your graphing calculator and simply graph and see which one is sort of above all the other ones after X = 1 right and that's one of the things that's a little bit tricky here because for x equals 1 right perhaps this is the largest one perhaps it's not actually it still is which is kind of funny but we won't go there you know when X is zero it's not the largest one but when X is one it's still bigger um anyway let's move on to question number eight Mike uses the equation B = 1,300 * 2.65 raed to the X to determine the growth of bacteria in a laboratory setting the exponent represents what right so this is trying to understand all the components to an exponential function right and there's really three components well or four components there's the output which is the number of bacteria present there's the 1300 which was the original number of bacteria there's the 2.65 which is known as the growth factor or the multiplying factor or whatever and then there's the X and what the X always is is the number of time periods in whatever units you're talking about so maybe X is the number of minutes that have occurred maybe X is the number of hours that have occurred or days or weeks or years it all depends on context but one way or another the exponent in that type of a model is always the number of time periods okay let's take a look at a unit conversion problem also not particularly in the standards but let's take a look at how that one does it number nine a company ships an average of 30,000 items each week the approximate number of items shipped each minute is calculated using the conversion what now one thing that's kind of interesting about the new formula sheet you know if you were doing this problem and you were like oh my goodness I need to have conversions right there are some conversions they expect you to know notice there are no time conversions here right they expect you to know that there are 60 seconds in a minute that there are 60 minutes in an hour that there are 24 hours in a day that there are 7 days in a week they might not need you to know how many days there are in a year or how many weeks there are in a year and side note there's not a you know like an even number of weeks in the year even though people will say it's 52 weeks it it's not quite um but anyway the point is this these conversion factors wouldn't particular particularly help you on the problem that we're about to go over right now so let's take a look at it first let's make sure that we understand this 30,000 items each week now I would have loved it if they had used real rate language like 30,000 items per week right because whenever you have that kind of per blank it should be sort of an immediate an immediate symbol or signal sorry that this goes in the numerator and this goes in the denominator right that's what you want okay which immediately means choice three and choice four are out that's weeks per item not items per week okay now how do we do this conversion well we don't have to convert the items we just have to get from weeks to minutes now what you're going to do right is you're going to eliminate the unit right that you don't want so what first I'm going to do is I'm going to go from weeks to days so what do I know I know that one week is equal to 7 Days right and that is good because the week's cancel I don't want that right now I'd like to go from Days remember we want to get down to minutes I want to go from days to hours so what do I know I know that one day yes I'm going to run out of room I know is equal to 24 hours and then the days cancel and now I have items per hour and finally right pretend that this is all in one line I would then do 1 hour is 60 Minutes the hours would cancel and now I would be left right with items per minute so I've got 30,000 times here it is and again you can really just see it weeks cancel days cancel hours cancel and you're left with items per minute right things in the numerator cancel things in the denominator in fact if you look at Choice one which is the only other one that's even mildly realistic when you see weak in the denominator here and weak in the denominator there you know it's wrong it just it just has to be right because those weeks aren't going to cancel likewise hours here and hours here aren't going to cancel right think about cross- cancelling in fractions except do it using the units all right I think that's the last one on that page let's keep going me just turned my notes a little bit and let's take a look at number 10 and boy this one I think I need to make a little bit smaller that's better might be a little bit hard to read but we'll go to the board in just a moment okay a function is graphed below a possible equation for this function is which of the following all right great so let's talk about this a little bit and I'm going to try to like zoom in a bit just so that we have well I don't know that's probably not going to do me much but yeah there we go that's good all right so let's talk about this particular function right to begin with we know it's not a parabola right we know it's not a parabola because it's got two turning points so typically right those types of functions are what are known as cubics now don't get me wrong you've got three hours for the algebra 1 test which means not a single person watching this would should miss this question simply because you could put each one of these things into your calculator and see if it gives you the correct answer right you could just see all right now what are they getting at though what's the actual content that they're getting at well they're getting at how does a graph of a polinomial relate to its zeros this is going to come up a little bit later on we have Zer in this particular problem of x = -3 and x = 2 all right x = -3 and x = 2 which makes you think then that you should have factors of x + 3 and x- whoops and x - 2 which would then PL you towards choosing Choice two except that thing only has two factors which would mean it would be an X squar function which means it would be a parabola right it would just be kind of a u-shape either pointing up or pointing down so what's the real key here and I think this is a bit Advanced for Algebra 1 but who knows right which is that this thing because it is what's called tangent to the X AIS that is what's known as a double zero all right a double zero it means that we'll actually show up in the factors twice all right it's almost like we have x = 2 and x = 2 so we have X = or Y = X + 3 * x - 2 squared all right and again I feel like that's a little bit more Algebra 2 than Algebra 1 but but keep in mind that you could definitely put each one of these things into your calculator and check to see if they're correct right now one thing I can say for certain is that it could never be Choice one because this thing has zeros at -2 and pos3 and those aren't the zeros here right it can't be Choice four either because this would have zeros at two Nega -3 and also pos2 but that's this graph is never going to come back down and hit 12 12 so that one's out the real problem is that they desperately want you to Choice choose Choice two they do and that's a little bit unfortunate because typically in a problem like this they'll actually have it have three distinct zeros as opposed to one of them that's a double zero but that's what's going on here all right let's take a look at number 11 if G of X is equal to x^2 - x + 5 then G of-4 is equal to which of the following all right so this is classic function notation right they give us a function rule right here algebraic Rule and they want to know what G of-4 is in other words if I take -4 and I substitute it in for X in each location what will the output be what will be the Y value when the x value is -4 now you could certainly do this the oldfashioned way all right and the oldfashioned way is literally like this all right and I'm going to go through this really quick but there's some trickiness number one you have to square that -4 and get 16 but then you have to negate it and you'll get -16 then what'll happen is you'll have -4 and that'll be pos4 + 5 and then you'll have -16 + 9 and that will be7 for Choice two all right but look at how easy this is is to do on the calculator all right now again I'm I'm a big fan of calculators sometimes and in a situation like this I would not hesitate to tell a student and by the way you're going to be able to do this whether you've got the ti 83 plus 84 plus or the ti inspire all right actually let me just make this bigger there's no reason why I can't do that um remember I'm going to do4 and what I'm going to do is I'm going to store it in the variable X now I can't really go through all of this right now if you haven't learned how to do store and whatnot but there is a store button where you can like take a value a numerical value and put it into the variable X and then what happens is right if you type in any expression involving X it will then give you the value based on X being equal to -4 well for us what we wanted to know is we wanted to know the value of - x^2 - 4x um + 5 right we just wanted wanted to know the value of that expression when X was equal to -4 so now I hit enter and that is not the answer x^2 oh no it's not - 4x it's just - x boy I was It was supposed to be a flourish and you were supposed to see7 there and I was supposed to be like hey look at how cool that is that that which clear didn't didn't happen the way I wanted it to but whoops thank God thank God five is not a choice otherwise I'd be like oh it's Choice five there it is -7 it's right down the paper I don't know why I can't read that all right so there it is forget about this um there it is right and so I I don't really need to know almost anything and you know I just need to know how to store a value in a variable and then type out the expression hopefully correctly and it will tell me what the value of that expression is when X is that value now some students get a little bit concerned they're worried about oh what happens if I graph later will it think X is only -4 nah not at all it only is on this sort of like evaluation screen that really doesn't have any effect on anything else whatsoever all right and I can go and I can change the value of x anytime I want and then kind of throw in another expression involving X and figure out what it's equal to you can do that actually with any variable whatsoever but X is the most common one all right that being said let's head right on back to our test I like having the calculator on that big picture all right exercise number 12 I would claim this one's more geometry than Algebra 1 but we here we got a movie theaters popcorn box is a rectangular prism with a base that measures 6 in x 4 in and a height of 8 in to create a larger box both the length and the width will be increased by X in the height will remain the same which function represents the volume v of X of the larger box all right so you know again to me yeah it's algebra but it's more geometry than algebra and it's not very much of that any anyway you know so what do we got you know we got some box right that is holding popcorn that apparently you can also see through um you know that is like 6 in by 4 in by 8 in okay and then what they're going to do is they're going to leave the eight alone and they're going to increase both the six and the four by X in so that's easy enough we just do plus X and plus X right so now we have 6 Plus X and 4 plus X and 8 all right and now they want to know what the new volume of the box is and this is where the geometry comes in right you're supposed to know by this point in your life that the volume of any rectangular solid is length time width time height you've been working with that formula since sixth grade I know I wrote a book on sixth grade math so I know that I had this formula back then you saw it in sixth you saw it in seventh you saw it in eth and here it is right but it's easy enough right so we just have to say Okay the volume is going to be the length which is 6 + x time the width which is 4 + x time the height which is 8 so there it is and it's interesting too right because if you look at the other choices choice one tries to get you they know they they realize that you know length times width times height but they try to get you with that 8 plus X right as and that that would have been the case had we also increased the height by X in and then choice three and choice four they're trying to get you to think that you calculate the vol of a box by adding its length its width and its height I don't even know what physical property that would give you but it doesn't give you the volume so let's keep going all right choice 13 the expression 300 * 4 x + 3 is equivalent to which of the following all right this is really cool because it's testing a particular exponent property and let's talk about that exponent property there are three main exponent Properties or rules or laws it all depends on like you know how you want to put it but the one that they're testing here is this idea that if you have a to x * a to the Y you get a to x + y all right you know and you use this property all the time when you do something like x^2 * x 5 is equal to X the 7th when you multiply two things that have the same base you add their exponents but you see equality is a two-way street right and most people think about these properties as only being applied in this direction but we can actually apply this property also it's not going to let me do that also in this direction what do I mean by that what I mean by that is that I can take that 300 which has really nothing to do with this problem and I can now write this thing as 4 to the x * 4 the 3r and in other words I'm literally taking 4 x + 3 and rewriting it as 4 x * 4 