Um, good questions but a good question, but no. Ankle II is not the vertical angle of angle II.


Angle II is down here on the left side between line A AD and uh and BC.


Oh


this, well, let me show you.


Ankle a 3 is on the right side between the lines CD and BC. They both, ah, along the same line AD, but they're on different sides of it, and they're not formed by the same intersecting gang a lines so they can't be vertical angles like you asked.


Vertical angles happen when two lines cross each other, and the opposite angles they form are equal, but here angle 2 and angle 3 and from the same intersections, they're formed by two different pairs of lines, so instead of being vertical lines angle 2 and angle 3 are actually alternate interior angles, like I said before, and, and that's why they're conquered in this roof.


So the red circle is angle 3?


Um, uh, if you don't understand, uh, think of it like this. Vertical angles are across from each other at one intersections, but angle 2 and angle 3 are at different corners of the shape, so they can't be vertical angles.


Um no. Uh, that's not a a Ankle II. You asked the angle a 3 if angle 3 vertical angle of 2, I just want to show you that um uh and go to BC angle 2 not 3.


Um angle 2 is labeled on the right side tho


Oh, sorry. Good observation. Um, uh, in the diagramangle 2 is actually on the right side, not the left. So look, an angle one is at the top left, where lines AB and AD meet. That's a cool one. Ankle 2 is at the bottom right, where lines CD and BC meat and angle ui is right next to it, not next to Angle B. It's at the top right where uh lines A, B, and


BC meet


Ohh


Is that it? You have another problem.


or or question


No


Alright hope I helped


Wait so


Go on


I just say angel 2 is congruent angel 3 and angel 1 is congruent to angel 3


Wait no


Okay thank you I have to go


hi


OK


share your Qs


I am happy to help


ok


volume?


Yes


I need help finding the LCM


Also the the total of the cubic inches


LCM or volume


both


volume is len x width x height


width = 3/2


1 + (1/2) = (2 + 1)/2


good?


does that answer your Q?


are you there?


hello


can you help me solve this


Hi Alex,  what do you think?


i already know how to solve the frist part which is pluggin in -1 into the eqaution


but the other is what im having trouble with


Good, what is j(-1)?


Good, if j(-1) = 0, what does that tell you about x+1 being a factor?


it is the factor


Yes.  Do you  want to verify that?


um sure


Have you learned synthetic division or polynomial division?


yes i did


In order to find the remaining zeros, we need to factor the resulting polynomial after dividing by x+1, correct?


yes


Or find the remaining factors.


Which do you prefer, long division or synthetic division?


synthetic


OK so far?


What does that tell us?


not sure:(


Because the remainder is 0, x+1 is indeed a factor.


ahhh got it


What is the polynomial after factoring out x+1?


The remaining polynomial is what I have drawn in red, yes?


im a little lost


oh


48 is canceled


i see


Now we need to find a factor of 2x^3-x^2-35x+16


do we do synthetic division again?


Yes, but what factor do we test?


Do you know how to find all the potential factors?


no i dont think so


Potential zeroes are the factors of 16/factors of 2, remember?


Uhhh


demonstration maybe?


What are the factors of 16?


All the potential factors


1,2,3,4,8 ans 16


16,8,4,2,1 plus and minus...


not 3


yea


3 is not a factor of 16, that is correct


What about the factors of 2?


2 and 1, yes


So the possible factors of our cubic polynomial are 16,8,4,2,1, plus or minus


ok


Which one to try first?  Or use DESMOS?


Plus or minus?


plus


OK?


OOPS


i have a question if i have to do all the numbers one by one with plus and minuses it will take a long time right


Yes it could, but you can stop when you find one zero.


but dont i need to find all the zeros?


Yes, but once you find one, you will be left with a quadratic (power of 2) which is easier to solve.


I made a mistake.  The polynomial after division is at the bottom.


is the other division easier for this eqaution


Synthetic division is usually quicker.


The polynomial after division by x+1 is 2x^3-3x^2-32x+48


So unfortunately there are MORE possible zeroes.


