Or if it was like E to the X that to the zero, well, I'm just saying like if it was a number to the X and X is equals 0, then it would equal 1. But for the problems that have shown you so far, just remember that the Y intercept is when X, you said X equals 0.


and it would be that value


All right.


OK, I really do not like grafting.


graph is shown below. What are all real solutions of FFX.


equals 0


Would that be like the X numbers?


Yep, it would be the X intercept.


Well, I, it was negative.


So is it asking you what are they? or is it asking you how many are there?


What are


What are they? OK, so where would it cross the


X-axis


negative 20, and 4.


Good. And


-204, you see how there's one more?


Oh


Yeah, it would be every time it crosses the X axis.


8. I, well, I always forgetting that.


No problem


OK, what is a conjugate?


Uh, what is the conjugate of, all right, so this is for imaginary numbers, right?


Mhm


So do you remember how they always come in pairs of two.


I kind of forgot


No problem. So basically, when there is imaginary numbers, right? It would be a number minus that Y value and then that same number plus that, that y value.


So


Hmm? Like, can you repeat that?


yes, 4 negative numbers when they come in conjugate pairs.


Mm.


Like you know how the quadratic formula is plus or minus that number.


Yeah.


So it would be that's it, so like for here it's -8, right?


Mhm.


So it would be -8, but then it would be plus and minus 5i. So if it's -8-5i, that would mean that there has to be a negative.


What?


We're just changing the sign?


That's all it is


Yes, you would be changing the sign of whatever value, yep.


Oh


If it's a conjugate pair, it just means that


it is


like that


He is scry.


So what would this one be? -5 + 4i, that means there has to be a -5-4i. Good.


I hate grapping


Oh no


I don't know. I don't know how to find like equations from graphs like this, like.


Oh, no problem. Here. Uh, can I see the equation for a second? I also apologize, the laptop that I'm currently using is half broken, so I have to reserve you to one side of the screen.


OK


And it can, can you scroll up so I can see the question?


There we go. The which Aras in choice could represent the graph below. All right, so, so do you know what type of graph this is?


Um


uh, a quadratica.


Good. So then what would it be?


Like what would the degree be


or the leading term be in this?


Not 2, but you remember, so if it's a square, if it's a quadratic graph, that means the highest it goes is 2, right?


Yeah


All right. So if we scroll down to the answers.


OK, and then for this, right?


So you see how


you see how it crosses at -6 and then 1.


So like if it's expanded like that, it, it basically already tells you the zeros of the function. So yeah, you would just find which ones it would equal -6 and then 1.


This is not part because


cute.


Now, that would be a cubed graph. That would be if it also crossed at 0


Does that make sense


Yes, I'm, I made a lot of sense.


Good


So let's have you on this one.


So where does it cross at


The good face


8 and


Mhm.


will be


So do you see how it says X minus 64?


Yeah, that's 8 and 8


Yeah, because that would be a perfect square, right?


Yeah, but like,


what?


um, what's his name?


Like, what is this like, like why is there like, how do you find what the zero is like.


Well, you said that it's -8 and 8, right?


Yeah, but the 0


So what would the 0 be? Look on, look on the first two.


You see how it's just X


Yeah, but like, wouldn't it be cute?


Yeah, that is a cubed graph


Oh, this, oh.


Yeah


You see how it goes up and then down and then up. You see how the, the um degree of the graph is how many times it crossed or how many times it can cross the x-axis.


Oh, I see um.


So if it if it can cross the x-axis 4 times. That means it's a


The the


do more


or power.


But see how this can only cross 3 times, like no matter what you put it as.


Oh, it is a cubed graph


and it crosses at 88 and then that X on the outside is 0.


Which equation represents the graph shown below. OK, so you see how it, um, dips and then comes back up, not crossing the x-axis, but touching it.


on -3 and 6


you see how it like does a quadratic on top of it.


Can you hear me


Oh, sorry. You're, it seems like your mic cut out a little bit. Can you hear me?


Mm


One second, I'm gonna go down.


Hello


It seems to use hello. Yes, now I can hear you.


I might, I might cuddle.


No, no problem. Uh, you did stop sharing your screen though.


Oh.


I mean, it's that


So just tell me when you're sharing your screen again.


Hello, I can hear you


Oh, you are kind of muffled though.


Yes, I'm trying to


you know.


but it's not


po pp ing up.


s


OK, no problem. We can do it from here. So you


I sit up and, and come back to it.


mhm


So you know how, one moment, I shall return shortly. I want to see


if cats are OK


Yeah


I'm


interestingly


Hello, sorry about that. I have returned.


You know seems like you either have bad internet or like you're far away from your mic, but I do see your oh, I just want to wait, where'd your screen go?


Can you hear me


Yes, now I can


OK. I'm


OK, I can see your screen now. Can you try talking?


Yeah, can you hear me?


OK.


OK. So you see how it does a dip and then comes back up at -3, like it's a quadratic.


Yes


All right, so basically when it


Wait, how is it quadratic? Isn't it exponent since like it goes like 2 times.


Yes, this would be to the 4th.


but you, not like that would, that's what the graph entirely would be. You see how it goes.


Oh he's cute.


No, cubed only goes, um, it, it comes from down and then goes up, down up. This one goes up, down, up, down, up.


So.


Like a cubed would only have it go like that once.


But you see how at


um -3 and 6, it does a thing like a quadratic would do.


such is the, the thing like this.


Yeah, it touches the graph once and then comes back up.


Yeah.


Basically, it is a quadratic at that


location. So the multiplicity there, remember how we were talking about multiplicities before?


What like our first question


Uh oh.


it appears you have


left again


Hello


Hello, I can hear you now


I, I don't know why. It's


yeah.


No problem. But either way, I was explaining that, you know how we talked about multiplicities at the start of the session.


Yeah


Well, there would be a multiplicity on those two spots, and the, it would be a multiplicity of 2.


Basically, it would become 0 there twice, which is why it bounces up off of the graph.


Does that make sense


Can you hear me


I am. Yes, I can hear you now.


And I have your screen again


You are sort of muffled, however.


Oh wait, can you hear me now


Yes, now I can


OK.


All right. So you see how it bounces up off of those two though?


Yes


So that would be a multiplicity of two there, because it's basically a quadratic, so it reaches zero there twice.


OK


And so that's why there's a squared there, because if it bounces off of the graph like that, it means that it reaches that function as a zero twice.


Oh, so like, if it doesn't cross it, is what you're saying?


Yeah. So if it doesn't cross it and it bounces back from where it came, that means there's a multiplicity there of 2.


So what do you mean bounce back? Like, do you mean like


But you notice how at -3 it doesn't cross the x-axis, it goes back up afterwards.


Yeah, yeah. Yeah, yeah.


That would be a bouncing back.


Like, like it stays like on the past like it stays on like.


it it didn't and it like it didn't like it doesn't cross this.


that


Correct, yeah. You see how it goes back up


Yeah


Yeah, so that would be why.


It's quadratic


so it's 2 times


Hm.


I can't hear you


Hello


Are you talking


Uh


hello?


Can you hear me? I can't hear you.


Hello, can you hear me


Yeah, yeah, now I can hear you


OK. I, so sorry. I tried joining on my phone because the screen wasn't working, but my phone doesn't allow the mic.


but I am back now and I have your screenish.


OK


So it will be


so 3 would beX + 3 and X + 6.


So would it


Would it be X + X


oh no, X minus 6.


Yes, and then they'd both be squared because you see how they bounce.


It bounces back


Yeah.


Good job


The degree of X, bro, how do I know the degree of the headaches?


Well, you see how, remember what I said, the degree of X is how many times it would.


it can go across the X-axis.


right


Yes


All right. So, can you count how many times it would? Remember to take into account that it did a bounce.


OK, so, first,


the first one


um


would be twice but they will be 2. And then this a bust bag, so like,


was it cute


so how many would be there. I believe that would just be squared because.


Yeah, cause, cause the


if it's cubed, it would go down again.


OK, so this square, so


And then


so what is this


Would it be cute


that's just one more, isn't it? Cause it crosses it once again.


No.


So like, it, it, they're all square


No. See how, here, I don't know if I can. So you go to the first one, right?


Yeah.


So that's one because it crosses the X-axis once.


Yeah


and then go to the point where it crosses again, that's 2, right? Cause it just goes all the way through.


Yes


And then the third one, you see how it bounces.


So it's really going


So it touches, oh, it touches the excesses 4 times.


and then there's one at the end for 5.


Wait, so is it 1


Yes, but I noticed you moved it a little bit for 3 and 4. I want you to know that the 3 and 4 would be the exact same point.


But because it


Oh, that's why.


Mm.


Mm


because it's X squad there, cause it's squared there, like, you know how it was -3 and 6 on the last one.


because it has a multiplicity of 2, it counts as 2.


All right, so it would be 12, and then that point in the middle would be 34, and then that 1 at the end would be 5.


Does that make sense


Mhm. So what does that mean the degree is?


Oh


So that would mean the degree is 5 because it has the potential to touch the x-axis 5 times.


OK


right?


Yeah, hold on


All right


Something is stuck in my teeth.


Odd


And the leading toy fishing.


How would we know the leading coefficient is negative or positive


Uh, let's see


So you see how the graph is facing up towards the negative and then down towards the positive.


Mhm


So if it's X the 5th.


that means that it has to be, so if it goes towards the negative.


Yeah


what, what would it, what would it go to in an X 5th function.


Normally


if it's positive


Like what's -1 x 5.


The answer would be -1, right? Because -1 multiplied by itself 5 times, is still -1.


you


But you see how that's going to the positive direction.


yeah


So that means that it has to be negative in the front.


this one


because, yeah, because you see how it's becoming positive, even though it's going in the negative direction.


the lead in


Yes.


Which one is lead in


The leading would be the one at the highest degree.


I thought they all had the same degree.


No, because


like you see how it would be, no, because it the maximum amount.


of degree it can have is the amount of times it crosses the X-axis. So the highest one there has to be x 5.


So how would we know which one is exceed.


You don't need to know which one's x 5.


Oh


You just have to know that the entire graph, it's leading one is X.


Oh, so the lead coefficient would be 1.


So


Well, it wouldn't be one, but it would be because I guarantee when you open that box, it's going to say positive or negative.


Right


Yeah


Open it


I did open it


Oh, OK, sorry, you're slow.


You can't see? Oh, it's, it's lagging.


I'm literally opening it right now, but it does say pa positive or negative.


Yeah.


Right? Yeah, so basically if it's going positive, even though it's increasing in the negative direction. What does that tell you?


It's positive


It has to be negative, right?


Oh, I think


Because the negative times negative is a positive.


I'm too confused. I don't get this part.


OK, no problem. All right. So you, let's make it X cubed. Does that make it easier?


So you know how X cubs nor when it's positive, it is going down to the left and then up to the right.