3r I'm running this property in Reverse so to speak and again I bet that most of you would feel comfortable by saying 4 x * 4 3r is 4 x + 3 the question is are you comfortable with saying that 4 x + 3 is 4 x * 4 3r there it h there we have it and it's Choice one right they're really trying to get you some of these things though like this one they're trying to get you on the distributive property they're thinking oh they're going to distribute the 300 over the X+ 3 I don't know it's all kind of interesting but just running that exponent property backwards all right let's take a look at 14 Ashley has only seven quarters and some dimes in her purse so we know the number of quarters but we don't know the number of Dimes she needs to have at least $3 to pay for lunch which inequality could be used to determine the number of Dimes D she needs in her purse to be able to pay for lunch all right so to begin with right let's just understand what money she has she's got seven quarters right so she's got 7 * .25 which is going to be $1.75 so if you look at all these choices and you're wondering where does the 1.75 come from it comes from the fact that she has that ready now the question is how much money does she have in dimes well keep in mind that d right the letter D does not represent the amount of money she has in dimes it represents the number of Dimes that she has not how much money she has in dimes but the number of Dimes she has right so how much money does she have in dimes well if you take the number of Dimes and you multiply it by 0.1 which is more properly put as 0.1 * D that is the number of money she has in dimes right and they put the zero there so I'll put it there as well just so that we kind of see that 10 cents idea 10 pennies right so that's the amount of money that she has in dimes so her total amount of money is the amount of money she has in quarters plus the amount of money she has in dimes right that is the amount of money that she has now she has to have at least $3 to pay for lunch meaning that she has to have three or more dollars right three or more right in other words the amount of money she has has to be equal to three or greater than three and that then is going to be our answer and notice again what they do right they give you an answer here which looks almost identical except it's just 1.75 plus d is greater than or equal to 3 and a lot of students choose that because they mistake what the variable d stands for they think oh D is how much money she has in dimes but it's not it's the number of Dimes that she has does she have five dimes does she have 15 dimes that would be weird to have 15 dimes in your pocket you know but but it could be the case right so D is the number of Dimes so that ends up being that 0.10 D has to be in there and then of course they have the greater than or equal to three and they have the less than or equal to three to try to test to see how many students still don't have you know the greater than or less than symbol down all right let's take a look at number 15 the formula for the area of a trapezoid is a = 12 * the quantity B1 + B2 * h the height h of the trapezoid may be expressed as which of the following all right so this is one of these problems where you're given an equation that's got a bunch of stuff in it and you have to solve for one of the things all right I'm going to rewrite the equation here so that we can see it just a little bit brighter all right now before I solve this I want to just like kind of really get at what the structure is that we're looking at here we're saying that a equal 12 * something * H and what we want to do is we want to solve for H this is the thing we want to get all alone I want to have that all by itself okay now what is happening to H well I'm multiplying it by 1/2 and I'm also multiplying it by this thing notice how I keep just saying this thing I don't care that it's B1 + B2 it could be B1 + B2 it could be a cow dived by a moose I don't care H is being multiplied by this thing and it's being multiplied by 1/2 now to get rid of multiplying by 1/2 you could divide by 1/2 but pretty much always you multiply by two so the first thing I'm going to do is I'm going to get rid of I'm going to get rid of that 1/2 by multiplying both sides of my equation by two all right and you have to feel comfortable with that at this point now how do I get H by itself well I divide by the thing that's multiplying H and again I don't care that the thing that's multiplying H looks a little complicated B1 plus B2 again cow divided by llama I I don't care what's in there do you know what I mean llama squared it could be anything cubic llama the thing is I'm going to divide both sides by B1 + B2 that's what I'm going to do and that puts my H all by itself and there's my answer 2 a / B1 + B2 all right let's take a look at the last problem on this page and we're going to take a little break number 16 the function f ofx equals the absolute value of x is multiplied by K to create a new function G of x equal K time the absolute value of x which statement is true about the graphs of f ofx and g ofx if K is equal to 12 all right so this is a problem that's talking about Transformations taking what's known as a parent function a very simple function f ofx equals the absolute value of x and we're going to see this parent function later on in this test all right the classic V right and what happens when we multiply that thing by a positive constant that is less than one right now what that does is it literally takes every one of the Y values on this graph and it multiplies them not surprisingly by 1/2 and you end up getting kind of a graph let me just change colors really quickly you end up getting a graph that is just half as tall G of X and literally like if you had a point on here that was let's say right here and let's call it 4 comma 4 because that would be the point then on this particular graph the mapped point would be 4 comma 2 right so what has happened well we can say that g of X got wider than F ofx I don't really particularly love that description I'm not saying that it's wrong it certainly did get wider right but I would much rather them say something like it was compressed towards the x-axis by a factor of 1/2 something along those lines right what it certainly isn't is a reflection across either one of the a es nope right and it certainly didn't get more narrow it would have gotten more narrow if what we multiplied by was a constant greater than the number one all right let me change back to Blue take a little bit of a drink break so I don't lose my voice at any point and of course we take a drink of our favorite beverage lemonade now I've heard the conspiracy theories that say that I only do this as a shill for the lemonade industry you know and that really I'm being paid by big lemonade right in fact the entire emath instruction Venture is just a ploy by big lemonade to get in front of you you know at this time you know is it a coincidence that my last name is Wier and that there's a lemonade company out there called wers maybe maybe not I mean it's spelled differently that one is with a Y actually I'm not even sure that company exists anymore but they're using used to be a a wers lemonade company it was w y l r s you know was kind of great apostrophe s that is but is it coincidence I don't think so do you know what I mean actually this review has been brought to you by the fine folks at eath instruction so I would encourage all of you teachers students and whomever to visit emath instruction for those of you that are in Algebra 1 this year of course you're going to find lots of great Next Generation geometry material there for next year hey Summer's right around the corner what a better time to get ahead on your geometry work so head over to emath instruction at some point check out our website if you haven't before we've got great instructional videos and things over there and always enjoy your lemonade all right oh that was good and hydrating hydrating big lemonade I like that idea I like the idea of big lemonade we're going to have to use that a lot I know right now my my daughter is inside just rolling her eyes she's not watching this why would she be rolling her eyes she's an algebra too all right let's get back to it problem number 17 some adults some adults were not all adults some adults were surveyed to find out if they would prefer to buy a sports utility vehicle an SUV or a sports car both please the results of the survey are summarized in the table below all right so we got this table that lays out you know SUV versus sports car male versus female it' be great if we got a little uh non you know gender in there but yeah eventually we'll all catch up all right so you know what this has done is this is called a two-way frequency chart and it really kind of lays out you know the numbers and where we're at and things like that in terms of who preferred what and a very easy question of the number of adults that preferred sports cars so only the people that preferred sports cars approximately what percent were ma males ah bad English there it should be approximately what percentage were males but that's okay so as soon as they say of the number of adults that preferred sports cars then the only thing we're worried about right now is those that is it I don't care about the SUVs or anything like that that is my entire what's called sample space at this point now approximately what percent were males well there were 36 out of 84 right 36 out of 84 that were males so I take my calculator out that's not my calculator take my calculator out I think it's 38 oh no not 38 was it really 38 it is 38 bad eyes bad bad thank you Joey 38 out of 84 so we just simply do 38 divided by 84 I have to hit the approximately or the control Enter key I've got 0.452 Etc right hopefully we're all very comfortable at this point of looking at that and going well that's that's 45.2% you know plus or minus cuz those decimals keep going on so this is 0.452 dot dot dot which is approximately 45.2% and there it is wouldn't it be nice if they put the percent symbols after it sure would all right number 18 the solution to 2x^2 = 72 is which of the following all right so one of the many ways that you can solve a quadratic equation an equation where one of the variables is squared is by simply using inverse operations all right so if I've got 2x² is equal to 72 the first thing I'm going to do is undo the multiplication by two and I'm going to get x^2 is equal to 36 now without even doing the whole square root of both sides then thing right I just want you to think about this what this literally says is is I'm looking for a number that when multiplied by itself gives me 36 and there are two numbers that fit that bill right and those two numbers are both positive 6 and -6 which we can summarize as plus or minus 6 now of course all of you have probably learned that you should just take the square root of both sides and hopefully immediately put the plus minus there which would then give you that but ultimately right you know a lot of kids get confused ah when do I put a plus minus in front of the square root when do I not nah man just think about it to literally what it literally means to solve this equation is to find all the numbers that when you multiply them by themselves give you 36 6 * 6 gives me 36 and -6 * -6 gives me 36 so they're both Solutions all right let's take a look at one of the I think trickier problems on the test number 19 three quadratic functions are given below number one FX = x + 2^ 2 + 5 number two quadratic function given in a table form and number three a quadratic function given in a graphical form all right let's see what the actual question asks us though which of these functions have the same vertex all right well there's only one of the three where it's really super duper easy to figure out what the vertex is and remember vertex is the same as turning point so this one easy right here the vertex is at -2 comma 5 all right so let's now examine Choice one real quick I'm going to bring this guy out here f ofx = x + 2^ 2 + 5 now this is a quadratic function that has has been shifted all right specifically it is y = x^2 that's been shifted to the left two units and up five units right and that means right it's been shifted to the left two units because it's x+2 it's been shifted up five units because of the plus five and that means the turning point of this thing which was at 0 0 has been shifted to a turning point of -25 so these two at least for now have the same turning point so what about Choice two does this quadratic function have the same turning point or the same vertex well it's very tempting to say yes because as you look through it very quickly you see the point -2 comma 5 so it seems like yeah absolutely Choice one choice 2 Choice 3 except here's the problem the point -1 comma 5 also lies on this graph all right and notice right I've got the two fivs the two tws and the 23s we're seeing what's called the symmetry of the parabola or the quadratic function there which means that its turning point is actually halfway in between there it has a turning point or a Vertex at -1.5 now I don't know what the Y value is okay I don't know right but I don't need to know what I know is is that its axis of symmetry its vertex has an x coordinate of - 1.5 whereas this one and this one have vertices or vertexes that have x coordinates at -2 so it's only choice one and choice two oh sorry Choice One and choice three that have the same vertex Choice two has a Vertex that's close to it but probably is more like - 1.5 comma I don't know you know again I'd have to like think about it a little bit more but - 1.5 comma 6 maybe I don't know right it's somewhere in that range it's probably not that