Have you learned about DESMOS?


nope we dont use desmos in class


Too bad, that would make this easier.  Maybe next year.


hopefully


The possible zeroes are the factors of 48/factors of 2


but how do you know


48 has lots of factors


The possible factors of a polynomial are the factors of the constant term divided by the factors of the most significant coefficient.  48 and 2 in this case.  plus and minus


Have you learned this?


now im not sure


If not, how would you find the possible zeroes?


umm by using synthetic division again


Synthetic division will test a possible zero and determine for sure if it is or isn't a zero.


But to identify possible zeroes, you use the factors.


oh so we need to find the factors of 48 and 2


Yes, do you have the time for this?


not really but can you show me for at least 2 factors if you have the time


or one


Using DESMOS, -4 and +4 are zeroes so x+4 and x-4 are factors.


Which one do you want to try using synthetic division?


Your choices are +4 and -4  :)


oh


Does the synthetic division make sense?


where did you get 8, 20 and -48


ahh


oh ok it make sense


4 times 2 is 8, 5 times 4 is 20, and -13 times 4 is -48


-12 times 4 is 48


where did you get the bottom numbers


Adding the columns


bring down the 2, -3 +8 =5 etc


ok it make sense


2x^2+5x-12 is the remaining polynomial


do you think there is an easier way to do this problem on a test


Can you find the zeroes?


Hopefully your teacher won't give you a problem this long on a test.


this is on the review sheet ;(


If you can do this problem, you could do any reasonable test problem.


then im falling this test:)


Im joking


Good.  I said reasonable


so the zeros for this problem is -4 and 4?


and -1 from the start and one more


1.5 according to DESMOS


-4, -1, 1.5 and 4


okk i think i understand


tyyy


sorry for taking so long


You're welcome, don't be sorry this is kind of complicated stuff.


have a good night


You too Ben


Hi Sarah!


How can i help


I need some clarification on how to do this problem


Of course


find the distance between the line and the point


Could you give me some context on this problem?


Oh ok


All right lets get started


Can you tell me what you know about solving for distance between point and line on the basic level?


You need to use distance formula


yes that is one of the steps


could you write the distance formula on the whiteboard for me


That's right!


This helps you find the distance between two points


Now, there are two ways you can approach this problem


You can find the line perpendicular through the point, find the intersection and use the distance formula between the two points


the second one uses a formula that calculates the whole thing for you. have you learned that formula yet?


but one is a line


Yes, it's for point and line


ok


This is the most useful formula to find the distance between point and line without doing multiple equations:

 |ax+by+c| over sqrt of (a^2 + b^2)

where ax + by + c = 0 is the equation of the given line, you identify the values of a, b, and c, and you substitute the point into the equation on the numerator.


Does that make sense?


I put it on the whiteboard


y=ok


who where do you sub


x and y?


if we convert y=x+1 into the standard form


you get x-y+1


so what are the values of a, b, and c


then you substitute those values into the formula, and then you put 2 and 2 in place of x and y


so how do you find a, b and c then


well if the formula for standard form is ax+by+c


and your equation is x-y+1


what are the a b and c values in this case?


well is c=1


Yes


Absolutely!


And what about a and b?


wouldn't it just be 2


Not quite


what multiplied by x gives us x


and what multiplied by y gives us -y


there is a one next to the x and y variables


x=1 and y=-1


Sorry, a=1 and b=-1


Now can you substitute all these values into the formula?


is that right?


Yes but remember a=1 and b=-1 so that also applies for the denominator


not 2


ok


is that good


Yes~


so now what


now you substitute x and y as your point


2 and 2


and then simplify the whole equation


i got 0.71


Yes that's correct!


Just note that even if you leave the answer as sqrt of 2 over 2 it is absolutely correct too


Do you have any other questions?


ok thanks so much


I will save equation for later so helpful


You're welcome!


If you have any other questions don't hesitate to let me or other tutors know, remember we are available 24/7!


Have a wonderful day!


Hello!