Hey, this is.


Yes.


No, that's z. Go to X cube.


Oh.


You notice how it is down to the left and up to the right.


yes.


That is because if you plug in a negative number in X cubed, what would you get?


uh


You did X to the -1


But you see how you deduct the -3.


wait, oh, what do you mean plug in a negative number to X cub.


Like, like


That would mean you change the X value to a negative number. So what would -5 be cubed?


but that would be -125. So do 2 because that's easier.


this


OK.


-2 cub would still be -8, right?


Yeah


Because it would be 2 times 2 times 2.


Yes


So you see how


Cause it's cubed


00, so like.


yeah, he's, you see how that's a negative number.


yeah, cause when you do a negative number, it always ends back up negative when it's cut.


Yeah, but you'd notice how it's going up towards the positive number when it's


going towards the negatives.


Like what


like go back to the original problem.


You notice how it's always, you know how it's going up and not down from the negative direction.


That would mean that it has to be negative because the negative times a negative would be a positive.


One is going up, it's a negative.


No, it, it would be going like the Y value, the Y value is positive.


Mhm


And the X is negative.


That's because the negative times a negative is positive, yes.


So if X is negative, how does a Y value become positive?


FX is like.


If it's multiplied by another


negative value


Yeah.


Does that make sense


Yeah


Are you sure?


Oh


a little bit


But


but it's fine. I'm gonna, I'm gonna put like a just that. Well, so 30


All right. Here, let's just finish this one. So there are, are you sure there's 5 different real zeros?


because remember it says different, right?


Oh, so 4


Yes, because one's a multiplicity.


And 3 relatives minimums.


Well, let's look. So there


02, is it 2?


I think it would be 2, because there'd be only two values where it's minimum.


It


I wanna try a different one


Yay, right, so let's stay on here real quick.


OK


All right, I just want, so you notice how


uh so that's basically saying as x goes to negative infinity


it would be going up towards positive infinity, right?


So


wait, can you resay that? Say that again.


So if X is going towards negative infinity, basically you plug in


all of the X values that are negative.


You notice how Y is still increasing to positive infinity.


Yeah.


So that, so if it's a positive value, like if we're going X, right?


Mhm.


Like normal X to the 5th.


That means if we plug in a negative value, that would be a negative times a negative times a negative times a negative times a negative, which would still be a negative, right?


Yeah


But that would mean the Y value has to be negative.


But you see how it's going towards the positive direction.


Yes


That means that there's another negative to make it positive.


because then it would be negative times and negative times and negative times and negative


But then there would be times of negative, which would make it positive.


Does that make more sense to you?


OK, good. So


So the degrees odd when an arrows point opposite directions.


yup, odd is basically one.


Yeah, yeah, yeah


Yeah, basically


and like, you know how the quadratics and like X^2 point the same direction?


Mhm


Yes.


All right, we can move on now.


Well, can I read this, the leading coefficient is negative.


when and behavior is, what did I mean write down, look up.


Oh. Yeah, yeah.


You notice how the right side is pointing down, and on this side pointing out, yes.


That's why


Uh-huh.


Wait, can I, can I say one more question on this just to make sure I got it.


This would be uh, this would be a, um, even one though.


So negative. So, I mean, odd.


since it's both like.


So, well, first of all, let's find the degree.


OK


So how many times does it cross the X-axis.


Um


OK, I have noticed a new thing, so I'm just gonna tell you this now. You see how it's going into


a


X value at a curve.


right here


Yes, that would mean that that's cubed, the multiplicity would be 3.


Oh.


You see how it's going into that like swooping motion.


Mhm


Yes, that would mean that the X value.


to the power of 3


So take that into account when you're counting the times it crosses the x-axis.


So that


Oh, OK. So when


mhm


Good


Yup


Yes


so even


The leading coefficient, uh, negative.


Good. And do you know why it's negative?


cause they're facing now


and there are different zeros.


Yup, because there's a multiplicity of 3


on this


and to maximums.


Is there 2? Yep.


Good job. See, you do it.


Was, OK. OK, I got it. OK, so, OK.


So I'm, I'm trying to write this down.


No problem. Take your time.


OK


All right. So, plot the X intercepts and make a signed chart that represents the functions shown below. All right, so you see it's negative, X + 2 x minus 1.


and x minus 6


a negative X minus 2.


So what would that be


Would that be equals 2.


I think so


or


I've never seen it written like that, but that seems correct.


or X equals -2.


Right, negative X


minus 2


Yes, so the X would equal 2.


OK, so


hold on


Well, it'll be negative too, negative X minus 2.


Yeah, well, yeah, if you distribute the negative, it would be negative X minus 2, which would just be 2, right?


X minus 2


Yep.


Oh, I thought it would be, um, I thought it would be.


Well, well, what would negative X minus 2? How would negative X minus 2 equals 0.


1 X is.


Oh wait, no


Yes


So it was you


Look, because X


-x - 2 = 0


equals 0, right


Yeah


And so, if X is equal to -2


that would just be


Mm.


2 minus 2 equals 0. So yeah, you would just basically ignore that negative because it's not attached to an X or anything.


So


and you just focus on the functions.


this will be X


equals 1


Yup


Wouldn't this be 2 times, so.


yeah


It would be a multiplicity, but I'm pretty sure if it's a number chart, you would just do it once.


And now we do a sign chart


uh.


So do you know how to do this?


no


All right, so a sign chart is basically you would plug in numbers that are.


next one


less than and or greater than a certain amount. So let's say, so what is the lowest value here. It would be 2, right?


Oh yeah


Because X + 2.


Yeah. Oh wait, but I put 2 right here.


Is this wrong


No, no, yes, that is wrong.


Cause it would be negative too, right? Cause X + 2, well, how would you do that equals 0?


Oh, cause I was doing negative X ma'am, so.


Yeah, no, ignore that negative sign until we do the sign chart, because if that negative sign isn't attached to anything, the zeros would be the same.


OK, so


right? So if it's negative 2, that means pick any number


that's lower than -2, and plug it into all of them, and you would just find out if it's positive or negative.


so negative 2, so like -4.


Yeah, sure. But even if so, so you just know that -4 + 2 is still negative, right?


Yeah


2 negative


Oh.


And then so it'd be negative and then -4 minus 1 is negative and then -4 minus 6 is still minimum, but it's squared, so that would be positive. So it would be a negative times a negative times a negative times a positive, which would be


So negative times the negative times the negative.


times a negative would be positive.


no, it would be. Where'd you get the 4th negative from? because remember it's negative and


cause it's, it's squared.


yes, which means it would be positive.


Well, I thought when it's squid, it's by the same thing, so right


oh yeah


So then click that area.


and then that would be minus, right?


All right, and so now choose a number between 2 and 1. I like 0.


Yes


OK, I'm confused. So, from the negative2,


it will be minus because negative


because we choose a number that's lower than -2, and we plug it in each functions.


and then when you, so it was negative here. It was negative here.


So was it, what was it? Was it positive here?


Yes, because even if it's -10, the final one, it would be square, which is always positive.


Oh, so, oh, so was this


this was the first negatives


Yep.


yep


This one right here. So negative times negative times negative, times a positive positive, it's still negative.


Yeah


because it would always be positive because it's squared.


Yeah, OK.


All right, so what would this one be then? If so let's choose a number between 2 and 1. I like 0 because 0 is easy.


0, yeah


So that would mean that if it's 0 + 2, that's positive, right?


Positive.


And 0 minus 1 is negative, and then


negative


X minus 6 squad is positive.


which would be


So negative times of positive, times are negative, times are positive, negative.


would it be negative


Would it be, would it? Cause negative times and positive is a negative.


and the ne.


positive


But then it would be negative times a positive again, which would be a negative times a negative, which is positive.


times of positive, so positive.


Yup.


And then we could do too.


And now we have to choose a number between 1 and 6.


to, no one.


Well, it has to be greater than one, yup.


Yep


Oh, between, between. So it's 22 + 2 is 4.


A


so positive


4, and then that would be 1, and then that has to be positive, right? So it'd be negative times positive times positive positive.


So positive. And then the other one would also be negative times positive, positive, positive.


This one


Yep.


because it still has to be positive.


Oh.


Well, yeah


right? So let's just plug it in. Let's plug in 10.


low


OK


10 + 2 is 12, positive, and then 10 minus 1 is positive and then 10 minus 6 is positive, squared is still positive.


So it'd be negative times 3 positives would be a negative.


Uh oh, what did I do


Mm


Sure see the examples.


wait, what? OK, so let's try it again. So


that'd be negative times negative times the negative times a positive. So that's negative. That's right.


Do we get the numbers wrong


Huh


Do we get like the plotting one.


I don't believe so. You did, um, what, what, what spaces did you do it? It's too small on my screen.


uh we did what, right?


-2, 1 and 6.


Yeah.


Yup, that's right


And so now let's plug in 0. So that would be


negative times A positive times of negative times of positive. So that's positive.


And then we plug in 3, which would be negative times of positive, times of positive, times of OK, so I see.


So if we plug in 3 here, that'd be negative times the positive times the positive times the positive, right?


which would still be negative


Does that make sense


And that's why it's also negative for the last two, because it'd be negative times the rest positive.


Um


right?


is it a 3? You did 3, right?


Yeah, I, I mean, yeah, 3 would, yes, 3.


So it'll be negative times of positive, that will be positive.


So that


yeah, so they'd all be positive. So that means that


Times are positive, that'll be positive, kinds of positive.


so that means it's 1 x 3 positives, which is negative.


And that's why the last two are negative.


Wow, I thought, how is it negative if it's one negative by 3 positives.


because it's not multiplying itself by another negative, so it doesn't become positive.


So is that always the case?


Yep


As long as it's not a negative times a negative.


It will never become positive.


Like it can be a negative times 20, it would still be negative because it's not negative times a negative.


So


Does that make sense? Cause if you put a negative in front of any number, it's negative, right?


to leave on the


ne g ative negative.


ice and they.


OK, let me see this one.


To be honest, I still don't get how this is equal to negative because if it's negative times of positive, it's positive. Times of positive is positive, times are positive.


Now, a negative times a positive is a negative, right?


Yeah


So what would be a negative times a positive.


negative, right


negative


Oh


That there would be a negative and positive, which is negative, and then negatives of positive, which is negative.


Yeah, but if it was a negative times a negative times a positive would be positive.


All right


identify the real roots of the function, then use those real roots.


to fill in the side table below


So you see what we plug in here.


What would, what


Well,


only see one.


well, let's see. So there's an X on the outside, right? So how would x equals 0?


huh


What, well, so if there's an X on the outside, how would you equal that to 0?


I don't know


X times what is 0.


From 0


Yeah, so it'd be 0 there


because you just, if X is equal to 0, that means it's equal to 0.


Wait, so because they're an egg from the outside.


that's 0


Yeah, if there's an X on the outside, it means that the root, that one of the roots is 0.