exactly but it certainly isn't -2 comma 5 parabas don't do this right which is what this thing would be doing if these two points had the same right something like that all right let's talk about Pro problem number 20 the domain of the function f ofx = x^2 + x - 12 is which of the following all right well I don't know if this ranks is the most obnoxious question on this test but I don't love it you know so let's talk about the domain right with all functions there are two sets there's the domain and there's the range all right the domain is all of the inputs all of the X values that will result in a y value all right all of the X values that will actually give you a y value out all the X values for which the rule applies the range is all the Y values all right that's a much simpler definition right the domain is all the inputs that are allowable and the range is all the outputs now let's take a look at the function itself all right let's go to the board real quick for this so that we can really see it well the function is FX = x^2 + x - 12 now this is yet again another quadratic function and we've seen a lot of quadratic functions in this right and they're really saying what x values are we allowed to put into this function well I can put any x value into this function any x value whatsoever I can put in -2000 I can put 1,152 in I I can put zero in I can put negative pi in I can put in any number I want so its domain is all the numbers from negative Infinity to positive Infinity all right and that's again a little bit tricky because normally we're going to see a domain problem literally on this page the last problem on this page is a domain problem all right and we're going to see that thing but and then the domain we'll have to like kind of get into a little bit more but here all real numbers number 21 we don't have to do thank you Brett Whitman that is a question that it would no longer show up on the next generational aligned Algebra 1 region exam so we're going to skip that one aren't you happy all right just so that we can get to an amazingly ugly inequality problem let's take a look at number 22 what is the solution to the inequality below 4 - 25 x is greater than or equal to 1/3x + 15 all right well let's take a look at this and don't let the fractions intimidate you remember whether you're using the ti Inspire the ti 83 Plus or 84 plus or hulet Packard or CIO or whatever calculator you're using all of them all of them all of them will do fraction arithmetic for you okay so don't let that throw you off too much that being said right let's now start to solve this now remember when you solve an inequality something with a greater than or a less than symbol in it everything works as it normally would except if you multiply or divide both sides of the inequality by a negative the inequality needs to flip but let's start going through it without further Ado let me just rewrite it down so that we can see it a little bit better here we go now you know some people like getting all the numbers on the same side of the equation first some people like getting all the X's I'm going to get all the sort of normal numbers on the same side of the equation so I'm going to subtract a four from both sides all right now remember subtraction's perfectly good all right we don't switch the inequality symbol there now if I want to get the x's on the same side of the equation there's no getting around this I'm going to have to subtract 1/3 x from both sides and again this is where I it's very understandable it can be very intimidating you've got -25 x - 1/3x what is that right well I mean again if you don't want to get common denominators and stuff like that I totally get that so you just do negative grab your fraction bar or just division that's fine two right and I think it was then minus 1/3 let me just make sure of that like 90% positive but yep minus 1/3 that's all I'm trying to do -25 - 1/3 and I enter and I get- 111 15 now again we could get common denominators and do all of that sort of stuff right now that'd be fine I'd be cool with that great uh but I'm going to just go -1 15 x is greater than or equal to 11 fantastic all right now how do I get rid of the 11 15 from both sides well you remember that problem with the trapezoid and the area formula and things like that well to get rid of -11 15 from both sides I'm going to multiply both sides by5 11s time -5 11s and that's going to trigger the one rule of inequalities that are different than equalities now I'm going to have to flip-flop that that inequality symbol okay but this literally just all cancels and I get X is less than or equal to the 11s will cancel -5 x is less than or equal to -15 make sure that's still the case yes still the case all right so let's take a look at number 23 which statement is correct about the polinomial oh my goodness about the polinomial oh my goodness sorry about that folks but I'm going to have to erase it all right which statement is correct about the polinomial 3x x^2 + 5x - 2 all right and then it talks about third degree polinomial constant term of -2 leading coefficients all these things so let's talk about a little bit of terminology all right there's a lot of terminology about pols including the word polinomial all right but one of the terms is the degree of the polinomial now the degree of the polinomial is simply its highest power so you have first-degree polinomial those are linear functions second deegree pols those are quadratic functions third degree pols those are cubics blah blah blah this is a second degree polinomial because its highest power is squared it's two all right which immediately means Choice One and choice two are out all right now there's also something called the leading coefficient of a polinomial a very important idea the leading coefficient of a polinomial is the number that multiplies the highest powered term so this thing has a leading coefficient of three now you might be it's the first number I see well it is the first number you see if the polinomial is written and what's called standard form which is decreasing exponents all right but if it's not written in standard form you got to be careful about that so watch out because that's one way they try to trick you they're not really trying to do that here we have a leading coefficient of three and then polinomial is always have a constant term sometimes that constant term is zero so it won't be written but they always have some kind of constant term that doesn't have an X associated with it in this case the constant term is -2 so it's a second degree polom with a constant term of two no the constant term is -2 it's a second degree polinomial with a leading coefficient of three you bet excuse me You' think the lemonade would have taken care of that all right let's take a look at our second domain question of the page number 24 a store manager is trying to determine if they should continue to sell a particular brand of nails to model their profit they use the function P of n where n is the number of boxes of these nails sold in a day let's highlight that n is the number of boxes of these nails sold in a day it's very important n is the input variable a reasonable domain for this function would be what so again another domain question after we just had one a couple questions ago all right but this one right has an applied function behind it they didn't even tell us what the function was they just said it's called P of n where the input is the number of nails sold in a day the number of nails okay so the domain are the inputs what can the inputs be well the let unfortunately the right answer is Choice one here again but let let's talk about like some of the wrong choices right integers integers are whole numbers but they include both positive and negatives so integers are things like -3 -2 -1 0 1 2 3 those are the integers well that wouldn't be a proper domain because the number of nails can't be negative now real numbers are literally all numbers that you know about they aren't all numbers but they're all the numbers that you know about those include numbers like - 10,000 3.8 7. to Pi 5 right real numbers are everything and again the N the number of boxes of nails it can't be you can't have a fractional box of nails right you can't have a negative box of nails so that one's out right rational numbers rational numbers are the whole numbers as well as the fractions both positive and negative all right but again you can't have a fractional you can't have a fractional box of nails sold in theory right plus the rational numbers would also include negative fractions and you certainly can't have negative boxes sold so non- negative integers right and what are the non- negative integers the non- negative integers are zero 1 2 3 4 Etc and that makes sense right because we can sell zero boxes of nails one box of nails two boxes of nails Etc all right so the correct answer here Choice One non negative integers all right let's pause again for a moment you know let me grab a little bit more lemonade no conspiracy theories this time just an excellent beverage we've been doing these Live reviews for quite a few years so if you need um more review material as if you wanted to do even more before your exam there are prior years Live reviews up on our up on our YouTube channel I would encourage you to go and check those out as we head into the next part of this test we're now going to start doing part two questions there are eight questions in this part all right each one of them is worth two and only two points but you can get partial Credit Now that cuts both ways though you know multiple choice questions are All or Nothing they're two points a piece they are all or nothing these you can get zero points one point or two points but I tell you you know you can do like 95% of the problem right and if you make just a little mistake you're only going to get one out of two points and of course if you make more than one little mistake you're going to get zero out of two so let's begin right away with question number 25 solve the equation algebraically 4X all right so this is a standard linear equation right right you could have really solved an equation that was very similar to this back in 8th grade the only difference is in eth grade they likely wouldn't have had the messy decimals so let's take a look at the equation -2.4 * the quantity x + 1.4 is equal to 6.8x - 22.6 all right fantastic I have no doubt that 95% of you would solve this thing beautifully if that was -3 if that was 2 if that was seven and if that was 23 right so don't let the decimals throw you off don't forget you have a calculator right so the first thing I'm going to do is remove the parentheses here by multiplying everything by -2.4 so I'll have -2.4 X and of course I'm going to use my calculator to do -2.4 * 1.4 I'm not going to break out the calculator and waste your time showing you that but it's going to be - 3.36 and there's nothing I need to do over here right now sometimes they like to combine both fractions and decimals in these problems those are very ugly all right now just like that inequality right I've got some variable on both sides of the equation and I've got some constant terms on both sides of the equation you know which you want to move when is really up to you first thing I'm going to do is I'm going to add the 3.36 to both sides of the equation and again don't let let all of the horrible decimals get to you you know take out your calculator if you need to and I probably would to do that kind of addition but when you do you'll get - 2.4x is equal to um of course I did it the other way around on my solution sheet so now I'm going to have to do it let me grab my calculator so many windows um 22. 