Hi


I have a Algebra Test on Thursday and I dont think im going to do well.


hmmm, well hopefully I can help you understand some more before Thursday


do you have some review problems to post?


uhhh I have topics that I dont understand


ok, what are they? maybe we can do some examples


quadratic formulas


and those things that are negative square roots where you get the i, I need help with them and simplifing them in a quaratic formula


ok. I am guessing that you know the formula?


yeah


oh and determining the mising square roots


not sure what you mean by determining the missing square https://example.comn you give me an example? Also, do you want to try and solve this equation?


yes i will solve it then give you an example


ok


is that right?


its close, but the negative in front of the first 4 should not be there, then you should simplify what you have (it can be reduced)


all 3 of your numbers can be reduced by 2


that's it


yay


for your next problem, you probably learned that radical answers come in conjugate pairs


see what I wrote? does that make sense?


can you explain it  more


if we are given that square root of 5 is a root, we automatically know that negative square root of 5 is a root. have you noticed that when you use the quadratic formula, when you get answers with radicals there is always a plus and a minus?


oh


so the missing root is


- square root of 5


right


good!


I am guessing you have to multiply it out? not sure what your teacher expects


i have no clue, i took a test on this today but its gonna be on the test on thursday


i think mulitply it out to find the degree


yes, so go ahead and I will let you know if its right


ok


is that right so far?


the part I underlined is not correct


do you know the term FOIL?


kinda


i use the box method to multiply it out


https://example.orgtice when you multiply the terms I marked in red, they will cancel out since they are opposite signs


ok, but what is square root of 25?


yes


now you get to multiply that stuff


what is the degree?


4 is correct


what do you want to work on next?


im looking for questions from past tests


dividing polynomials


and remainder theorm


wanna try this one? can you do synthetic division?


sort of


do i muliply or add


bring the 3 down first, then do 2 times 3 and add to the -6


help


make sense?


can you give me another one


sure


hang on a sec


okay


good


so could you say that x+2 is a factor of the original polynomial?


no


because it has remainder


exactly (you basically just used the remainder theorem)


okay perfect


on that first problem up above, can we say that x-2 is a factor?


yes


because no remainder


right. and then we can also say that 2 is a root or zero of that polynomial since x-2 was a factor


that's basically the "factor theorem"


okay. i understanfd


understand


what about how do you determine whether or not its positve by the degree and if the function is even or odd


not sure what you mean by the first part of your question (positive by degree?). but to tell if a function is even or odd you have to find f(-x) and compare to your function to see if you get the same thing or the opposite. I will do an example on the board.


see how we got the opposite of f(x) when we found f(-x)?


this means we have an odd function


okay


here is example of even


does that make sense?


so you have to remember to replace x with -x in your function, work it out, then compare with the original.


okay i got it


as far as "positive" and degree, are you referring to end behavior of a graph?


yes!


can i clear the board?


yes


can you tell me the degree of this polynomial and whether the lead coefficient is pos or neg?


degree is 4


leading is pos


yes. so we can just focus on the x^4 part and think about what happens to the y values if we plug in large negative numbers....will the y's be positive or negative?


yes, and if you plug in large positive values?


I think you are saying positive infinity for both, right?


ok. so notice when we have positive lead coefficient and even degree, the left and right behaviors are both going to pos infinity


if we had a negative lead coefficient, how would that affect the end behavior?


it would make it negative infinty


right


what do you think end behavior is for this one


odd degree always has opposite behavios


r


I have a few more minutes...what else do you want to ask?


i think thats everything thank you so much for helping me!


ok, you are welcome! good luck on Thursday!


thank you!


have a good night


you too


Hello! I would like some help on my assignment please


hi Lily, how can i help?


I would like help on my ixl assignment please


statement 3 and 4 have a lot like the choices and I couldn't screenshot it


but let me do it in pieces


OK, so you want to start with #3?


I'm not sure what your available options are but...


let me put them on the board


those are the options


and those are the same for statement 4 too


i sort of see it...  for #3 I think you want something like SSS


sorry Mr. Smith gotta go, have a nice day!


its the vertical angle theorem


Hey


Hi


I just want to check an answer real quick


Sure


I got 21


Thats perfect. Do you have questions on how to get that or did you just want to confirm the answer?