Oh, OK


because you just plug in 0 and it's 0 and a route is when X equals 0. So if the equation is X, when would it equal 0?


0, no, no, no, no, no, no, don't, don't do it yet.


And so the next one is 2 x + 3. What would X have to equal there for it to be 0.


It would be


-3 halves, right?


Oh yeah


Cause


yep


-3 halves


-3 over 2, because then it would be -3 times 2, 3/2 times 2 is just -3, and the 3 + 3 is 0.


-3 over 2 times.


3/2 times 2 is 3.


Yeah, because you see how it's divided by 2 and then multiple 2.


0, 1.5 plus 1.5, OK.


Uh-huh.


So it'll be like, it will be 3 halves.


yep, no, -3 halves.


because it's +3 still, right?


Mhm


And the last one is easy.


Yup. So click next


And so,


Oh my God, I'm not like I charts.


what's the, what's a, what's a number less than -3 halves, 2, right?


Oh yeah, make sense


OK, so it would be, so how would you do this?


So


we're gonna input -2 into the function.


So this will be a negative.


Yup


negative times to in a good 4 + 3. This will be a negative.


yup


and this would be a negative. So this would be a negative.


Uh, yup


We do always do less than.


because a negative times negative is a positive, negative, is negative.


Huh


Yes


00, so like when it's going this way, we always do less than the number and then we do win the numbers.


and then we do greater than the numbers.


Yup, because it would equal 0 at those numbers.


Yeah


um this will be, so


we could use one, right? No negative 1, right?


Yup.


So negative, this will be a negative.


Mhm


Uh


this should be positive


Yup


This would be 0


No, cause it's -1, -1 -1 is


00, negative so, so negative.


yup, so what would it be


So negative, positive, negative.


Mhm.


Negative from positive is a negative so positive.


Good job.


Wait, how are we gonna use a number, or do we do 0.5?


Between 0 and 1, 0.5 would work.


Yeah.


So this will be positive.


Yep


This will be positive.


Uh-huh.


OK.


And then that number is less than 1, so that'd be negative, right?


So what


negative, yeah, so positive and positive is a positive, some negative, it's negative.


Yeah, good job


So


And so the next one is greater than one, so it would be a positive number.


so this will be 2


oh with these two, right? So it's 2 positive.


positive


and positive


Uh-huh


Good job


Well, let me see how much I have left.


bro


Hm


You seem to be doing well so far.


hopefully I don't forget this on the sense.


draw a glass in the time.


Mm.


All right. I'm just just doing this.


Let's see. So, draw a graph of a polynomial with the x intercept. so we know what the X intercepts are, right?


Yeah.


So, mark those down first.


Yeah


oops


there a way you can


there we go


So what is it? 2/3?


but.


it's a lot of


It's like you have to draw the graph, I think.


Yeah, but it has to be negative below that number, right?


So 6


everywhere else


So below -6 would be like


and then a both father 6.


Wait, did we make sure that that 3rd 1 was positive.


they


So would it be like this?


like


Which one


Wait, are we sure that number, that 3rd number is positive. It was


Um, between -4 and


What


Oh, number one, OK. So make the X intercepts. So what would the X intercepts be here?


64 and 1.


Mhm. And so that means it would pass through -6, right, because it's going negative to positive.


Yeah


And then it would bounce off -4.


because it's positive to positive, and then it would bounce off again at one because it's positive to positive.


Does that make sense


That looks right


Was that not


It's a -6, so pass through is a -6.


It would bounce off at is that 4, yep, 4, bounce off.


And then it would bounce off again at, does that say one?


and then go up from there


That should be right. And if it's not, that's not on you.


This because it's not touching before.


That's why


You should be right there. I don't care what that says.


There'll be, oh wait, what?


Yeah, that's


It's high value


huh


Why is there a white a white barrier.


I


don't know


also neg


ative


Yeah, you were right on the graph, so I think that was just the program being stupid.


So negative 6


and then neg


causes to 0


Mhm


So it would have to cross the graph at, what does that say -6?


And then bounce off again at -3.


and then cross at 0


Maybe make it like taller. Maybe that'll work.


OK.


I'm just saying that cause that's what it looks like the problem was last time.


So I just redo that


So, started off low, yep, crossed through at that point, go up at the.


started off low, passed through at -6.


bounce off at -3.


I just say make it like taller and bumpier


Yep. Oh, that's, that's good enough. And then go down, and then bump at -3.


Good.


And then pass through at 0.


and then pass through again at 3.


Yeah


Does that make sense? Why we're doing that?


because it says like it goes.


Yeah.


And then pass through again at 3.


And then go positive from there.


And if you're wrong, I'm blaming the website.


Yay


since


OK, let's move on to the next one


You understand why we did that?


So you remember how to do the Y intercept?


All right


Would it be 0


Are you sure


It would be when X equals 0, right?


there was all because of the flesh.


So 0 + 3


times 0 minus 22.


times 0 minus 5.


0 minus 2 is -222, no, like 2 square is 4.


Good. So it would be 3


So if you have a 585.


Well.


3 times 40 sorry, sorry, what would the function be then?


12 times 5 is -60.


Good.


I think that would be the right answer.


OK, so the root roots are


Good


and 5.


Mm.


Oh, bro, I hate sign charts.


so less than negative, so -4 + 3 is negative.


Well, do you wanna know a trait?


Yeah


If it's lower than all of the functions.


So I


then it would be negative, right? Because all of them would have to be negative.


so if it's low, it's negative for, if, if 4 is lower than 32 and 5.


Yeah, so basically, if it's all the way to the left, then every function is going to be negative if it isn't squared.


Does that make sense?


So this will be negative, positive, negative.


So it'd be have to be negative, negative, and then positive because it's squared negative, right?


Yeah, so, so negative, positive, negative.


Negative, negative, negative, positive.


huh?


right?


Oh, you said, you said.


wait, so if this is negative,


Or sorry, I guess negative, negative, positive, negative.


Well, yeah, but you still, you, you still quirks causes.


Oh.


Right, cause there's a negative. Oh, sorry, that's not, I'm sorry, I, the equal sign only has like one black thing at the top. So I thought that was a negative sign. So no, it would be, yes, sorry, negative, positive, negative. So the answer would be


negative, negative.


positive is the negative, so positive.


yep.


OK. So, in this


in general


And so that one, another trick is basically, it would only change one, right? Because it's in between them.


it will only change what


It would only change one of them negative. It would change the X minus 3 negative, right?


Or it would change it to a positive, right?


Like it would flip its sign from the last one because it's sna.


What, wait, what will, what, what was the, the song?


because originally we're doing less than -3, right? So that would have to be negative.


Oh


But now we're doing more than 3, so they'd have to be positive running.


because anything that's more than -3 + 3 is positive.


Yeah


So that would be negative times of positive, times of positive because it's squared times a negative, which would be negative.


Well, how is it negative? It's positive first, right?


Sorry, I keep thinking that equals sign is a negative sign. So positive, positive, negative.


positive, positive, negative, yeah.


Mhm


So that'd be negative, right?


negative, yeah


And so this one, it would originally change the two, but because it's squared, it would stay positive, right? So that would be the same as the last one, would it not?


Good, so do that. And then it would have to be positive, positive, positive for the last one. If it's the last one, it would always be positive, or all of the ones inside of the function would be positive, right? Because it's the largest value.


So negative, oh, so are the, oh the Y intercept is 60, so 60.


and the roots are -3


Mhm. So if it's positive to negative, it would go from positive to the right.


go down, pass through 60 and then bounce at whatever that value is, is that 2?


Yeah 2


Yeah.


So it bounce at 2 and then go to 5, right? And then cross the 5.


So do that


Oh, it's, it's it.


bro, which point is plotted in currently.


I think you just gotta make it like taller, cause it's weird like that.


Oh, also you're, you're doing positive 60. Remember it's -60.


Oh


Oh, that's fine


So we're done


Oh, we have to draw the graph.


Mhm


OK, so it then will go from here.


Yeah, and it would cross at what number?


Wait, does it touch the Y, the Y intercept?


It would touch it, yeah, it would touch it at -60.


because that's the Y intercept. It's when X equals 0, so at x equals 0, it would be -60.


processes all


to 5


Yeah, it would bounce off there and then where to cross over at?


I would, wait, wait, wait, wait, wait, wait, wait.


Oh.


I just think that's too short. I think you got to make it bumpier because the graphic is weird, right?


where, where would I make a bump here


At the, you see where it bounced, I would just make it more obvious that it bounced.


OK.


Yeah


So I know it's nitpicky, but I just don't want you to get it wrong and then you have to do it over again.


So like, what do I do here? How do I make you bumpy?


Like this


Uh, yeah, I always make it like more visible. Oh, no, no, no, no, no, you wouldn't like do a straight line. You do it like basically


you make sure it touches it and then goes back up and then you make it go across.


Like you want to like make sure that the bump is visible and then go up again.


Yep. And then touch it and then basically you go like all the way back down. They go down, no, no, no, that's a straight line, not a straight line.


I'll go back. Oh.


You got to make it sure it touches it and then goes back down.


Good. And then pass through at -60.


and then


touch it


I


and then when you touch it, go back down


Mm


like down


more down, like, make it, yes, and then go back up.


Sorry, I know that's nitpicky, but I just want it to not fail.


And that's like a quadratic graph, right? So turn that in.


Yay


And you see how it like went like a bump there too.


See, that's almost like an exact graph.


All right, you ready for your next one


Like.


cannot hear your mic if you're talking


Hello.


Hello, I can hear you now


so


All right, you know how to identify the real roots, right?


Yeah


All right. So what would they be


0 times 1 times 0 is 0, right?


So what would negative X have to be to equal 0.


Uh


Where would it equals 0


negative X. Well, what do you mean negative X? You mean X?


Well, it's negative X there, but yeah, -1 times X equals 0.


Mhm


So what would X be


Yup.


OK


So that would be one of the routes. And then 2 x minus 7


what would that one have to be?


OK. So,


2 times.


Yup, or, or um seven halves, either one, it's up to you.


OK, so please not 5.


And then this one will be


Is there a way you can write a route


like a square root


I'm just writing here for now.


I would do the square root of 10 or, oh, actually that would be a negative number. So it says


Oh wait, we're identifying the Y intercept. What are we doing?


Oh yeah. Oh


Oh yeah


So 0 times 2.


So


So the first number X is 0, so 0. The first one is 0 plus.


I mean times


well, no, we're, we're we're multiplying, right?


Yeah, times


So,


0 minus 7, so times -7.


times 0.


well, times


No, 10


+ 10. But what would 0 times anything be?


So what would the Y intercept be


Mhm.


Oh, cause, oh, cause it starts at, at 0.


Makes sense. OK, so do we.