68 plus + 3.36 that's going to be 9.32 all right so 6.8x I've already forgotten what it was 19.32 lovely uh let's see - 19.32 all right now what I'll do is I'll subtract 6.8x from both sides of this equation all right that'll get rid of that that'll be 9.2x is = to 9.32 then I'll divide both sides by 9.2 not doing it in my head whatsoever just doing it on my calculator and I'll find X is 2.1 all right now when you have solutions that are even relatively nice and by relatively nice what I mean is that they're not they don't have decimals that go on forever they're they don't have square roots in them certainly not irrational numbers don't forget you can always let me just see if I can I can't oh maybe I can let's see clear history yeah there we go look at that all clean right now remember in that problem right I found that X was 2.1 so let's say I store that next right that store command remember that equation right the equation was -2.4 X oh no no it wasn't sorry -2.4 * x + 1.4 right there's an equal sign here I stored what I2 point nopeus 22 68 good thing I checked right and now if I hit enter notice it says true here what that means is that this value of x makes this equation true which means it's a solution had I botched it right and i' had gotten let's say a slightly incorrect answer maybe 2.2 and I stored it for X let's say for some reason that was the answer I got instead of 2.1 and I typed in this equation it would say false telling me that 2.2 is not a correct answer your calculators have the ability to tell you whether nice answers and again by nice I mean answers that aren't irrational generally speaking are correct by just telling you whether they're true or false okay let's go back and let's keep going all right number 26 the function f ofx is graphed on the set of axes below uh first part of the problem State the Zer of f ofx all right great simple enough now this is basic terminology the Zer of a function are where y = 0 right in other words the x axis the zeros are right here so x = -2 two and three those are the zeros and this is just straight up terminology right you have to know that if you're identifying the zeros of a function graphically you are looking for their X intercepts the place where or place or places where the function touches the x-axis now I love this second part explain your reasoning in other words if you wrote down -2 2 and 3 and you didn't do the second part you're going to get one out of of two points that's the best you can do explain your reasoning I found the X values where the function touches the X AIS or you could say I found the X values where the Y value is equal to zero that would also be great but you have to explain how you found those and it can't just be I looked at the paper that's not going to fly and no argument coming from a teenager is going to win like the state graders over just not going to happen all right let's talk about number 27 Briana creates the pattern of blocks Below in her art class a that sounds fun a friend tells her that the number of blocks in the pattern is increasing exponentially is her friend correct explain your reasoning all right great so first part of the problem is yes no second part is explain reasoning kind of similar to that last problem except one of the big pieces here is if you just say yes or no and you don't say anything down here and there's no work whatsoever at all they're not going to give you credit for just guessing it correctly all right so the first thing you want to do in a visual pattern like this is is literally count right and I'm going to use sequence notation right I'm going to call this A1 there are four blocks A2 six blocks A3 hello red pen eight blocks and A4 is 10 blocks all right so my sequence now is 4 6 8 10 10 the friend says that it's increasing exponentially exponentially and geometrically are really the same thing and we talked a little bit earlier about arithmetic versus geometric sequences all right but what's really happening here what's happening here is that it's increasing by adding two each time well that's not exponential right if there's a common difference you can say that that's either arithmetic or you can say that it's linear all right because arithmetic sequences are just special types of linear functions geometric sequences are just special types of exponential functions is her friend correct and the answer is absolutely not no right explain your reasoning well there's a couple ways to explain it you could just say it is increasing linearly or arithmetically all right but you could also explain very clearly why it's not increasing exponentially to increase exponentially you would be multiplying by the same quantity each time to get to the next term that's that would increase exponentially multiplying by something so instead of saying it's increasing linearly or arithmetically you could say or it is not increasing exponentially because there is isn't a common multiplying Factor now the technical term in sort of sequence terminology is that there isn't a common ratio right the first ratio is 6 to4 the second ratio is 8 to6 the third ratio is 10 to8 those are not equivalent so that's another way to to do it you could say 6 ID 4 is 1.5 8 / 6 is like 1.3 I think um 4/3 yeah or 1.3 repeated Etc right so it's not increasing exponentially because there isn't a common multiplying Factor keep going here we go oo 28 the data set 20a 36 comma 52a 56a 24 comma 16a 40a 4A 28 represents the number of books purchased by nine book club members in a year and hats off to the person who bought 56 books and 52 books that's a lot of books to read in a year all right anyway um it says construct a box plot for these data on the number line below all right in order to construct a box plot what you need is what is known as the five number summary here so the five number summary now what is the five number summary the five number summary is the minimum value the first quartile value the median value the third quartile value and the maximum value all right and those values will then help us create the box plot so what are we going to do right what are we going to do what we're going to do is we're going to put this data into our calculator now I've already put it in the calculator hopefully I can figure out how to open this thing let's see let's open a document there it is this is algebra 1 June um 2023 number 36 no 28 no did I not put this one in uh I didn't put this one in okay we're just going to we're going to do it right now here we go uh so we're going to do a new file new document thought I had already entered this data but I didn't I did that for a different one and let's put the data in okay here we go I'm going to enter it just down here so that'll be quicker we've got 20 36 52 56 24 16 40 4 and 28 now what I definitely want to do always always always when I enter data is I want to check it over 20 36 52 56 24 16 40 4 and 28 great all right so I've got all my data in all right now what I'm going to do is I'm going to go to menu I want Choice four statistics I want statistic calculations now this is a very simple data set right it's just one data set so all I want is one variable statistics I can just hit okay I can hit okay and it's going to give me everything I need so let's take a look by what I mean by everything I need um somehow these things got out of order here we go right so what I need right is my minimum I need my my my summary my five number summary so my minimum is four my first quartile is 18 my median is 28 my third quartile is 46 and my Max is 56 literally there is our five number summary Min quartile one median cortile 3 and we're going to need quartile 1 and quartile 3 later on and Max 56 so let's go back over to our test and write all of those down right my minimum is 4 q1 is 18 median is 28 Q3 is 46 and Max is 56 now typically this is where it's easy you know you're just like draw a line above where the minimum is and where the q1 is and the median and the Q3 and the max all right but check this out can you imagine who made this question 020 460 seems good but what are each one of these marked off in let's take a look how many spaces are there between Z and 20 there's 1 2 3 four five you might be like oh five cool awesome but that means they go by fours this person whoever made this scale scale marked it off in fours not twos not fives not tens no they marked it off in fours all right this problem is as much about the idiotic scale that is on this particular number line as it is about all of this because you could get every bit of this right and then if you think that this is by fives or twos or something you get yourself in trouble and look at how ugly it is right it's 4 8 12 16 20 24 28 32 36 40 44 48 52 and 56 that's what each one of those things represent right 10 and 30 and 50 they're not even on here but now the rest of it's actually pretty easy all right so the minimum is four I'm going to draw a line right above four q1 is 18 which is halfway between 16 and 20 the median is 28 so I'm going to put a line right above 28 Q's 3 is 46 so I'm going to put a line halfway between 44 and 48 and the max is 56 so I'm going to put a line right above 56 how do I complete my box diagram I will put a box around the middle 50% of the data right that goes from q1 to Q3 you don't have to put those on your graph although you could and then I'm going to extend out like that and that is my box plot right and again it was funny because when I first did this you I cranked out the data I got my five number summary I started to go down here to plot it and I'm like wait a second what is the scale there oh my goodness they went by fours do you know what I mean and you pretty much have to I'm sorry I think you got to like write the entire scale down here otherwise when you're thinking about like like particularly nasty when you're thinking about things like 18 and 46 you know where is 18 on this thing right you know that's tricky a lot of kids I would assume would think that the 18 is where the 16 hash mark is because they're thinking about it going by twos or by fives or something anyway horrible I don't know why they would do that test exactly the wrong thing okay well that that was weird oh just stop there we go okay let's take a look at 29 a equal x + 5 b = x^2 - 18 Express a^2 + B in standard form all right great now this is not unusual they take a couple capital letters or whatever and they set them equal to two simple polinomial and then they manipulate them they might multiply them by a constant they might Square them you know stuff like that and then they ask you to combine them so what they're literally asking us to do here is they're asking us to take x + 5 which is what a is square it and add it to x^2 - 18 all right and they want it in standard form which means it's got to be a polom with decreasing term terms in terms of their exponents and coefficients and all that stuff now maybe the hardest part of this is just being clear about evaluating x + 5^2 right so so you've got to be able to multiply out x + 5 and this is critical and you probably learned lots of different ways of doing this I'm just doing the foil method right plus 5x plus whoops plus that should be a plus 5x + 25 so that's x^2 + 10 x + 25 all right so that's just what a s is that's just a s right so I now can say I've got x^2 + 10 x + 25 + x^2 - 18 now notice right besides that like little red there those parentheses really aren't doing anything at this point right because there's no numbers sitting in front of them that we have to sort of distribute through if you want to think about it as being a one in front of it that you're Distributing through great but ultimately what that means when there's nothing in front of them no subtraction or anything like like that then I'm really just dealing with this so now I can combine like terms I've got an X2 and an X2 which is 2x^2 there's nothing that combines with that 10x and then I have a positive 25 and a -8 which will combine to be a positive 7 so 2x^2 + 10 x + 7 okay let's keep going oh hey next up the most irritating problem of the test for me I don't even know that I'm going to like do this problem per se but we're going to talk about it problem number 30 the two relations shown below are not functions awesome explain how you could change each relation so that they each become a function I cannot tell you how much I dislike this question and how much I disagree with the state putting this question on this test all right functions are function and what that means is that for every x value there is one and only one y value let me put that another way for every member of the domain the input set there is only one corresponding member of the range now there could be repeated members of the range repeated yv values but no x value can get repeated so I would have no problem with this question whatsoever if they just simply said explain why each one of these is not a function right but you can't just you know be like oh these aren't functions let's change them into functions like in that case there's an infinite number of correct answers for instance it could be like oh relationship one let's just replace it with that line that's a function do you know what I mean I explain how you could change it I would just do this okay great it's now a function does it look at all like the original no but they didn't say explain how to modify each relationship so that it is as close to possible to the original relationship but now happens to be a function that's not the question the question is just how could you change each relationship so they become a function and I'd be like well change this one into y = x^2 change this one into Y = 5x they're both functions now follows the direction I guarantee I'd get a zero for two on the problem but that's not my fault that's the fault of the test cre anyway yeah that's right let's talk about why each one of these is not a function and then talk about what answer they probably would want all right so the first one the reason it's not a function is because of these two points literally literally this is the point 4 comma 20 and this is the point 4 comma 30 and that's a problem because that means that that that input xal 4 has two outputs both 20 and 30 so probably what the state is looking for on this one is for you to say you would take either 4 comma 20 or 4 comma 30 and of course it makes a difference which one you do but you would take one of those and redraw it as an open circle kind of like here because that open circle means that that Point's not actually on the graph right so you would take one of these two points and make it into an open circle all right now let's take a look at relationship number two why is that one not a function well let's take a look we have -5 comma -2 -4 comma 0 -2 comma 1-1 comma 3 and -4a 4 well again the only problem here is these two and it's a problem for exactly the same reason that this is a problem right which is that I have an input of -4 that has two different outputs in this case an output of zero in this case an output of four okay and that's a problem so what the state probably wanted you to do was just say get rid of one of them so get rid of one of these the funny thing is again you could be like get rid of both of them it would make it into a function right they're both kind of problems shoot I could have two Open Circles here and here and that would be fine it would still be a function right but that's not what they're looking for anyway it's a horrible question it's just terrible um shame on whoever wrote that question that that's just weird anyway 31 straightforward no complaints here 31 Factor 2x^2 + 16x - 18 completely completely completely completely all right great great so factoring means to break something up as the product of other quantities right to break a polom