Yea I'm kinda confused on how I got that when I don't really understand it much


I assume the confusing part is writing out the percentage as a fraction.


Yea


I meant decimal... but percent is written as "hundredths". for example 37% = 37 hundredths which is 0.37 The 1/2 adds the confusion probably


With the 1/2, we know we are halfway between 0.37 (37%) and 0.38 (38%)


So it's 0.5


Since percentages are in "hundredths", its actually half a "hundredth" with is 0.005 (1 hundredth is 0.01, so half is 0.005)


Ohh so that's why the 5 was in the 0.001 spot


Yep, a little confusing, but each percent = 0.01 in decimal, so a half percent is 0.005


That makes more sense now


Did you want to try another similar one, or do you have other questions?


No I think I'm good on the questions


And thanks for showing me/explaining it better


No problem, hope it helped. You got this.


Thank you


Have a wonderful day!


You too


Howdy !


hi


So you've some math stuff to work on?


yea


my internet might be rlly slow tho


Is it something you can type in this dialog box?


yea ill put it on the board


Okey dokey.


So you are to graph this line?


So there are a couple ways to do this...


1. Find two points on the line and then "connect the dots"


2. Use the point-slope format of y = mx + b and sketch the line based on the values of m and b.


Which way you wanna do this?


i think it would be easier to find the two points :)


Yup, that's what I normally do (but sometimes the teacher wants something else).


Okay, so let's write a little table on the board like this:


So the idea here is to pick a few points for x (values of x to be precise).


Then we plug those into that equation and find the resulting y value.


And we'll have our first point (x,y) to plot.


So you can pick any value of x but I would keep the number less than 10 since that's all the paper we have on the graph.


How about x = 0 ???


oki


So when you plug in 0 for x, what do you get for y ?


would it be based on the y = 9/4x - 1 or just in general?


Yes, it's based on that equation.


You plug in zero for x like this...


So we've got our first point - yippee !


I'm gonna plot it in red.


Okay so far?


yea


Now let's pick another value of x (again between -10 and +10).


I'd recommend 4 - is that okay?


You'll see why in a minute.  Some of the choices of x will make our lives easier.


Yippee !!!


Point # 2


So now I'll connect the dots and we're DUN.


oki :)


Something like that.


alr


Anything else for today or are you all set?


yea i have a few more graphing ones


Can I erase?


So connect the dot method again?


So I'm just gonna plug thru it, same as last time.


Hollar if you get confused but I don't think you will.


when we do this should we always start with zero ?


I'd recommend that as it's usually an easy choice (again, let's make our lives easy).


And the second guess for x on this one is gonna be 4 again just because we have a fraction in front of the x term with a 4 in the denominator.


oki👍


Yeah, something like that.


Still making sense?


yea


You wanna try the next one, or are you done for now?


i have 2 more


So make your table and find your points.


Again, I'd recommend to use x = 0 and also x = 4


I have the hardest time writing on this whiteboard too - it's like writing with a crayon stuck between your toes.


ik


it rlly sucks :(


Perfect.


so the 4's cross cancel right ?


Yep.


so then i do -11 + 2


You're crushing it.


Well done !!!


Wanna do the last one?


sure


Perfect !!!


Now you're probably bored with these...


not rlly i just wanted to finish these ones on my math book :)


Well you're doing great.


Best of luck to you...


u too :)


Take care.


Hi Liam


Hi


What do you need help with


can you help me what is on the whiteboard


Which one


all of them we are plugging them in


Okay


where should we start


you tell me


okay lets start here


perfect


what do you think this is


linear change


Correct


yessss


ok anything else i need to add


Try this one


Actually your first one is not linear


It it exponential


sorry about that


yes i was going to say


What about the one i just underlined


i just said my answer


which is?


the oen with arrow


The graph


oh ok


The table shows that every month it is increasing by 25, not 50


so this is correct


Not quite


listen you have to help i can't keep guessing meaning that you have to at least show me


Yes, I am wondering your thought process so I can help you


yes not going to keep guessing as i stated


Okay highlight, the question you would like to solve and we can go through it


nvm


Hi Sarah!