Yeah, 0


So now we find the real roots. So there would be 10, right? And then one at 3.5. Now, you see how


How do you meet this -10.


because it's


see, that would be an imaginary route because it would be x + 10.


So because it's imaginary, we wouldn't have to write it down.


Oh


Does that make sense?


Yeah, that makes sense


Because it can't be equal zero. It can never equal zero unless it's an imaginary number.


OK


All right, so plug in any number that's


negative


So you


Well, let's do -1 because it's easy.


yeah


So,


omega one


positive


Uh-huh


yep


So negative


time, sorry, positive and negative times of positive is.


Good


OK, let's do one.


right?


Yeah


Negative times and negative is so positive.


OK


Negative


negative


OK, I have a question


Wait, negative. Are you sure that one's negative in the middle?


This


No. Are you, for the last one, are you sure that 2 x minus 7? because we're using a number that's bigger than 7 halves, right?


00 yeah.


Oh, so next you's positive, positive.


So that'd be a negative kind of positive times of positive, which is right.


yeah


Uh-huh.


negative. Well, I have a question. You know how you were telling me how exqui I when it's squirt, it's always a positive. Is it the same for when it's scored inside the parenthesis.


Yup


Like one


because what's -12?


One


OK


Yeah, and so if it's -52, it would just be 25 plus 10, which is 35. So


yeah, the, if it's a, unless it's imaginary, if it's squared, it's positive.


Mm


it


So if it has an I in it, then I'd watch out cause that would be the square root of -1.


oh OK


It's


So what's the, so we know, we know the Y intercept, right?


0, and then we're looking for 3.5, right?


So it goes from negative to positive at 0, and then from positive back to negative at 3.5.


So negative to positive.


Oops, so


A negative


to positive as 0, right?


Uh-huh


And then it comes


here.


and then back to negative, right?


Cause it's positive to negative. Yeah, looks good to me.


Yeah, I don't think I touched it though.


That's up to you to decide. I can't see from here.


OK.


like this


Oh no.


Yup, that looks good to me. Doesn't look good to you?


It was crooked at, it was like straight lineup so.


I don't think it needs to be exact. I just think it um wants to be a bit bumpy.


so.


Mhm


Oh


Oh my gosh, I hate inequalities.


Let's see. What does it say? 5 X and then 2 minus 8 is less than 0 or equal to.


Yeah


All right, so when would it equal 0?


At what two numbers


Well, it's


At what


oh, we're trying to find the boots.


Yeah, we're plotting the roots. So what, what two numbers would it equal 0 at?


Uh


Where'd you get -5


Oh.


Yeah, because 5 times what is 0.


Yup


And then the second one, would it be 0 or no?


2 X minus 8. Then with 2, yep.


less than or equal to. That means it's as close.


Well, we are, we're just finding the roots right now.


Oh yeah.


So


Yeah


side choices


So plug in a number less than 0.


negative.


OK, so this is gonna be negative.


And then this is gonna be


negative


yep


Do we need to do this


Do we need to do what


No


Does it


No, I wouldn't bother with it yet cause now we're just making a sign chart.


Because wherever it's negative, it has to be less than 0, right?


Wait, negative times negative is positive so it used to be positive.


Are we sure on that


Yup.


Right? Negative times the negative.


I'm just asking if you're sure on that. I know it's, I know you're right.


Oh


OK, 1


positive


and um negative.


Yep


Uh-huh


So negative.


And then this one positive.


and positive


Good.


Oh.


interval notation


How do you write the answer? So let me see example.


Sorry.


like affinity


0. How do I write? You see how they're writing it and like it's about.


Yeah, so let's go look at our normal problem.


I don't know how to write it


All right. So, it can't be, so it's greater than or equal to 0, right?


Mhm. It's less than working.


So do you know how to make it, so you know um how to make inclusive brackets.


So you, you


She what


you know how to use parentheses to exclude things, right?


Yeah


So brackets would be inclusive. And so when is it um less than or equal to 0.


Mhm


Well, positive can't be less than 0, right?


Oh yeah, yeah.


And so, when would it be less than or equal to 0 at what numbers?


Wait, what


Well, when does it equal to 0?


Why isn't it this one?


Oh


what


And yeah, those two numbers, it's equal to 0, right?


Yeah, 084


And then everything in between those is less than 0, right?


0 and 4


Yeah, everything


Oh, cause it's negative


less, yeah, because it's negative, so it'd be less than 0, right?


Yeah


And so, if it, if we're including 0 and 4,


and it's everything in between there.


We would use the double brackets, right?


OK.


The bracket is this, I'm gonna send it in chat.


Like that's a bracket


Oh, are we thinking the double brackets because of the equal to?


because it's equal there at 0 and 4.


I you.


over


So it would be from, from what number to what number? Is it less than or equal to before. 0 to 4, right?


Good.


So we just put that


That makes sense, right


Oh


So do you know the roots here?


If it's getting too late, you can let me know.


I, I know. I'll be fine. I stay up late.


So.


so.


So 2 times what the 40?


0, right


If you're talking, I can't hear you


Hello hello.


Yeah, I can hear you


OK, so 0. So this will be, we're trying to get 3, so.


I mean, yeah, I


Yup, 3. So 0 and 3, right? Are they the only two roots?


Why is it cubed and quadrupled.


Yeah


Those are multiplicities, but the roots would still be the same. It just means that it equals 0 multiple times on those numbers.


So we're doing less than


Yes


I guess


So -1


negative once the 4th power.


is -1


-1 to the 4th power


No, it's 1


It.


positive


Yes, it's 1, good. So that'd be positive and then we plug in the corner to the other one would be


Would this matter, the 3?


yep.


OK, so,


because it'd be -8 cubed, right? And that would still end up being negative, right? Because 8 times 8 times -8 would still be negative, right? Because negative, negative, negative.


oh


the time, oh.


So if we do less than 0, that would be a positive number times a negative number.


Does that make sense


So good


So negative


OK.


OK. And then for the middle ones, we are plugging in a number between 0 and 3.


So let's do 1


It would be


OK


positive, right


And then


2 minus 6 is what, -4, and then 4 cubed is negative or positive.


negative, right


Yeah, and so it would be a positive times a negative, which would be


good. And then if it's greater than 3


what would it be?


This one also.


Positive


Good


and this will be positive signs are positive, positive, so positive.


Uh-huh


Oh, in Boston.


I don't not know how to write Invernotations.


Well, let's submit this one first.


How do we do this


Uh, and what was asking


Inver notation


In


Ah interval notations.


interval not. OK. Yeah, so.


The one who didn't make the one was


so it's asking what is it less than 0, right, less than or equal to?


Yeah


So when does it become


more than 0


This one right here


Well, what number


but it's still equal to 0 at 3, right? So we'd include 3.


Oh, it's equal to.


Yeah, cause it's less than or equal to 0, right?


So it's equal to 0 at 3, right? Because it's a route.


So the ending point would be 3 in a bracket. Do you know what we put? cause every number before 3 is negative, right? So that'd be less than 0.


I you know 43 is less than 0.


huh?


You said every number before 3 is less than 0.


No


Uh.


Yeah.


Yes, because if we plug in every number before 3, it's negative, right? We see that on the chart.


Yes


And so, what would it be if every number before 3 is equal to before is equal, is less than zero.


Um, is it bracket


is negative infinity, right?


And see, actually we would put negative infinity in a parenthesis because we cannot technically reach negative infinity.


I get.


or sorry, in a parenthesis. So it'd be parentheses and then brackets.


So every time we start with negative infinity, it always has to be a parenthesis.


Yes, if it's going towards an infinity, it's a parentheses because we cannot technically reach infinity.


OK


And then this would be


is it the bracket


And but then it would be a closed bracket, yeah, because at 3 it's still equal to 0.


3, it's still equals to 0, right?


Yep. So that means we include it.


So when will it be both, when will it be both parenthesis.


Um, if it's negative infinity to infinity.


Um


or if it said it was just less than 0?


Like if it said it was just less than 0, it wouldn't equal 0 at 3, or it would equal 0 at 3, which wouldn't be less than 0, so it would be a parentheses there.


But because it includes the number 3


it would be a closed bracket.


Does that make sense


Yeah


All right


How much more do we have


We got a lot, man.


Mhm.


You're doing well so far.


So plot the roots


It should be 1


Uh-huh


to get


what


Good


Then make a sign chart. I'm going to leave this up to you.


but.


uh


Just making sure you're still there


All right, just making sure. I believe you got this.


Cause you gotta be able to do it on yourself on the test tomorrow.


I also, your mic's cutting out so I can't hear you that well.


Can you hear me now


A little bit, still cutting out a bit.


Can you, can you hear me now


Yes, much better


All right, so what would you do next


OK, so


um


times 4 is 0.


Those never full. So negative.


0, so negative.


She sounds for.


positive


Yay


OK. Solutions that


uh


OK, this


Are you sure that last trip would be positive.


Yeah


right.


Yup, you're right


Because


OK


I'm just making sure, testing you.


OK, well that I take into notation.


OK, I'm gonna, I think you can do this, but remember, look at the sign.


You see how it's not equal to zero.


Oh yeah, negative infinity


So remember, it equals 0 at 5 and 1 or at -1 and 1. Do you know how to combine things together.


to combine two things together


It equals your -1N1, right?


Oh, sorry, wait, it's looking for things greater than 0.


Oh, you're miserable


right? And not greater than or equal to. So what would the lower number be?


Negative and


No


If looking for things greater than 0, right?


Would it be 0


When does it start becoming positive?


Look at your chart


6 bright clear. Isn't this one?


Yeah


a one


And you know why it's a soft bracket.


Yeah, because it's not equal to.


Good.


So one, you know, it woulder than 01 infinity.


Yup


That's what I got


Yes


OK, I'm gonna not say a word on this one.


OK


3 times, 2.


it


How do I do this


How do you do what


Because it's -8. So how do I turn 5 into 8.


-8 + 3 is equal to 0, right?


Um.


So what would -8 have to be multiplied by to get -3.


Well, you said negative X what is there?


8 times what is 3, or is -3.


It would be a fraction


810. What is -3?


Mhm


Oh.


It would be 8/3, right?


Oh sorry, 3/8.


Because a divided by 8


times 3


times -1


I equal negative 8


Uh-huh.


or it would equal 3/8.


because 3/8 times 8 is -3, right?


So


so how would I, so what equal eggs.


Hm.


Would it be -38?


Well, it would it be -38 or 38?


Because it's already -8


It's already +8


Oh, which one are you looking at


This one


Oh, I'm sorry, I thought you were looking at the other one.


So, 5 X + 8


So 5 times what would equal -8.


but it's fine


It would be


-8 over 5, right?


Because you would multiply


thought, you, you, you'd solve for it.


You'd have to get that equal to 0. So 5 x + 8 equals 0. Do you know how to do that?


Oh yeah


So.