into an equivalent product now when it says factoring completely what it means is that you're probably going to have to do more than one factoring technique now there's three factoring techniques that you're supposed to know in this course one of them we've already seen is the difference of perfect squares all right another one is What's called the greatest common factor and you've been doing that one for years and then the third one is a trinomial okay what you should always look for if they say to factor something completely is you should always look for the greatest common factor first and when I look at the numerical coefficients 2 16 and 18 I noticed that each one of them can be divided by two so the first thing I'm going to do is Factor out a common factor of two now I factor it out but I don't want to lose it because if I lose that two then I will lose a point on this problem all right and then if I botch factoring this I'll lose two points and I'll go over for two all right so now I think about this trinomial okay and you were probably taught in many different ways how to factor trinomials generally speaking right when the leading coefficient is one which it is now we can look and we can think about two numbers that have this sum and this product so I want two numbers that have a sum of NE of positive 8 and a product of -9 and those two numbers are postive 9 and -1 right if I do 9 + -1 I get 8 and if I do 9 * -1 I get -9 and that means that this thing now factors into 2 * x + 9 * x -1 and that's it complete factoring simple great not as simple 32 solve 3D ^2 - 8D + 3 is equal to zero algebraically for all values of X or sorry sorry for all values of d rounding to the nearest 10th all right let's talk about a couple Flags in this problem that I want you to be aware of one of them says solve it algebraically and the other piece is to the nearest 10th all right to begin with they say solve it algebraically because in theory you could solve this graphically how would you solve it graphically well you'd take the left hand side of this you'd put it in as a function you'd graph it and you'd use your zero command to find the zeros of the function and if they didn't say to solve it algebraically that would be fair game you'd just have to draw a little graph you'd have to label it some things like that but would be fair right the other piece is this nearest tenth piece now that is telling us that the answers are very likely to be kind of ugly all right implying that what we should be doing in this problem is using the quadratic formula all right now the quadratic formula which is most certainly on your formula sheet right right here x = b plus orus the < TK of B ^2 - 4 * a * C / 2 a what is the a the a is the coefficient of x^2 what is the B that's the coefficient on X what is the c that is the constant term a is the leading coefficient B is the the the coefficient of the linear term x to the 1st C is the constant so let's go back over to here right oh my goodness now they have D's instead of x's well solving this is exactly the same as solving this all right so hopefully the fact that they have D's there isn't going to throw you off so what is the quadratic formula well let me write it in terms of D D = B plus or minus theare < TK of b^2 - 4 a c all over 2 a all right where a a is equal to 3 B is equal to8 and C is equal to 3 all right so I've got all the numbers this is now what we call a plug-and chug exercise all right so that's going to be8 plus or minus the square < TK of 8^ 2 - 4 * 3 * three oh that's nice all divided by 2 * 3 all right now8 is POS 8+ or minus square root now in a problem like this where they say to the nearest 10th or even if they said simplest radical form which could also happen sorry about that the most important thing is getting the number underneath the square root correct so what I always say to students is look take your calculator right take your calc calculator and just put it in and what do I mean by putting it in you put in What's called the discriminant b^2 - 4 * a * C so this expression with the parentheses around the8 really critical that you have the parentheses around the8 without the parenthesis around the ne8 you're going to get the wrong answer all right but if I do this and I hit enter then that tells me the number that should be the square root that's called the discriminant of the square root or the discriminant of the quadratic function so that's going to be plus or minus 28 2 * 3 is 6 so I want all values of D so what are my two values let me label them D1 and D2 one of them is going to be 8 plus the S < TK of 28 over 6 and one of them is going to be 8- the < TK 28 / 6 now some of you at this point might have the urge understandably to simplify the square < TK of 28 to break it up into the < TK of 4 * the < TK of 7 and then make the < TK of 4 into 2 giving you 8 + 2 * < TK 7 / 6 and 8 - 2 * > 7/ 6 there'd be nothing wrong about that at all whatsoever but you don't need to because they're not asking for the answers in simplest radical form they're asking for the answers in terms of a decimal so all I need to do at this point is just come in here I love using the fraction bar for this I'm going to do 8 plus the square < TK of 28 wow I don't know how that became squared there apparently I didn't quite get the oh you've got to be kidding me okay it just keeps giving me the square let's try it one more time controll there it is o all right so I have 8 + the < TK of 28 / 6 enter and I want that I think to the nearest 10th so that's going to be 2.2 and I can go up now instead of typing it all in the same I can just highlight the thing hit enter kind of use my back arrow keys delete that and put subtraction in remember subtraction not the negative button but the subtraction button and there's my other one right I don't have to retype it all in and now I can kind of come back go into my test and I've got this let me just write out some of the decimals 2.25 Etc it says to the nearest 10th make sure I got my rounding right and again this is another horrible thing you could do everything correct in this problem beautiful work just beautiful work and if you round to the nearest 100th instead of the nearest 10th you'll get one out of two points one out of two which which is ridiculous right because this question is really testing do you know how to use the quadratic formula but they're going to zap you on the rounding because these things are irrational numbers so you got to round to some level of accuracy if you're expressing it as a decimal this one is 0.451 4 Etc so to the nearest 10th it's 0.5 and there are two answers and again it's really frustrating because I've seen so many kids that will do problems like this and they know what they're doing they know what they're doing they know the quadratic formula they know how to manipulate it and then they do 2.22 and 0.45 or something like that and they get like half credit taken off even though they knew 95% of what they were doing right they just botched the rounding all right we're up to part three so let's take another moment let's take a little break little more lemonade for Kirk it is now a quarter till 8 we've already been going for almost 2 hours but the good news besides how tasty this lemonade is and how fantastic the emath instruction website is besides that right the good news is we've got four of these part three questions we've got one part four question and then unfortunately we've got a couple extra questions I made up just because it's a new test all right so so let's jump into the part three questions now we've got four of these each one of them is worth four credits all right so there's a lot of chance for partial credit here that's the good news the bad news is that often times these questions are actually just a couple two-point questions squished together that have some kind of unifying theme to them and we'll see that starting right away in question number 33 now one thing I wanted to do uh that I should have done right away and I didn't let me just open up Smart Notebook really quickly because this question is a little bit better hopefully it's not going to take all day and if it does we'll just do the problem in here come on SN notebook you got it you got it you got it all right real quick open hopefully I'm going to be able to find these things there they are all right here we go nice all right so here's question 33 it's just going to be easier for us to do it sort of in this view okay so let's take a look graph F ofx equal the absolute value of x + 1 and G ofx = x^2 + 6x + 1 on the set of axes below all right so the first thing that you've got to be able to do in this problem is you've got to be able to graph functions when they give you equations now the easiest way to do that is to put it into your calculator and generate a table of values and graph and we'll go through one of those in a moment but this one F ofx equal absolute value of x + 1 I'm going to do that from a shifting perspective this thing whoa this thing that is way too thick now um this thing is just y equals the absolute value of x that's been shifted up one unit all right so you should feel very comfortable with the absolute value of x graph that's just that V Gra graph and this thing then would kind of look like this and I'm not going to we could put it into our graphing calculator and get a table of values but again short on time kind of thing and of course you want to draw these nice straight lines with a ruler as opposed to what I'm doing and what you definitely want to do is label them with their equations boy is that important they just love taking points off when you don't label your graphs okay that being said now let's really quickly review how we get a table on our calculators and again unfortunately this is got to be done with my ti because it's the only calculator I've got right here so I'm going to just really quickly open up a new document nope I want to graph all right so here I'm just going to put in my - x^2 + 6x um + 1 all right so it's going to be a parabola now I don't need to see the whole thing it wouldn't be bad to see the whole thing and I could put in my My Graph grid here which is -10 to 10 and -10 to 10 but what I really need is the table right so I could go into my menu and get my table all right and I want a split screen Table and there we have it right so now I have my table and you know I want to kind of like find my points and this is the first one that's going to graph on there right -1 comma 6 0a 1 1A 6 2A 9 3 comma 10 4A 9 Etc right so I'm going to be grabbing those off of here and then I'm going to just simply yeah see I want to be back in here um let's go view full screen so what I'm going to do is I'm just going to draw a table right now of my values for G of X and again this would be from my calculator I have -1 6 01 1 6 it's not bad to have these tables actually written on your paper um okay so those are now my coordinate points along with a nice squiggle here for no apparent reason all right and I can start to graph them all right so let me do it I've got negative 1 1 2 3 4 5 6 negative 1-6 01 actually let me do this in red I know that on the test you wouldn't be able to do it in red but let me just have a nice nice contrast there 1 1 2 3 4 5 6 that's 1 six 2 1 2 3 4 5 6 7 8 9 310 49 look at the Symmetry starting to occur 5 6 61 and 7 -6 all right draw that in all right make sure it's a smooth curve don't draw it with straight lines on this one this is going to be G ofx = x^2 + 6x + 1 label your graph label your graph label your graphs right label your curves that is okay very very important that you label your curves at least one of them when there's two all right and then that's it right that's not the whole problem but it's most of it let me really quickly go back to Blue back to Blue right and then the second part of the problem based on your graph determine all values of X for Which F ofx is equal to G ofx and boy I hope your teachers have hammered this home for you right this is what it means to solve an equation graphically right I'm asking for the values of X where f ofx is equal to G of X what I want are the x coordinates not the the Y but the x coordinates of the intersections all right so let's go up here real quick we intersect here at the point 0 comma 1 we intersect here at 1 2 3 4 5 6 5 comma 6 but those aren't the answers right the answers are just the X values how I how do I know cuz it asks me for the X values so x = 0 and x = 5 and that's it right now I didn't look at the exact grading key but I would be surprised if on this fourpoint problem it wasn't three points for this and one point for this but it could be two and two especially given that there are two values of X that are solutions all right but watch for this there will guaranteed be either a multiple choice problem War free response problem where you have to use a graph in order to solve an equation all right let's keep going whoops that's not what I want let's go back to the test we'll get to that one in a second all right back here we finish that one all right great let's take a look at number 34 34 is definitely a situation where it's a four-point problem and then they ask you three questions three questions on a four-point problem whatever 34 Gene recorded temperatures over a 24-hour period one day in August in Syracuse New York go Syracuse go orange her results are shown in the table below all right and let's take a look at this table for a second okay so for a second right when we look at this table let's come out of it um right if we look at the time 0 3 6 9 12 15 18 21 24 notice that those times are actually spaced evenly right they're every 3 hours and that's not critically critically important but it's kind of nice to just note in this table and