What do you need help with?


hello


can u see board


Yes!


Do you need help with the first question?


yes all


i got c


for first one


Yes! There is also another one that can be right.


Do you know which one?


hmmmm no


Not sure


Ok, so the term X 1 and X 2 can be used interchangeably as long as they stay with their corresponding Y value


Ok


Let's go through them in order


For A


ok


Do you know what the two points would be for A?


like (x,y)


like (x1, y1) and then (x2,y2)


well for x 14 y 5


-7 9


x


Yep! For A, the first pair would be (-7,9) and the second pair would be (-14,-5)


It doesn't matter if the order doesn't match the order they give you as long as they are the same two points


So that would also be correct


Ohh ok


Those are your two correct choices for the first question


Does that make sense?


Yes kinda so my answer c was correct


C and A


because A is the same thing as C, but the order of the points were switched


ohhhhhhh


C is (-14, -5) and then (-7,9)


A is the same thing but it is (-7,9) and then (-14,-5)


does that make sense?


yes


for theslope


we do y2-y1


right


Yep!


And you divide y2-y1 by x2 -x1


Let me know what you get!


ok


i got -12/-5


am i wrong


What did you plug in?


-7-5 and 9-14


Okay so for your x and y values that you are plugging in, let's use the answer we got for the last question


So using the x1 and y1 and x2 and y2 from C


let me know what you get


ohh i was just using it as thr pojtns


abouve


You can use the points above too! This just simplifies the process cause you already found out what you are plugging in


qair si now ut 9-5


and -7-14


right


Because you have to do y2 - y1


9 - (-5)


would be 9+5


Cause the negatives cancel


does that make sense?


so 14/-21


The same thing happens to the bottom so it would -7 - (-14)


yes dont leave though


I won't!


What did you get for your slope?


i got 7


so 14/7


Yep!


ohhh


i know your good at this just wanna make sure cause my homeowrk will make me restart if it wrong but i get it


No, you are okay! I understand


ok


the line goona rise


left to right


Yes!


should i simplfiy it


so 2


instead of 14/7


Yeah, I would simplify just to be safe


i will be right back gotta go


if u loeave it ok im gonna request again can u try to click e


Okay! I will try to stay on, but if I leave I will try and click you again


Hey, I have to leave in fifteen minutes


Just like a quick heads up


ok im back


Ok! Is there another question you need help with?


thus


Okay, do you know slope intercept form?


isnt it y=5/1x plus 10


or no


Slope intercept form is y=mx+b and you are on the right track for the question


ok


it would be y= (-10-x)/5


because you have to isolate the y so you would subtract the x


over to the other side


ohhhh I see


does that make sense? I can like write it out if that would make more sense


like dis


ohhh


and then you would simplify


i got -11/


wait what


i tgought u subtract the -10-x


then over 5


Ohh I see what you are doing


y= (-10-x )/5 is slope intercept form


you can also write x/5 as 1/5 x


ohhh i didnt


and slope intercept form is y=mx +b


and in that m is your slope


so the number thats with the x


is your slope


does that make sense?


yea u divided -10/5


so -1/5x -2


or plus 2


yeah but the slope is just the m value


ohhh


so if the equation is y = mx + b


do you know what m would be in this case?


what is attached to your x?


the the -2


try again, what is like right next to the x in the equation


yep!


That's your answer


yes ima graph it


and then the -2 is the x right


i mean y


that's your y intercept


so when x = 0


your y = -2


If you ever need to find slope, isolate the y value so you can get it in slope intercept form and then the number in front of your x would be your slope


and the number behind it would be your y intercept


ok


does that make sense?


do i just put the -1.5 for slope or the  whole answerr


yes


just -1.5


hey heads up I have to leave in 5 minutes


is my graph right


ohhh waiy


this is because it intersects the y axis at -2


that's what the -2 is for


and the slope goes down 1 and over 5


yea i see whe i msedd up


yep! you got it


okay so if x+7 =0


what does x =?


ummm 7


i think


we subtract right


yep


close, remember your signs


it wouldn't be +7


but instead...


right!


here no slope so udnefined


then how do we make one on graphb'


graph


exactly


for the graph it would be a straight line


cause no matter what y is x will always be 7


does that make sense?


yes


like  this


Yepp!