Yeah, so you, you're, again, for every route, you're just finding how it equals 0.


So 5 x + 8 equals 0. How and what would your answer be?


foods


85, you said?


Yeah


Good.


but all


good


Well, 85 is slightly more than one, right? Or slightly less than -1.


And so what would that one be


Do you know how to solve it? You set it equals 0, right?


so


So 8 x + 3 equals 0. Solur x.


it


You said


3 cases


38. Good


is a 3.


You did 3/8, cause 3/8 would be what, slightly less than 15, right?


with slightly less than a half.


Yeah, because 3/8, cause 4/8 would be 1/2, right?


Oh yeah


So 38 will be slightly less than 1/2. And then what's the last one? It would just be


Um


negative


Yeah, good


And sote chart, you know how to do this.


positive


Oh, wait, when it's


oh


I forgot


When, when it's um to the power of 4, is that positive as well?


Yeah, because it's even, right


Yeah, and then negative


Good.


So negative and positive is negative, times positive.


negative. So the first one is positive.


Probably, I don't know


Mm


I'm not doing it. It's up to you


Yeah


So


mhm


no


positive


symptoms


so sometimes.


wow


huh


0 times 3 is 0, so.


3 times 20 yeah, cause if it's not greater, if it's not greater than 3.


Wait, so


It's rarer than or equal to it looks like.


Oh, if it's not greater than if it's not greater than 2.


It was not greater than 2, then it can't be positive.


So that's negative. This is gonna be positive.


Mother, positive, positive.


we


such positive, positive.


positive.


Those nes


negative.


Yes, I got it right.


OK, now, inequality.


OK, so this will be


bracket


Uh, you wanna look at that one more time?


Um


It's everything to the left of 2, right? You just have to do it, um, from the lowest value to the most value. It's a like number thing.


OK.


It's, is, isn't it to an infinity?


Wait, is it


Not infinity cause you see to the right it's negative.


Yeah


So it'd be that all the numbers leading up to 2 and then 2.


So what's all the numbers behind 2?


Negative


negative


00, behind that?


Uh-huh


So like right here it's positive.


Yeah, cause it's positive from everything leading up to 2, you say, right?


So what would all those numbers be?


Any number lower than 2.


No, positive


It would be negative infinity, right?


What


Because all the numbers are lower than 2


You see, even if it's positive, the X value you're still plugging in is negative, right?


Mhm.


So it would be


what's going up 22.


So


yeah


Because if you plug in any number lower than 2, it would be positive, right?


to do.


Sorry, repeat that


greater than or equal to zero.


Uh-huh, that's what the equation says. So where is it showing that it's all greater than or equal to 0.


You see how it's positive, positive, positive, positive.


You see how it's positive all the way to the left.


That means that negative infinity.


up to up to 2


right?


Mhm


Does that make sense


So what would that one be


Remember negative infinity has to be in a soft bracket.


Oh


cause you can't reach infinity. And then 2 would be in a hard bracket because you can't equal zero.


Good


OK


Now, I don't know if you did the top thing right. I was just helping you with the bottom one.


So I would check that again cause I'm not sure what you did, cause I'm not doing it.


oh


positive.


positive


negative


negative test positive.


positive negative.


times negative is positive.


Uh


one


You're making sure to look at the squares and fours.


Yeah


this would be negative


positive, positive, negative.


which would be positive, right


Yeah, for all these numbers.


Actually what I do


So sorry


actually.


68 positive, positive, positive.


Positive to positive response. So we, I just rechecked in and it says, and I think it's right


Oh, strange. Well, if you believe it's right, and then what is it you said it's negative infinity of the 2.


Mhm


Is, is that 2? I can't see that number. Is that number


Yeah, it's too


Well, then everything should be right.


So you want to submit the answer and see what you did wrong or


uh well.


I'm just gonna restart


All right


So you got 2, right?


Yeah, yeah, that's super one.


Go ahead. So 2 for the first one and then it's


8/5, right?


Yeah.


So -8/5 is


negative 1.6.


slightly more than -1.


Oh, I think I got it wrong like right here, -1.6.


Is it? I don't, I don't know.


Yeah, it is


All right, so do that


It's been in -38. Well, no, it was 3 days.


Yep


Yup, 386, right


do.


That's just slightly less than 1/2.


And then the last one is just -9, right?


And so,


but let's just look at these one more time. So if everything's


negative, right


Mhm.


So that would be negative.


negative, which is, sorry, negative, positive.


positive


negative


Are you sure that one's negative


but it's 3 right


That's a 3, right


Yeah, so it's negative


Are you sure? that's a negative X, isn't it?


So plug in -10


OK.


What's negative, -10 minus 9.


negative blood count


1 cube is


-6. Yeah, so it's negative.


Are you sure? Look, so you're, it's -9, right? Is our 0?


Yeah


for the last one


-10? OK. So negative 10


No, because that's a negative x, isn't it?


Yeah, it is


But if we're plugging in -10, what's -1 times -10.


Oh


There we go.


Oh, OK


So


the answer will be one.


So this will be positive


Yeah, so that would mean it's


negative. Good


And so are you confident in everything else?


No


All right, so I'll just check again.


OK


negative too


Yeah


then


negative, positive, positive.


neg cause this is called.


a little.


a little


pain


one


the


So we're good for everything else


So the and change


All right. And so what would that be if it's greater than or equal to 0.


will


So it equals 0 at -9 and was that?


worse.


too


OK. So it would equal 0 at -2 and and negative not sorry at 2 and 9, right?


So would we, which brackets would we use or which parentheses would we use, hard or soft?


if it equals zero at 9 and 2.


parenthesis, right


Would it? Cause parentheses mean we don't include them.


So there's one you have to sign.


But it's greater than or equal to 0, right?


Yeah.


And so would we include or not include 9 and 2.


Sorry, repeat that


If you're saying things you're not


Mhm.


or your microphone's really muffled again.


Hello


Hello, I can hear you now


So, it, are we including or not including 9 and 2?


I can hear you


Oh, you want to.


Oh


OK, can you hear me now?


Yes, I can hear you


OK


So would we include or not include -2 and 9.


We would include


hello


This is your microzole


I was sorry I was, I was muted.


So if we would include them, what, what, um, parentheses would we use?


Hard or small


we would use the brake.


Yep


And so which one would we do first if we're going from least to most


Well, what happens to the -10?


I mean, what most of this


Well, it's negative at -10, isn't it?


Yeah, it is


So we wouldn't include it


The equation


Any more questions


So it would just be positive from 9 to 2, wouldn't it?


Yeah


All right


So that would mean it would be brackets


And


So can you do double brackets


I can't see if you're doing double res or not.


what you can see, I can share my screen.


I can see


but I just


do double brackets for me, please.


I do.


Do you see, um, it's show, it's showing on my screen. It's lagging.


Uh, maybe it's just slow. All right, well, I believe you. All right, so, and then what would the numbers be inside of them?


Yep


Can you see my screen


I think it's frozen for me.


I try refreshing because it works a lot.


All right.


Hello


Are you back


Hello, can you hear me?


Yeah


All right, I can hear you


All right, I see it now


Yeah.


So is this you want to submit?


Oh, yes.


Good job


This was so much work.


Yes, it is


Did you take AP Calculus


Uh-huh


Was it easy


Yeah, I did it about 2 years ago.


You got a 5


Ben? Yes, it was actually the first year they did that test, so it was pretty easy. Or so they probably um graded it easily.


but I did get a 5 on that exam.


You


That was 2 years ago. Now I'm in college.


All right, so


determine whether the function


is even, odd, or neither. You know how to do this, right?


Your mic is off, by the way


Do we work on this earlier


Oh, I can hear you


Hm


I said I can hear you now. All right. So, is this function even, odd, or neither? Do you know how to do this?


No.


of


All right. So, to determine whether a function is even or odd.


So you see how the signs of them are negative, negative and positive.


Yeah.


All right. So if it's an odd function, if you plug in


X, all of the values should be the same.


And if you plug in negative X,


then all of the values should change.


What do you mean plug in


Like instead of X, uh, let's just go with -1 or let's just go with 1 because it's easier, right?


Well, if you, if you plug in a positive number and then the negative.


But yeah, but like basically like split them up. So if you do


one here, it would just still be -9 cubed minus 5 plus


5, right


It would still be negative, negative, positive.


If we plug in a negative X


or sorry, if we plug in a positive X.


um


because it would be the same function, right


yeah, positive, negative


All right. And now if we plug in a and so if it's odd, if we plug in a negative X, every value should change sizes.


so neg


So instead of positive X, it's a negative X. So it would be 9 X cubed minus


5 to the X, which is, which would be plus 5 and then plus 5 to the negative x to the 5, which would be minus X, right?


probably be


So you see how every single sign changes.


Yeah.


So if we plug in the normal X, everything should stay the same. If we plug in negative X every sign changes.


Does that make sense


it


I know


All right, so this, so that means that this function would be odd, right?


because everything changed


Yeah, because everything changed


Oh.


OK. And now if it's even, if you plug in a negative, everything should stay the same.


right?


No, it wouldn't be odd, because everything would stay the same, right?


If we plug in a negative, it's still going to be negative t, cause it's


basically look at the degrees


You see the degrees


Your mic is off, by the way


Yes, I see that.


All right. So if it's all even, then, it should be


even if it's like 13, and 5, it should be odd. And then if it's a combination of the two, it's neither


That's a trick


Oh


donate


So go to the next problem for me.


You're confused


which, yeah, cause how is this event?


Here, I'll show you. Go to the next problem.


All right, so you see how it's -4 -4 X cubed minus 2.


It's positive 4


You notice how there's a degree of 4 and 3.


Yes, but you see how it's 4 and 3 and 2?


Yeah.


So that means it's going to be neither, because


if it's even then every exponent would be even.


Correct.


Mm.


Does that make sense?


Oh yeah, but it's a mix of both.


And then if it's odd every exponent would be odd.


But because that has both, it would be neither.


And so look at this one. If it's negative 4 and 6.


Yup, see


Um, what are the options it gives you?


I click the options, I think you can't see it on his face. I'm gonna say.


Yeah, I can't see your options.


The first one is 23, and 4.


Mhm


And the second one is


first differences in Y are constant second differences in white or across 3 differences in right across it.


4th differences in y are constant.


OK. So the function must be a degree of degree blank because over equal length x intervals.


I'm gonna be real, don't know this.


Let me see this


OK


So I will, I'm gonna be right back. I'm gonna use the bathroom. Well, you figured that out because I have no clue.


It is negative 15.


and 18.


minus


I it is 2


47 minus 222.


Hello, I'm sorry about that.


Yeah, it's OK


I, uh, still have no clue what's happened though. So, did you figure it out?


OK, I looked at the example, kind of makes sense.


All right.


Yeah


So I'm just trying to get out and I'm gonna see if I got it.