then we've got the temperatures great so here is that temperature graph you know that she showed us with a nice little axis break right here awesome so now let's take a look at this first question State the entire interval over which the temperature is increasing fantastic right so let's just look at the graph right at Zer hours to 6 hours the temperature is definitely decreasing then from 6 hours up to 12 hours the temperature is definitely increasing and then from 12 hours down to 24 hours the temperature is definitely decreasing now a lot of Algebra 1 teachers understandably get caught up in talking about intervals of increase and intervals of decrease should I include the six and the 12 should I exclude the six and the 12 the crazy thing is the state doesn't care it really doesn't so you could say all sorts of things on here you could literally and remember it's it's here it's from 6 to 12 you could literally say now I got to go back to the thicker line thess all right you could say 6 to 12 that's a great answer they love that they even took this 6-12 which kind of drives me nuts because it looks like you're doing 6 - 12 but I guess context is everything here um if you said something like T is greater than 6 less than 12 that would be great if you even said 6 is less than or equal to T is less than or equal to 12 that would be great all of these the state will take all right you know now there's some Math teachers that are like absolutely not you can't include the six or the 12 you that that's terribly wrong and I would disagree but that's not the point here the point here is that you basically just have to say from 6 hours to 12 hours the temperature is increasing one way or another you have to get that idea across now let's take a look at the next part of this problem State the 3-hour interval that has the greatest rate of change in temperature in in other words how quickly the temperature is changing now really to do this really scientifically what we would want to do is we'd want to look at each one of these and kind of go all right that was -5 that one was5 this one was positive 8 this one was pos4 this one is -3 this one is -4 this one is -5 and this one's Min -6 right well clearly the interval at which the temperature increased the most right was from 9 to 12 right when we went from 78° to 92° we increased by 14 all right now we can just look at those changes because these time interval are all spaced evenly if they weren't then we'd have a tougher call right but you can literally say from 9 to 12 hours or just 9 to 12 something like that right that is this section another way to judge that is by the steepness of the line how steep is it this is the steepest portion of the line meaning that it's got the greatest rate of change all right you can really visually tell that right that is steeper than this portion and it's steeper than these downhill portions as well right okay finally one that we can actually assign some numbers to last part of this problem State the average rate of change from hour 12 to hour 24 explain what this means in the context of the problem all right average rate of change now you've seen this quite a bit over time from 12 to 24 well let's just take a look at this table right at 12 hours we're at 92° let's write that as a coordinate point and at 24 hours we're at 74° all right now average rate of change is always the slope formula the change in the output minus the change in the input but you have to be consistent all right so the average rate of change or the arc is 74° f - 92° F all divided by 24 -2 okay now you can totally do this all on your calculator and all at once but boy it is really important to have 74 minus 92 and not 92 - 74 all right is absolutely critical because the temperature is decreasing and that's going to be part of the explain piece all right so we we'll get there in a second but like literally you can take your calculator and just go all right I'm putting my fraction in I've got uh let's see what was it 74 - 92 in the numerator and in the denominator I have 24 - 12 if I hit enter I get -3 hves right if I did control enter it's - 1.5 if you want it as a decimal either way is completely fine I like the decimal personally but -1.5 or -3 haves is fine right and now the question is what does this mean in the context of the problem well this is where units are really helpful that's degrees Fahrenheit per hour that looks like the word of degrees Fahrenheit per hour so how do I interpret this well I interpret it as from 12 to 24 hours the temperature is decreasing by 1.5° F per hour all right and everything's important there the term decreasing really in critical I mean you would definitely get away by saying the temperature is going down by 1.5° Fen per hour right but you have to indicate that the average rate of change because it's negative is telling you that the temperature is going down by this amount per hour all right o oh man systems of inequalities gosh why can't we get away with not having one of these on the test just once well maybe you won't have one on tomorrow's test you just never know all right let's go back to here we've got another one of these things um whoops come on let's take a look at in Smart Notebook 35 solve the following system of inequalities graphically on the set of axes below label the solution set s and then there's another portion of the problem and of course to make matters worse not only did they give you a system of inequalities but they gave you one where you have to manipulate both inequalities both inequalities and then graph them all right so let's talk about manipulating these inequalities first thing I'm going to do besides changing my my line width is I'm going to rearrange this one all right now what I want to do is I'm going to subtract a 2X from both sides and I'm going to get 2 Y is greater than or equal to -2X - 6 now I'm going to divide everything by two and I'm going to get Y is greater than or equal tox - three that's that's a three y Kirk not a two y that makes a big difference that's just ah okay um let's see so that's going to be a 3 y and a 3 y everything's everything's bad everything's wrong let's try this again 3 Y is greater than or equal to -2X - 6 divide by 3 divide by 3 divide by 3 notice I haven't divided or multiplied by any negatives so the inequality stays exactly the same all right so that first one is y is greater than or equal to -2/3 x - 2 so how do I graph this well what I'm going to do is I'm going to first graph the line that my equality has as its border which is -23 x - 2 so I'm going to not use my calculator on this what I'm going to be doing is just my old fashion go down to my Y intercept -2 I'm going to go to the right three and down two to the right three and down two right I'm just graphing a line left three and up two left three and up two left 3 and up two now remember we always have to think about whether the line that connects these points is solid or it's dashed right because this is a includes the equal sign we have a solid line right so I'm going to draw that line in solid all right now I have to think about shading because this is greater than I'm going to shade above it right so I'm shading above because it's Y is greater than all right finally I definitely want to just label my inequality I'm just going to kind of erase this and I'm going to label this Y is greater than or equal to -23 x -2 all right so I've got one of my inequalities graphed let's take a look at the second one all right the second one's almost as bad maybe it is just as bad what is that that's X maybe it's worse is less than 3 y + 6 all right so what I'm going to do is I'm going to subtract six from both sides x - 6 is greater is less than 3 y I'm now going to divide everything by three okay and I'm going to get3 x - 2 is less than y or better yet Y is greater than 13x - 2 so they've thrown all sorts of irritating things in here you know putting X on the left side thankfully what they didn't do is make us flip the inequality at any point all right so let's now do this one again I'm going to go down to a y intercept of -2 and I'm going to go right three and up one right three and up one Etc then left three and down one all right now we have to think about solid versus dashed because this doesn't include the equality we're going to make it a dashed line okay because it's greater than we're going to shade above it again so so oops let me move this out of the way don't want to overshare here all right and notice right it says label the solution set s well keep it in mind right the solution set to a system of inequalities is where the shading overlaps right so there's no shading down here but there's shading here and over here that doesn't overlap and there's shading here that does okay so you just got to make sure that s that giant S no matter how messy your picture is is somewhere where the shading overlaps all right so there's my my solution set s okay but there is a second part of this problem and almost always is remember this is a 4o problem here's a yes no problem here's an explain your answer problem so the shading all of that is just two points is the point 4 comma -2 in the solution set explain your answer well let's take a look right um I think the easiest way to do this personally is to graph the point 4 comma -2 1 2 3 4 1 2 there's the 4 comma -2 right now for me it's straightforward the answer here is no explain your answer it doesn't lie in the overlapping region right another way to do it is to show that it doesn't make each inequality true now by the way you can tell that it does make one inequality true it lies in the shading of the first inequality so if I took 4 comma -2 and I substitute it into this thing it would make it true but if I tried to substitute it into this one it would make it false which would mean it's not part of the solution set and that's another way to answer the question is to Simply show that the point either makes both inequalities true in which case it does fall in the solution set or it makes one or both of them false in which it does not fall in the solution set and again specifically take a look at this we've got X Less Than 3 y + 6 it's this one that it doesn't work for and in fact you can see that if I put four in for x and I put -2 in for y I'd get 4 is less than -6 + 6 and 4 is less than 0 that's no which means that's a no in that case you would show this work and then you'd say explain your answer you'd say well it doesn't make the second inequality true and that would be your answer okay great let's go back to how do I get out of here where is my I don't see my way of getting out it's very weird all right well fine I'll do it that way I'm like I can't find the little X that's very strange and I have no idea why um okay that's right I think we'll live excellent this problem made both my wife and my my daughter very very happy let's take a look at why number 36 Suzanna collected information about a group of ponies and horses ah it's not just about the ponies and horses which they love she made a table showing the height measured in hands and the weight measured in pounds of each Pony and horse all right so then they give me a table of the heights of horses in hands and their weights in pounds and you might be like height in hands well for any of you that have ridden horses especially competitive you know that there's an old system literally for measuring how tall a horse is by how many hands high it is and you can imagine in the old days you know this like Medieval Europe or something they're like how high doth this horse be you know and then they're like well I don't know it's you know eight of my hands you know and that would be a very short horse you know nowadays I think a hand is exactly 4 in but I'm not sure anyway not the point let's take a look so they give us this table we've got hands here and we've got weight here and the first thing they want us to do is write the linear regression equation for this set of data round all values to the nearest hundredth all right so this is a regression question and it's not a surprise given that we've got a table of values I'm not in any way shape form surprised that this is regression all right so this is pure calculator now this is one where I actually did I think yeah number 36 I did put the data in my calculator already so let me open that up file Open document June 23 number 36 open all right so I've already put the data in I didn't want to waste your time and notice I've actually labeled my first colum height and my second column weight okay and it's really really helpful when you do linear regression on the ti Inspire this is a little bit different than on the 83 or the 84 plus but on the Aspire it's really helpful to have the height and the weight or whatever the column A and column B labeled all right but now this is just button pushing right I go into my menu I go into statistics I want statistical calculations and I specifically want choice three linear regression MX plus b so I hit choice three now it asks me for what my X list is well I want my X list to be that height right that height in hands all right my y list on the other hand is weight in pounds I don't need to worry about any of the rest of this totally irrelevant so now I'm going to hit okay and it's going to spit it all out right title linear regression I don't care right the regression equation M * x + b the m is 184. 892 Etc the B is 1,76 something and I think we need to know the point something because that's right they want it to the nearest hundredth right and it's funny because on this it's actually hard to see the hundredth unless you go down and you highlight it but that's basically it right there's nothing more than that I go into here and I simply write down y = 184. 