It's, it's OK


OK, good. Sorry, I can't help you with this one, or I, I just


never learned it


Yeah, it's fine


That's


Oh, I OK, I think I remember how to do this a little bit.


OK, yeah, I remember how to do this now.


that


All right. Um, can I see the original problem?


So they're going by ones.


and then it's


plus


let's 33 minus 18.


or


33 minus 8C. I won't be the 15.


what's 33 minus 1815?


Yeah


OK, so that'd be


plus 15 and that one's plus.


65. Yeah, and then you'd do that again.


And then you do it again until it equals 0.


So I think that would be a degree of like 4, probably.


because over


intervals of whatever, it would equal 0.


or so fast


Huh.


How do you know it was forced so fast?


Oh, I guess, but it just looked like that was the way it was going.


Oh


Yeah, see, you see how it's done. And basically, you would do it until it's adding the same amount to each one.


yeah.


So this doesn't really matter?


Um, I think it prob it I think it does matter, but I forget why.


I just remembered how to do it


Yeah.


It, it's been uh like 2 years since I was in university or.


All right


Determine whether the degree of the function is odd or even. OK, so you know how to do this, right?


Well, yeah, this is gonna be


or on


Well, each of the arms is facing opposite directions, right?


Yes.


Yeah


So then why, so what's the degree?


And do you remember what a function is?


A function basically just means that for every x value there's one y value.


for


So I just can't overlap.


Can you hear me


Yes


It just literally took me out of screen sharing, so potentially use something flagged, so I have to re-screen share.


Oh, OK, no problem. Uh, it's still showing for me though.


OK. So which one


OK. So


It's, it should be an odd, right?


uh-huh, so that looks like a function, right? Because every X value, there's one y value.


passes the vertical line test.


Do you know what that is


I go, I forgot what that was.


Um, basically, just a few.


Is it like when, when you put a line down and it's too like


make sure it passes through one time.


uh-huh, yup


So it's


And so it's odd and function, right? Because there's only one X X value for every y value.


so if it's an even fraction, it'll be there'll be like multiple values for one.


No, that would be if it's not a function.


If it's even, it just means that both of the lines are pointing the same direction at the end.


But it's not an odd function.


Oh, whoops, um


What is, what do you mean by, what is, what do they mean by um?


Oh, OK. So the point of symmetry is OK. Sure.


I


I really don't notice what it goes like.


huh


I don't notice, but it's fine.


Mm


Basically, the point of symmetry is just, if, if you reflect Y equals x, it should be the same function.


Oh yeah


But if you do it, if you flip it diagonally, it should look the same. That's why.


yeah


All right, and it, it's not.


All right, let's go to the next one. You should be almost done.


Yeah, got some more.


So you can


It says how much you have right here.


even on or neither. What does it look like? If you flip it on its origin, it should look the same, so.


So that


is


looks odd to me.


and then because


it doesn't say if you flip it on its orig or if you flip it like y equals X or something.


Um, this is what it said.


rotate the graph on its origin or


Yup.


wait, it's a symmetrical about why accents.


that looks correctish.


probably


is it symmetrical about the origin, symmetrical over the axis or about the origin, symmetrical.


I think, yeah, symmetri about the origin would be an odd function.


If it's symmetrical over the Y axis, it would be even.


um


last one


OK.


The table below shows select values from the function F, which is odd, and whose domain is all real numbers, finds negative X of -9.


So, let's f-9?


um hmm


If we're finding


Hello


hello


Yeah.


I can hear you


You said it was fine. Yeah, you said it we're finding what?


So we're finding negative F of -9. I would show an example because I don't know how to do this.


Oh, OK, because it's, I get it. So click X out.


Uh-huh


So it's an odd function, right? Which means that it's reflected over the origin.


Mhm


So if negative F of -9. So that means at 9 it's positive or at 9, it's positive 16, right? Because at 9, it's -16.


because the like just the the um symbols in front of them would switch, right?


Yeah


So if it's negative F of -9, wouldn't it still be -16?


Because it would be positive 16 and the negative.


Does that make sense


Wait, I


wait


How do you do negative F? I've never seen that before.


Uh, it would just be a negative answer of what the answer would be.


Oh


So, OK, so you know how it's flipped over the origin, right?


yeah


So if the X value is equal to 9,


Mm.


So what would be, and if it's -16, then what would it be if it equals -9


No, look at the Y value now. What would the Y value be if it equals -9.


Mhm.


You know how like the X cubed function is mirrored on both sides.


Yeah


Yeah, so you see how X equals 9?


So what would be if X equals -9, it would equal 16, right?


Oh, you're right here. Oh yeah, I see.


Yeah, that makes sense, but it's negative, so the answer would be a negative version of positive 16.


which is


So negative?


-16. Yeah, that makes sense.


And if it isn't, the question's wrong. No.


What


I'm kidding


But


find F of -9.


uh-huh, yeah, so what would F of -9 be? If F of 9 is -16.


Yep, and so it would be negative 16 because it's negative F of -9.


Congratulations


Thank you so much.


No problem.


So now I gotta go practice for a push.


Uh, I remember a that. I did that last year.


Was it easy for you


It Wasn't hard. I got a 4


OK, I took AP Psychology last year and I got a, I got a 2.


Yeah


Oh, I got a 5 to many people.


Well, thank you so much. You


you helped me a lot


No problem. Have a good day.


You too


hi:)


great it worked!


let's finish this homework!


yes:)


lets do question 1


ok great :)


so to do a percentage problem you take the percentage, turn it into a decimal, and then multiply it by the non-percentage number


so it would be 18% turns into 0.18


than we would multiply .18 by 500


ok


yes correct!!!


so it would be 18/500


i meant 90/500


no it would just be 90


because 500 is the whole number


and then we took a portion of that whole number


(the portion being 18%)


and then the portion of the original number ended up being 90


oh okay


like if 18% of the students were boys and there were 500 students in total


then 90 of them would be boys


does this make sense?


yes it does


great!


do u want to try it first or have me explain it?


i can try


amazing :)


almost, that is the amount that the was reduced by


were looking forthe new price of the tv


yes great job!!!


that is correct!!


do u want to go first or me?


i think 528


yes that is correct!!!


great job Sarah!!


you're doing so good!!


thanks:)


question 29


question 29 has to do with factoring i think


let me solve it real quick and then i'll get back to u on the best way to do it


ok


thanks


ok so yes i solved this one with factoring


first i moved every term to the left side so the equation would be equal to 0


is that making sense so far?


kind of


what is factoring?


factoring is when you take an equation and split it up so you can easily see what the solution of the equation is


oh ok


so if the answer is 0 i can just type in DNE?


not necessarily, we just made it equal to zero so we can factor


having the equation equal to 0 makes it easier to factor


can you see the equation i wrote on the whiteboard?


yes


great so i'm goingto show you how to factor it


so do u see how i labeled each term with an a, b, or c?


great!


then we will draw a big X


at the top of the x we will multiply the coefficient of a and c together


so what is the coefficient of x^2 in the equation


ummmm


the coefficient is the number in front of the variable


and since there is no number in front of x^2


that means the coefficient is 1


is that making sense?


you mean the 1 written under c?


not exactly, see the x^2 written under the a


yes i see


so the coefficient for a would be the number in front of the x^2


do we get what a coefficient is?


yes


so basically there is no number


so we just made it to 1?


yes because the coeffifient is the number that is being multiplied to the variable, like how for 2x the coefficient is 2 because we are multiplying 2 and x


the coefficient for x^2 is 1 becuase we are multiplying x^2 and 1 to get just x^2


are we understanding?


ok


great!


so we multiplythe coeffifient of X^2 by whatever c is


so if the coefficient of x^2 is 1 and c is one


then we do 1*1 which just gives us 1


so we write 1 at the top of the big X


i get it


and then at the bottom of the big X we write the coefficient of the number under b


which would be 2


does it make sense how we got 2?


amazing!


now for factoring


we need to see what two numbers multiply to the number at the top of the big X and add up to the number at the bottom of the big X


so for this question it would be 1


because 1*1=1, which is the number at the top of the big X, and 1+1=2, which is the number at the bottom of the big X


is it making sense so far?


yes


ok great!


and then we take each of the 1s from the big X and bring it back to the equation


since the 1s are positive we add them to the x


ok


because if you look at the new equation if we expand it it will go back to the old equation


so now all that is left is to solve the equation


so for the example equation, the solution would be -1


does it make sense how we got there?


kind of yes


ok great so now i'll demonstrate it with #29


do we get how i got there?


yeah but its kind of confusing


but you got 0 as your answer


my fault then i'll do it again but show u each step individually


no its not your fault


its my fault


but


can we please do a different Q


yes of course


and then come back to this one later


thanks:)


yup we will do whatever is good for u


thanks


question 25


ok do u want to go first or me?


i can try


yes good job!!!


that is right so far!


yes that is right!


great job!!!!


is this one correct?


yes that is the right set up!!!


yes great job Sarah!!!!


that is correct!!!


do u want totry it first or me?


is it correct?


yes it is!!!


great job!!!


almost!, look at the negative sign in front of the 9


is it incorrect?


it should be a positive 9


because first we multiply 5 and -1 to get -5


then we add 2 to get -3


and then we do 6-(-3)


oh ok


which is another way of saying 6+3=9


i got it


but the denominator is correct!!!


so the answer is dne


yes!!!


great job


question 32


for #32


t is for childrens movie tickets and n is adult movie tickets


for each time someone buys a childrens movie ticket it is 4 dollars


so we multiply the number of tickets bought (t) by the cost of the ticket ($4) to get 4t


and then we do the same with the adult movie tickets 9*n to get 9n


and then to see the total that Maya spent we add then together to get 4t+9n


did that all make sense?


yes:)


great!


question 20


do u want to try first or have me explain it?


you can please


ok great :)


to start it says the quotent of so we will need to divide somewhere in the expression


ok


and then it says "16 more" which indicates that we have to add 16


and then it says the cube of a number


and for "a number" we will use x


then it says "and the number" which means we divide by x


oh ok


is that all makig sense?


yes


i accidently pressed enter


oh no 😭


how am i suppose to get the dividing line ?


maybe start with the fraction button and then type the 16+x^3


thank you so much:)


of course :)


question 18


do u want to go first or me?


you please:)


ok perfect :)


so Alex's score is "15 more"


so we have to add 15


and then it is 3 times more than Ben's score


and Ben's score is x


so 3 *x is 3x


ok


and that is the expression :)


is that makeing sense?


yes:)


amazing!!


sorry Ben's score is t


so instead of x use t


its ok


question 19


do u want to try or me?


i can try


is it correct?


not quite, it says "13 less" so you are correct to subtract but you would subtract from the end of the equation


since we are removing 13 from half of Chris's debt


and then it says "half of Chris's debt"


so we have to divide y by 2


oh ok


thanks:)


question 21


do u wanna take a go at it or me?


you please


ok great :)


so it says 3 less


so we have to subtract 3 from the end


and then twice its width


so we multiply the width by 2


is this making sense?


yes


yay :)


yay:)


question 2


do u want to go first or me?


you :)


ok cool :)


so 39 is not prime because it can be divided by 3


so if it can be divided by 3 it is not prime?


if a number can be divided and the answer is a whole number then it is prime


for example 39/3=13 so it isn't prime because 3*13=39


but for 53 there are no whole numbers that multiply together to make 53


does that make sense?


if a number can be divided and the answer is a whole number then it is prime or it is not prime?


then it is not prime


ok


now for 57


do u think it is prime or not prime


(try dividing it by 2, 3, 4, 5, 6, 7, 8 ,9, 10, 11, 12 to see if any divide it nicely)


only 53 and 67


are prime


yes great job!!!!


thanks:)


question 3


do u want to go first or me?


you:)


ok cool cool :)


so to factor a number, we have to find the simplest numbers that when multiplied together they make up the number


so we have 60


and two numbers that make up 60 are 2 and 30


and two numbers that make up 30 are 2 and 15


and two numbers that make up 15 are 3 and 5


and now you see how each number at thebottom of each arro is prime?


yes


thats how you know we've factored the number enough


and now we can count


how many 2s are there?


and how many 3s and how many 5s


there is one 3 and one 5


yes!


and now we simplify the equation to have 2^2 instead of 2*2


oh ok


and that is how you factor a number


ok


yay:)


yayyy :)


question 26


i can try


great!


yes great job Sarah!!!


that is right!!!


can you please help with question 27


yes of course :)


thanks:)


so the first one is asking which numbers from A and B are in BOTH A and B


the second one is asking which numbers are in A or B


i think its the opposite


wait you're right i'm wrong


its okay if you are wrong


you are my tezcher


teacher*


yup but thank you for catching my mistake :)


for a or b do i just write down one of them?


yes you only write it once


ok


don't forget the 2 in the


and the 5


and the 8


great job!!!!!


thanks


can we do question 28?


yeah of course :)


do u want to try first or me?


i can try


great :)


not quite, this one requires order of operations (PEMDAS) so first you have to evaluate what is inside the parentheses


so -2-7-(5-4) becomes -2-7-(1)


and then you can solve normally


yes


wait i just realized what u meant by 5-4=1


that is my fault


yeah


its ok


i thought u meant the end answer was 1


i am not sure what to call you


you can let me know please


just Ms. Evans is fine :)


no no


i can call you Ms. Ms. Evans instead


ok that works too :)


but anyway yes you were right
5-4=1 :)


i think


almost its -10


because -2-7 is -9


and then -9-1 is -10


oh ok


is 5 and 6 correct?


check number 6 again


make sure u multiplied 9*9 instead of adding them


now?


can you walk me through how u did #5


and yes 81 is right for #6 :)


okay


i know 5 is wrong:)


anyways i can do it now


yes correct!


yes you're doing great!


yes you're almost there!!!


yes amazing job!!!


you did it all by yourself!!!


All because of your help:)


aww thank you Sarah :)


can you please check 2 and 3


can u walkl me through how u got teh answer for #2


and #3 is almost right just look at the negative sign :)


hello Sarah?


yes


sorry Sarah*


i am writing on the board


are you able to see


oh my fault i didn't see it


i was looking at the shared screen


its okay:)


this is how i got 2


so for order of operations (PEMDAS), you have to multiply the 7 and the 9 before you do the -2


because order of operations is
1. parentheses
2. exponents
3. multiplication
4. division
5. addition
6. subtraction


almost, it would be -2-7(-9) then it would be -2-(-63)


since we multiplied the 7 and the -9


then we would get -2+63 to give us 61


oh ok


-36 for 4?


not quite, it would be -2(-7)-5(-4) to it would become
-(-14)-(-20)
14+20
34


yayyy:)


yayyy :) great job for powering through


thanks


I still have 6 more questions


so when is the next time you are free Ms. Ms. Evans?


i am free all day today except from 11am-2pm


Central time


this HW is due this sunday


so we can do another session whenever you have the time


so i wanna have a next session on friday


yup that works for me :)


what time do u want to have it?


around this time is okay for me


ok so 7am on friday?


or 9:30 am on friday?


oops i just remebered there is a wedding on friday


around this time


how about saturday?


ok so 9:30am on saturady?


sure


ok great i'll see you then :)


okay so you will messege me the same way ?


yes i will


All right thank you soooo much for all your help Ms. Ms. Evans


of course Sarah :)


i'll end the session now bye bye :)


I am so happy i found a tutor like you


aww thank you :)


bye bye now


:) Have a good rest of you day


bye:)


hi Sarah :) i'm ready for our session when you are


hello Ms. Ms. Evans:) I am not able to join right now  so can we please do our session at 6 pm today please?


yup 6 works perfectly for me :)


i'll see you then!


okay thank you so much Tutor Provider.orgMs. Evans


see you then


i'm ready for our session whenever you are Sarah :)


Hi


am late


i*


that's ok :)


but i am ready to start now


ok great u can start the session and then i'll pick it up right away


ok i am


Hello Ms. Ms. Evans


How are you doing?


I needed some help with my Homework


Are you available right now?


please let me know


hi Sarah :) i'm sorry but i won't be available tonight or this weekend 😢 i am busy moving into my dorm. but i will be available again starting monday afternoon to help you with your homework again :)


however i believe in the other tutors at Tutor Provider to help you with this weeks homework so good luck :) you're a smart kid and i know you will do great


Okay No problem at all Ms. Ms. Evans. No worries:)
I will go ahead and take help from the other tutors at Tutor Provider 192.168.1.1)


Thank you so much for replying:)


of course good luck :)


How are you?


I needed some help with my homework


please let me know?


hi Sarah :)


yup i am available right now


Alright


should i start the session please?


yup whenever your ready


:) ok


Hello Ms. Ms. Evans


I need some help with my homework


Are you available right now?


please let me know


Hi Ms. Ms. Evans


i am available right not if you still need help


right now*


yes Ms. Ms. Evans


I was waiting


I still need help


ok great :) then you can start the session and i can pick it right up


ok :)


Ms. Ms. Evans


im sorry Sarah i didn't pick it up fast enough


its okay I will do it again in 3 minuites


pick it up please:)


ok perfect message me again when you are about to start the session


I meant then you can pick it up please:)


and yes i will definitely be fast enought this time


okay :)


Okay I am starting a session now


is that okay?


yup that is good :)


Oh now :( you didnt pick it up


i am so sorry Sarah 😭when i clicked on your name it said that someone else had already picked up the session


Its okay I will see how the other tutor is helping me if he is helping good than i will just work with him for today or else i will messege you again


ok that works good with me :) good luck and i know you will do great


Okay Thank you so much :)


Hi example.comMs. Evans


Ms. Ms. Evans *


I am back


Are you still available?


The other coach only had time to help me with one question


Can you please help me if you have time?


yup i'm still available :)


and this time i promise i will be faster and pick up the session


okay thanks :)


i am about to start the session now


Good Morning Ms. Ms. Evans


How are you?


I need some help with my homework


Are you available to help right now?


please let me know


hi haleema :) i'm sorry but i won't be able to help you today because I have class. However I will be available tomorrow and weekends. But for now I have full faith in the other tutors here at Sarah :)


Ok Ms. Ms. Evans


No problem at all


i will see tomorrow or which other day i need help


i will let you know


No worries at all :)


Hi Ms. Ms. Evans


How are you


are you available to help right now?


hello Ms. Ms. Evans


how are you?


i need help with my homework
are you available right now ? please let me know


hi Sarah i'm sorry i couldn't respond to your previous messages but i'm here now and i can help you for as long as u need me to :)


No problem Ms. Ms. Evans
is it okay after 20 minutes please ?


or as soon as i am ready i will messege you please


yeah of course :) just message me when you're ready


thank you so much Ms. Ms. Evans :)


Ms. Ms. Evans I am ready


please let me know whenever you are ready


i'm available now too


thank you Ms. Ms. Evans for waiting for me i am starting a session now please answer it


Please say ok whenever you are ready to answer or else someone else will pick it up


yup i'm ready now


ok


Hello Ms. Ms. Evans :)


How are you?


Are you available to help me right now
Please let me know


hi Sarah :) i finally have my computer back so i can help you now if you still have time


yup


just 2 minuites please


that's cool just start the session whenever your ready


I am ready


i am startting now


Hello Ms. Ms. Evans
How are you doing?


I need help with my homework
are you available to help me right now


Please let me know


hi Sarah :) i'm sorry for the late reply i was in class. i'm sorry but i won't be able to help you today :( but i believe in the other tutors at Tutor Provider to be able to help you :)


Okay Ms. Ms. Evans No problem at all.
Thank you for letting me know :)


I need some help with my homework


Are you available right now ?


please let me know :)


hi Sarah :) i'm available now if you still have time/need help


yes pleaaase Ms. Ms. Evans I need help


Should i start a session please


yes you can start the session now :)


i am starting right now :0


Hello Ms. Ms. Evans
How are you?


I am so sorry I wasn't able to messge you at 11:00am


i am actually a little busy right now
as soon as i get free i will messege you


My sister needs some help with her math homework
are you available to help right now


Please let me know :)


hi sorry i'm available now if u still have the time/need help


Hi Ms. Ms. Evans


its okay
But are you availble now to help me


I need help right now
PLlease let me know


hi Sarah i'm sorry i'm here now do you still need help


I am so sorry Ms. Ms. Evans


I just saw your messege, I don't need help anymore right now but i will need help later so i will messege you


ok cool :)


:) thanks


Hello


Ms. Ms. Evans how are you ?


hi Sarah i'm doing good how are you :)


Are you availble right now to help?


I am good?


i am good :)


My sister needs some help with her math homework
Can you please help her?


yeah i can help you :)


Thank you so much Ms. Ms. Evans :)


I am starting a session please pick it up


of course :)


Hello Ms. Ms. Evans


How are you?
How is everything?
I need help with my Homework


Are you available right now?
Please let me know:) Thanks


hi Sarah :) i'm sorry but i'm not available right now, but I will be available in 30 minutes if you can wait that long


But i understand if you can't and I have full faith in the other tutors here at Tutor Provider :)


Yes No worries I can wait for 30 minutes


HI haleema i'm so sorry to have to do this to you but i just remembered i have a prior arrangement i have to attend :( i'm so sorry


Its Okay no worries192.168.1.1)'


hiiiiiiii I need help with number 4 if that's okay


hi, of course. give me one moment to catch up


no problem take your time


ok


im ready


OK


okay


Yeah, so


um


first


it's good to know


just what the, what are the equations, right, for the trapezoidal rule.


Um, we can just say big uh is equal 2.


the sum


of I equals.