89X - 176.007 and again you got to be careful on this I mean I hate this but this is a fourpoint problem and if you round these things to the nearest whole number or the nearest tenth you've just lost the point for this you know and I actually think this is worth two and then there's a one and a one all right so you probably only lost one point if you round incorrectly but it's a shame you know what I mean so just watch that I could easily make that mistake myself thus making me not get a 100 the next part of the problem State the correlation coefficient for the linear regression round your answer to the nearest 100th the linear correlation coefficient is also known as the R value all right the r value so let's go back to our calculator and see what our r value is here's our r s and there's our r value 0.985 etc etc really messy but there it is and it's really nothing more than just kind of getting that down to the nearest 100th now that's let me write it out and it's kind of gory decimal version Etc again rounding and this is definitely worth only one point I need to round to the nearest 100 which because there's a five there means that to the nearest 100 this is a whopping 0.99 all right and the last part of the problem explain what the correlation coefficient indicates about the linear fit of the data in the context of the problem I really don't like the answers that they wanted on this so essentially I looked at this and said well the R value is very close to one so there's a strong positive linear fit between the height of the horse and its weight and believe it or not that wouldn't have gotten credit right what they really wanted on this was for you to look at the R value and go oh the R value is positive and so not having anything to do with it being close to one they want you to then say um because R is positive as the height increases so does the weight surprise taller horses are heavier now to me you know I would want way more I would want students to say you know there is a strong positive correlation between the height and the weight you know meaning that we can definitely predict the weight if we know the height based on the linear equation right there's all sorts of things but what they wanted when we looked at the model response sets were as the height increases so does the weight the thing that's ironic about that is that if the R value had been 0.2 you would say the same thing 0.1 you'd say the same thing you know so it doesn't get at all in terms of how close that r value is to one and the fact that it is very close to one means there's an extremely strong positive correlation meaning that the equation actually predicts the height of predicts the weight of the horse quite well if you know its height in terms of the hands again weird but true all right and this brings us to the last problem of this particular test which would mean we were almost done but then we got these other questions we have to to do anyway the last question of any of these tests is always a Sixpoint question not sure if that's the case on the new format or not but at least on this format it's a six-point question it's often broken up into three two-point questions because hey why have a six-point question when you can have three two-pointers anyway little drink of lemonade and then we're going to get to this last question but first I'd like to like spill the lemonade all over myself that's that's important little lemonade After Shave that's good for the skin it's probably not good for the skin don't don't take my word on that I tell you you know one thing is certainly true when you stand in front of a camera in front of lights for three straight hours although it's only been 2 hours and 15 minutes right now you definitely get thirsty and you definitely get tired so let's take a look at number 37 the last problem on this particular test Dana went shopping for plants to put in her garden she bought three roses and two daisies for $31.88 later that day she went back and bought two roses in one Daisy for $18.92 if R represents the cost of one rose and D represents the cost of one Daisy write a system of equations that models this situation all right well this is a classic question and I hope you did plenty of systems of equations in your course in your algebra 1 course so you see this one and you're like autopilot okay now keep in mind R is the cost of one row whatever it is $33.25 $7.50 I don't know yet and D represents the cost of one Daisy maybe it's a buck 25 maybe it's 250 again I don't know yet but they're the costs of a single row and a single Daisy now what do we know we know that she bought three roses and two da da for 3188 so if I take three and I multiply it by the cost of one rows and I take two and I multiply it by the cost of one Daisy and I add those together I have to get 3188 right if I take the cost of one rows and I multiply it by three I get the cost of three rowes and if I take the cost of one Daisy and I multiply it by two I get the cost of two daisies and then when I add those two costs up I get the total amount I spent then she went back later and bought two roses in one Daisy for $18.92 same idea right if I take the cost of one rows and multiply it by two and add to it the cost of a single Daisy I'm now going to get $18.92 and that is simply the system of equations that represents or that models this situation right done now some students will write this down and they'll immediately start solving it because they're just like in this autopilot mode which I totally understand but the solution's now going to come here so let's take a look use your system of equations to algebraically determine both the cost of one rows and the cost of one Daisy awesome so in other words I've just got to solve the system of equations and I'm going to use the process that's known as elimination now to do elimination what I really want are what are called additive opposites right I want one equation to have a neg five in it and the other equation to have a five or one equation to have a neg -3 and the other equation to have a three or something like that now right now if I added these two equations together nothing would cancel 3 r + 2 R would be 5 R 2D + 1D would be 3D right nothing would cancel and I need something to cancel now I could multiply this equation and this equation by numbers and get the RS to cancel but the easiest thing to do will be to take this equation that second one and multiply it all by -2 so just for a moment I'm going to take this first I'm going to take this equation and I'm going to multiply both sides of it by -2 and what's that's going to give me is -4 Rus 2D is equal to 37.843220 right underneath it 3r + 2D is equal to 3188 88 and I'm going to add them together and when I do add them together I'm going to take -4 r + pos3 r and I'm going to get a net Nega R now -2d and positive 2D cancel each other out and then on my calculator I'm going to do - I'm going to get - 5.96 so R equals - 5.96 and you're more than welcome to divide both sides by Nega 1 but look if R is equal to 5.96 then positive R the cost of one rows is positive $596 a strange price for Rose yes but it is the way this problem is now I also need to know the price of one or the cost of one Daisy so how do I figure that out well the key is now I'm going to take that 596 and I'm going to substitute it either into this equation or this equation and solve for D now in my opinion it's easiest to substitute it into the second equation just marginally easier and it'll be easy then to solve for D so here we go I'm going to take this equ wh no lovely come on there we go I'm going to take that equation just rewrite it down here and I'm going to take this and I'm going to put it in for R so I'm going to have 2 * 5.96 + D is equal to 18.92% + D is equal to [Music] 18.92% anyway it sounds like a golden doodle um so we've got daisies at $7 we've got roses at $596 and this is a classic just like a classic classic kind of problem in terms of systems of equations I'm going to grab that solution set back all right so let's take a look at the last part of this problem if Dana had waited until the plants were on sale she would have paid $4.50 for each Rose and $6.50 for reach Daisy determine the total amount of money she would have saved by buying all of her flowers during the sale now the thing that's kind of interesting about this problem right is that there is just there's no algebra involved in this at all it's almost like they were like well we got to make it six points how do we get two more points now personally I would have liked to have just seen them make that problem previously be the whole deal all six points but but watch this this doesn't have anything to do with algebra right what did she pay originally originally she paid 3188 and 1892 so let's just find out how much total she paid originally originally come on man I cannot originally originally right she paid 3188 8 +$ 18892 that's what she paid and that was for a total of $50 80 right so we we don't we don't have to do anything for that other than just you know look and go well let's see when she first went uh she spent 3188 and then she went back and spent $ 1892 so she paid $508 for these roses and exceptionally expensive daisies all right so now keep in mind right what she bought she bought three roses Two Roses so she bought five roses and she bought three daisies so now let's figure out what she would have paid for on sale so on sale right she would have paid what $450 for each row so there would have been five times 450 and she would have paid 650 for each Daisy which is still too expensive right but I can just put that into my calculator and figure out what that'll be and that's $42 even right so that's not hard again we we know she bought five roses we know she bought three daisies they gave me how much of those things were right here and here so I just crank it in my calculator and do that and so what's my savings my savings will be my how much I spent or how much she spent $508 minus $42 which is $880 no algebra there like at all all right and that's that problem and that is that algebra test right and it really besides that one problem that we eliminated every other problem that was on here could have easily be on the test that you're going to take tomorrow right so the difference between the common core standards and the Next Generation learning standards isn't very large right but there's some topics that when we look specifically at this formula sheet when we look at this formula sheet specifically two things What's called the point slope form of a line and what's called the outlier test these two things are both problems that even though they're not particularly mentioned in the standards per se either the common core or the Next Generation standards even though they're not mentioned there because they're on the formula sheet the test developers have told us here that they could be on the test all right so I created a set of supplementary problems to kind of go over this all right so let's take a look at these supplementary problems and these will be pretty quick and then I promise I'll let you stop watching math for the night question number one which of the following is an equation of the line that passes through the points -3a 8 and 3A 5 all right and then we look at all of these things and it's all like y plus this number or Yus this number equals some number * X plus or minus etc etc right so here's the idea all right if we have a line if we have a line and we know it's got a particular slope let me call it m and we know that it goes through a particular Point let me call it X1 comma y1 okay then we can write the equation of the line by just simply doing y - y1 = M * x - X1 that's what's called the point slope form of a line and it is a super duper convenient form of a line here it is y - y1 = M * x - X1 now you might say to yourself but there's you know every Line's got an infinite number of points on it so wouldn't this mean then that every line has got an infinite number of equations and the answer is yes they are all algebraically equivalent and they're all algebraically equivalent to y = mx plus b but yes every line can actually be expressed in an infinite number of different equations due to the point slope form of the line and other algebraic reasons but Mo but let's just stick with that for right now all right so one thing that you have to know regardless is you have to know the slope of the line so let's figure out the slope of the line using these two points so I'm going to say m is equal to 5 - 8 over 3 - -3 5 - 8 is -3 3 - A -3 is POS 6 and -36 is the same as one2 now one of the good pieces there is that we can look at the four choices and we can eliminate Choice One and choice two because those two have a slope of -2 and not a slope of - one2 although of course we get plenty of students who will say that -3 / by 6 is -2 that's a little bit unfortunate but it is what it is and if you're not sure put it into your calculator now then the question becomes well which point of these two should I use as X1 y1 you might say well I've got to use this point well it could be either one of these now notice in this equation we subtract the coordinates right so let me write down the two equations that would be legitimate based on both of the points so in the first case we would have y oh boy that was weird um we would have y - 8 is = -2 * x - -3 right so y - 8 = -2 the slope * x - -3 but they will never leave it this way when I subtract a negative it will become a positive so this is y - 8 = -2 * x + 3 so that is one legitimate equation for this line all right the other one using the 3 comma 5 would be y - 5 is equal to -2 * x - 3 and that one is just done the way it is so then the question is which of those two is one of the choices and that happens to be choice three all right point slope form of a line now I topic that is harder to talk about but let's take a look at it question number two given the data set shown below identify any outliers for the data set justify your final answers if there are no outliers state so okay so outliers now you've been talking about outliers probably since about seventh grade statistics and the idea is very simple an outlier is a data value or more than one data value that lies way way way away from the mean you know or better yet way way away from the median all right but how do you quantify that how do you like definitively say well this thing's an outlier but this thing isn't an outlier all right well let's let's go back to that formula sheet really quickly all right and this is what it's all about so let's talk a little bit about what these formulas are right it says lower outlier boundary q1 - 1.5 I QR upper outlier boundary Q3 + 1.5 IQR what in the world does that mean right what what what all right I'm just going to go back over here real quick so we can talk about this right so when we have a data Distribution on a on a box plot all right the width of this thing how wide it is literally how wide it is this is the IQR the IQR is the width of the box it's Q3 minus q1 right it's the width of the box so you've got this formula that says q1 minus 1.5 IQR and they put this in parentheses whatever right so what what does that really mean well q1 is sitting right right there on a number line the idea is if I took this box and I multiply that width by 1.5 which makes it 50% wider right that's what multiplying by 1.5 would do it would make it 50% bigger and I subtracted it right you know maybe I that would put me way out here right so any value that's to the left of that thing that's to the left of q1 minus 1.5 IQR that's an outlier and likewise right if I add 1 and 1/2 times the IQR to the third quartile to this thing right that might put me way out here and anything to the right of that would then be an outlier all right and I put that one kind of like it almost looks like I put it right at the maximum which would mean there'd be no outliers maybe out here all right so how do we actually use the test well let's go into our calculator all right I've put this data in again I think I did course I thought that on an earlier one did I not get that let's try it again Open document there we go um here we go all right there we go so I've put all that data in okay and what I need is I need two things I need to know what my q1 is and what my Q3 is that's really all I need so I'm going to go into my menu I'm going to go into statistics statistical calculations one variable stat okay okay right and all I want are my q1 and my Q3 there they are right q1 is 82.5 and Q3 is 95 98.5 all right I wish I could write those down I can't 82.5 and 98.5 all right 82 998 2 and here we go all right so q1 is 82.5 and Q3 is 98.5 right now what that means is that means my IQR which I have no no room to write now my IQR is going to be 98.5 minus 82.5 which is going to be 16 all right so the width of that box is 16 okay so let's now substitute the numbers in here q1 is 82.5 come onus 1.5 * 16 so let's get that 82.5 - 1.5 * 16 is 58.5 all right so uh 58.5 all right and now let's do this that one's going to be 98.5 + 1.5 * 16 so 98.5 all right yep that's going to be 122.5 now what that means is that any values that lie in between these in between them are not outliers they are within the realm of like a normal data distribution right so it's sort of like like again kind of taking up a little bit of space from this problem 58.5 to 122.5 these are the normal values but anything out here or out here those will be outliers so anything less than 58.5 anything greater than 122.5 are outliers so let's see uh here's an outlier cuz that's less than 58.5 see do we have any other ones that are less than 58.5 here's another one so those are two outliers now let's see if there's any that are bigger than 122.5 here's an outlier and that's it right so the 57 and the 55 lie more than 1 and A2 iqrs below q1 and the 124 lies more than 1 and2 iqrs above Q3 and so those are outliers again if they ask you to identify outliers right come up with the first quartile and the third quartile unless they give those to you directly which they might especially if it were a multiple choice problem right then multiply or then subtract one and a half of those iqrs from q1 to get that lower boundary add one and a half iqrs to Q3 to get the upper boundary and anything outside of those two numbers or outliers everything inside that's quote good data you you know what I mean like normal data all right let's take a look at one that's way more straightforward and probably something that you did a lot of this year because it actually was in the standards that IQR test not in the standards weird that they would have it on the formula sheet but whatever let's take a look at one that was definitely in the standards just not in the Common Core number three solve the following system of equations algebraically show the steps that lead to your results all right so we have y = x 2 + 12x - 13 and Y = 10 x + 2 we've got a parabola and we've got a line and we want to solve the system of equations right and we're going to do this by substitution so what I'm going to do is I'm going to take this x^2 + 12x -3 and I'm going to substitute it in for that y giving me x^2 + 12x -3 is equal to 10 x + 2 awesome right so substitute tion is one good way of solving a system just as elimination is all right but we wouldn't probably use elimination in this problem now since this is a quadratic system one that has an X squ in it we want to use either the quadratic formula or completing the square or factoring or something like this I am always biased towards the zero product law and factoring in other words I'm going to subtract a 10x and subtract a two from both sides of this equation leaving me with x^2 + 2x - 15 is equal to 0 all right now we're looking for two numbers that have a sum of two and a product of -15 and that will be x + 5 and x - 3 right so that's how that x^2 + 2x - 15 is going to factor and now I use the zero product law and I say well x + 5 is equal to 0 so X is -5 x - 3 is equal to 0 so X is 3 now we're almost done and I can guarantee maybe not guarantee I can almost guarantee on tomorrow's test there will be a 4. problem where you have to solve a system that is comprised of a parabola and a line now they can do all sorts of things on this just like that one with the graphing of the system of inequalities they could put the Y on the same side with the x's and things like that but ultimately you're going to substitute you're going to get your equation equal to zero you're going to factor you're going to probably find two values of X and then you're going to take each one of those and find a value of y now which equation do you want to substitute them back into to get the value of y and the answer is it doesn't matter all right whichever one is easier for you I'm going to use 10 x + 2 so here we would have y = 10 * -5 + 2 y = -50 + 2 y = -48 here I'm going to have y = 10 * 3 + 2 y = 30 + 2 and Y = 32 now if you have your work kind of written like I do you're completely okay right somebody can follow this it's easy it's good but it doesn't hurt to put them as coordinate points especially given the last part of this problem what does your solution represent graphically right so what does this represent graphically the intersection points of the two functions or of the two curves or of the line in the parabola or something like that all right one problem left let's do it here we go problem number four write the following fraction that so that it has a rational denominator and so it is in simplest form all right so for many historical reasons that we are not going to even scratch the surface of uh it used to be that fractions that had square roots in their denominators were frowned upon all right so having the square OT of 6 in the denominator which is an irrational number right means that the denominator 2 * the < TK of 6 is irrational and I don't want that y again is a little bit silly and we're not going to go there but we don't want it so how do we rationalize a denominator how do we get rid of the square < TK of 6 well doing these problems rests on two very broad Concepts one of the concepts is the following that you can change any fraction sorry I'm not going to even write this down you can change any fraction into an equivalent fraction by multiplying the top and the bottom of the fraction by the same number right and you do that all the time when you take a number like 3 fifths let's say and multiply by two and two and get like 6/10 do you know what I mean right or you you know you take it multiply by 10 and 10 and you get 30 50ths right you do that all the time when you are changing denominators in order to get a common denominator so that you can add fractions so one principle is multiply the top in the Bottom by anything you want as long as it's the same number okay because that means overall you're actually just multiplying by the number one all right but the other concept that's important is that when you multi mly the square root of a number by itself you get the number under the square root so anytime I multiply the square root of a * theare < TK of a I get a so theare < TK of 7 * theare < TK of 7 is 7 theare < TK of 100 * theun of 100 is 100 the < TK of 32 * the < TK of 32 is 32 so when I see a problem like this and I can almost guarantee you will see one on tomorrow's test might be a free response might be multiple choice I don't know you're going to look at it and you're going to see that denominator and you're going to see that radical six and what you're going to do is you're going to multiply the top and the Bottom by the square < TK of 6 now in the numerator All I Can Do Is Write it as 3un 6 but the denominator now I have < TK 6 * < TK 6 so that becomes just a six not a 2.6 but 2 * 6 right so I now have 3 * the < TK 6 all / 12 okay and that is definitely a fraction that now has a rational denominator because the 12 is a rational number but it's not in simplest form because 3 12ths can be simplified right I can divide that by three divide that by 3 and now I have 1 * < TK 6 which isun 6 all divided by four and that is that fraction with a rational denominator and in simplest form that also brings us to a conclusion of this particular review right so we went through this entire test let me see if I can bring this all the way back up to the top right we went all the way through the June 2023 exam the exam that they took just about a year ago right plus we took a look at that formula sheet the new formula sheet for the algebra 1 test and and we also took a look at some supplementary problems that weren't on this test but very likely will show up on your exam tomorrow now I want to remind you of a few things you know one thing is that you've got three hours for the test tomorrow that is that tends to be twice the amount of time that a very typical student needs to take and don't get me wrong if you're one of those people that takes all three hours and you absolutely need all three hours more power to you right I think that that's absolutely fine but more importantly for the people that want to rush through this test get it done turn their paper in and you know kind of go home for the day take your time you've got three hours work through the test if it takes you an hour and a half great take another half an hour to hour to just look through it check through everything you can check things that you may have done algebraically but now you have your calculator use that store function to make sure you've solved equations correctly use your calculator to the best ability that you can right this is a test that you need to pass in order to graduate from high school in New York state so it's certainly one that you want to take with as much seriousness as you can right and I know it's stressful right big time tests like this always tend to be stressful and it's even made worse by the fact that it's being given two to 2 and 1 half weeks earlier than what it normally would be right you know it's not like they took topics off the test because they're giving it to you early they're giving it to you early so that they can create a curve for the test that then you know will like affect what your final grade on it is like in late June but you won't even know until then or probably until July what you actually got okay because the state has to take a bunch of data come up with a curve and then apply that to your raw score to get your final scaled score whatever the point is tomorrow take your time you have all the knowledge you need you've got a wonderful tool at your disposal in your graphing calculator right take your time work the test draw your graphs and pencil never forget that do all the other problems in non-erasable blue or black non-el ink pen you know I'm sure your teachers have gone through all of that with you all right make sure to use rulers when you draw straight line graphs on your on your graph grids make sure to draw any other curve nice and smooth don't draw straight lines between the points all right and you will be Absol absolutely fine I know you know this material those of you that have been working hard all year you've been getting into school or you've been watching our videos at emathinstruction.com I know you know this material so you can handle it tomorrow just take your time all right for now though I just want to say goodbye right this has been the emath instruction Algebra 1 Regent review June 2024 my name is Kirk Wier and as always keep thinking and keep solving problems