N


of


F


X


I


plus just the average of XI and its subsequent


post one


one


over


uh delta X.


right? So we have to um


and here they basically give us the steps


It's at uh 1.5 increments.


So your um


Delta T I guess here.


is just the um


the difference between


these two, these two, where each increment is 1.5.


So delta T would just be 1.5.


right? And your small F here is just equal to your velocity.


The only caveat is um


they give you the starting elevation.


So if we say s


is the um position function.


You would say F(X1) +


the sum


from i equals


as we would say it was 6 steps.


6 steps


of


I


X I


plus V of XI + 1.


over 2


Delta T


T


And you just have to consider each step.


right? So there's


one


23456 steps.


Does that make sense


I know I went a little fast


Um, if you need me to explain anything.


a little bit more, I can try to give more examples. I can


um,


I can explain to you how the trapezoid rule is effective.


you know, what's the purpose of it? How does it actually work?


It makes sense I just cant see the numbers on your bottow drawing good


Oh, you need me to expand my formula?


yes please


Is that better


And also, I, um


I see I got this, this, this X shouldn't be here. It's supposed to be a T.


And let me just rewrite this.


So it should be


um T of I and your TI + 1, right? Basically, so it's your current, whichever um


instance you're starting from, right, whichever, uh tea you're starting from, and then plus the next increment.


ok ok


Yeah


If you want


and this is all ddor part a so fart


I can also describe, that's a little too big.


far


and this is all for


parts so far. Yeah. And that's just for A. Yeah, this is just for A.


Um.


but if you've seen how


the trapezoid rule works.


ok im just making sure bc im writing this on my paper


um, as opposed to like a right rhyming sums or left.


Are you sure we can


Yeah.


we can use


say


X


I


all the way to


XI + 2.


XI + 1.


So as you know, um, or you may know that Riemann sums and trapezoid rule are all just approximations.


right? They're just attempts to


uh, figure out the actual value below the curve, right? The area below the curve, which is very similar to what you do with derivatives.


Right? Derivatives we had to take the limits, right, to approximate


uh


well, we took uh


steps to figure out what the


um I don't know if you remember, if you know the air function.


So for the dri a derivative, you would have your function and then you would


find that you'd find a point


you would take the tangent out of the function at that point. And then if you wanted to, based on that tangent, try to and not knowing what the actual value of what the actual function is. You just have a curve and try to approximate the next value


on the curve


you would have um


uh let's say you have the function


of X2, right, the point X1, you have X1 is your original point and you have X2 is the point you want to know.


You could say that F of X2 is equal to F of X1.


Um, let me get some space


By the way, are you following along? I don't wanna just keep running and running without, you know, you kind of being with me.


It's not really helpful to either of us.


no yeah im following


right?


So, um


we have this approximation. What we're saying is that if we have some tangent line Y with some function F.


right, that we could say why one


is equal to F of X1, right? So we're saying that the tangent to the point.


X1, F of X1.


is, you know, that Y1 is the same at that point, right? We're saying that the tangent value is the same.


But if we know the tangent function


right, we say Y is equal to


um the slope, which you can just say is M


uh


B +


X


right? So this would be the line, the tangent line.


Actually, hold on, that's wrong.


I forgot you got a


you gotta take your vertices X minus X1 plus.


right? So this would be the tangent line


at the point X1Y1. And this is derived from the um point form, which is Y minus Y1 is equal to MX minus X1, right? All they did was just move Y1 over.


OK


And if we


you know, use this funk, use this value, we would say that Y is equal to M


X minus X1 plus F of X1.


OK? And the next step,


will be to try to figure out what is this M? What does that mean?


right? So we know


M is


well, you know what the derivative is, but it's just the same what Delta is just the change in why.


over the change in X.


2 minus X1.


And we know, and we can say


that it's F(X2)


minus FOX1 because Y1 is just equal to F of X1.


over X2.


minus X1


right? So now the question becomes


um, what is F of X2.


And how is it related


to X1? Can we use X1 to approximate F of X2 if we don't know what F of X is.


right? Like the only, the only information we have about FFX


is the point X1F of X1.


and that it's occur and then we have a tangent line.


That's the only information we have about the function F


Um, and knowing that we want to know, is there, is it possible to find the value F of X2.


OK.


So going back to this equation, we say FX2 what is is equal to FFX1, which is just, you know, Y1


We're saying plus some uh delta X


times


the slope


which really is just some kind of


ok ok


right, which really this, this is actually just some kind of like error function.


ohhh


right? Because we know that um


that


Y2 cannot equal F of X2.


because the definition of a tangent line


means that it only intersects the function at one point.


So by definition, Y2 can't, there can't be a second


point


that the tangent line touches on the function because by that point, it wouldn't be a tangent line anymore. It'd be a secant line.


yeah


right


So, um


and what what happens is, is that if you can approximate


this delta X, if you take whatever step you have, you would find that, um, let's say F of X2


is equal to Y1, right? The, the point we know about the, the height of the point we know about, plus some


uh


I'm sorry, let me do that again. This is actually


not Y2. This would be Y1, right? So you say FFX2 because we know what F of X1 is in relation to why we can approximate say FFX2 is equal to Y2, let's say plus some error value.


right? Cause we know Y2 can't equal XF X2.


but if we, depending on how big our delta X is, we know that we can, um, kind of approximate


this value. This in all this right here is just an approximation.


right? Because we want to know what is this error value?


And we say it's an error value because we know, um, if you look back at the graph


right, we have Y2 here.


and this is F of X2, and you see there's this gap here.


right?


And that gap


I'm sorry, that's not a plus, it should be a minus then.


Uh


I guess it doesn't really matter


Yeah, it's, it's a minus


I guess the sign doesn't really matter because it could just be a negative value.


But basically, we want to know what is um


what is this error value


right? What is this gap between Y2 and X2.


And the, and the problem becomes if this gap


is too big


then we really have no idea what F of X2.


is.


right? Approximation could be way off.


And this is, this happens when um


this gap between X2 and X1. By the way, I didn't, I didn't explain this, but Delta X is just equal to um the difference between two points of X. So it'll be X2 minus X1, or it could just be XI + 1 minus XI, right?


That's just what Delta X just means there's a difference.


We're just comparing um the distance between two x values in this instance.


OK.


Does that make sense so far


yes


I know I'm going, I'm kind of going back and repeating um information you might have already known.


Um


so, right, so we want to figure out


no no its very helpful


so we've solved for the error value, we find that um Y2 is that the error values you go to Y1 minus


F of X2


right?


And we can also say


that Y2


or sorry, Y of X2.


is equal to


M of


X2 minus X1.


plus Y1 based based on our uh tangent line


right, which we have up here.


right? All I did was plug in X2 for X.


OK.


But we also know that X2 minus X1 is the same as delta X, right?


So we find that M times delta x


+ Y1


is equal to YX2


So what they're saying is that Y2 is just equal to Y1


plus the slope, right, the change.


times the change in X


which we also know that the slope M


is just equal to deltay over delta X.


right, which we found up here


Oh, I'm sorry, the Y2 here.


Yeah, I was a little ugly. Let me, um, you're correct, that is supposed to be Y2. The Y2 cannot equal X2


because the definition of a tangent line means that the, the line can only intersect with a function at one point


and we define that point as X1.


ok ok


Any more questions before I continue?


That's a very good question.


nope


Sorry, my handwriting can get a little bad


Please let me know if I need to, um, kind of enlarge something.


or rewrite it


Um


OK. So back to our um solution for Y2.


OK.


So if we remember back to our slope equation


right? I didn't finish writing it, but you know, uh, if you look at the denominator


this X2 minus X1. Doesn't that just look like our delta X?


yes


Yeah. So we factor this back in here.


right? So we find that M is equal to


um we'll say Y to uh


yeah, I will say F(X2) minus


F of X1 over delta X.


times delta X + F(X1)


Um, what does that give us


I guess it's just give us X of X1 is equal to X1, that doesn't help us.


But I guess we can leave this


leave that as is.


Um, cause I, I fly plug it back in. This is just going to give me that F of X1 is equal to Y1, which we already known, doesn't really help us.


but we could rewrite this in terms of


uh instead of this being Y2, we can rewrite this as M delta X + F of X1.


So that


if we


sorry, go ahead


Did you have a question


OK. So, if we leave it


of this form


um what Y2 is and we factor this back into our error equation.


right? We find that


uh


M delta X.


plus F of X1.


minus F of X2, right? And then we write it all out, we find that


um


the error function is just the, whatever our step side is, tell them to slope.


minus, we could say Delta F.


right, with the actual function is actually, let me, let me just try to solve for.


no F of X1


Yeah, we said


Yeah, that's fine. You can just say F of X1.


which is just equal to


M delta X is equal to minus.


RDF


All right, which is basically just the difference in a function is equal to whatever our change in why is, whatever our slope is.


Um.


and


if we can get


this is where we start taking, yeah, sorry.


Let me enlarge this.


Give me one second I think. I did this, I did this exact derivation


a couple of weeks ago. I'm think I'm trying to make sure I remember everything correctly.


Uh, don't forget anything.


ok take your time]


I found a tangent line um found the air functioning, found the error value.


I think we just want the error value in terms of


why one and S's.


Yeah.


Yeah, sorry. I feel like this delta, this, this error function is wrong.


Somehow it's close to being around. I feel like I'm missing something.


Cause once you consider that er cause like the air function


if you take a limit


as delta X


goes to 0, right? That we make our step size infinitesimally small.


right? We make our X2


kind of really close to X1, right? You'll notice that our air function


should equal 0


right? At that, at that point, you would say that Y2 is equal to f of X2.


right? If your, if your air function is absolutely zero. And this is where we can um


say that our approach using the tangent line.


is an effective approach to evaluating um


the next values in the function, right? That it can be used effectively and if you had um


if you had the


tangent, if you had the slope


at any point in X. So you had M


for every X


in the domain, right? All real numbers that the function lies.


then you should be able to approximate


all the values of F of X.


Um, there is


Yeah


All right, I'm gonna cheat


Actually, I can't, I don't know if I can find that.


Um.


anyway, and if you had, and if you did the sum,


or if you


if you did the set of all slopes for every X.


you would have the function F of X, which would give you all the information of the change in behavior.


of F


right? So what a derivative actually is


is just, it tells you the behavior of the function F.


which means that you know how the function is changing for any point in for any point X


inside the function


So you could predictively model


you would like just with the derivative function, you would know exactly how the function is changing at any point in time.


Yeah, that's, that's the all real numbers are.


So this is, so this is just to prove that like, you know, it's


an effective


inaccurate way to predict the behavior of F.


the function F, that this derivative, this derivative function.


Um, all that to say is that, you know, this is what we did the derivative. Now we want to figure out what is an effective and accurate way of predicting, of knowing the behavior


of the area under the curve, of being able to find the value under the curve.