I hope that helped, I have to go now!


can i requst you again


Have a great day/night!


Sure, if I am free


I will definitely help out


I hope that helped


Hey, I'm Sarah, how are you?


Hello


I'm good, how are you?


I'm good, are you just trying to solve for the zeroes here?


Yeah, I'm trying to solve this equation. I know it's non-factorabnle and I have to use quadratic formula.


Great! What do you think we should do first?


Great, so first we move everything to one side to set our equation to zero


Alright


What are our a, b, and c values for the quadratic equation now?


So, now we can just plug them into the quadratic formula right?


Yep!


I'll calculate these right now


360 doesn't square root evenly.


But we can simplify the radical right?


Yep, I'm confused how to


That's alright! Let's look at how we can do this


First, we should pull out our radical -1


And always remember radical -1 can be written as i


You got i by multiplying 360 by -1 right?


I got i by splitting radical -360 into radical 360 times radical -1


Ohhh, okay


Anytime you want any further clarification on anything, please just let me know btw.

But just always remember when you have a radical -1, it's the same as i


Now, what perfect square number can I pull out of radical 360?


Okay, thank you for clarification I get that part now.


Hmmm


Perfect!


What does this leave us with?


What is radical 36 equal to?


So I can rewrite this as 6 times i times radical 10 right?


Yes!


Now, there isn't anymore perfect squares to pull out of 10, so we have fully simplified our radical


Do we go back to the quadratic formula now?


Yes! Now we can rewrite 6i radical 10 in for our radical -360.


Okay! What is next?


Well, is there any way we can simplify our fraction?


I think we should divide by 2


For 22 and 6i


Exactly! Every number is divisible by 2 right?


Don't forget though, you also have to do it to the 6


The 6 that's before the plus-minus sign?


Perfect, so that is our final answer


However, just as a side note, you may see in some questions it's written differently a little


Ohhh okay, I understand it better now


Like that instead


Ohhh


Thank you a lot! That made it easier for me.


Of course, do you have another question you want to try?


I have one more that looks a bit harder, but I have to leave class on 10 minutes so not much time :(


Oh alright, that's okay, if you want to put it to just see what we can get through you could.

If not, just remember you always pull out radical -1 as i when simplifying, and then pull out your perfect squares.


Alright, thank you! I understand it now, so I can do it later. But I'll keep those in mind.


Great! You can end the session whenever you're ready.

Goodbye!


Thanks, Bye!


hello i have atest tmr and there just some questions that i dont get at all


and ineed help really understanding them before my test tmr


Hello


Which questions?


this question


can yu see it


hello


Yes


How do you understand this exercise?


What are they wanting from  you?


to find out howmany were red and the howw many were blue


Okay


How write ratio?


like this?


Okay,good:)


When we have ratio what do we?


im not sure what to d after this


Let's meet show to you


okk


Me*


what is this


We have rule


rule ?


how would we solve for pink on the right side


Yes the product of the numerator and the denominator is equal to the product of the numerator and the denominator


Rule of ratio


ive never heard of that be used for theses type of question


You have r,what will you do?


Ratio a:b=c:d the product of the middle terms is equal to the product of the extreme terms


And we can ratio like fraction


would i multiply by 7/1 to both sides


im not sure  what to do


You will have the same thing


But with the rule of ratio it is more easier


Are you understand about ratio?


coudl you show bc i dont understand youinstrucctions


msybe if you visually show me i might understand better


And we can write  ratio like fraction


Okay,sure


Unfortunately, I am on phone


i see


I am sorry,I don't expected that


okay maybe we can skip this problem i ts kinda late and i wanna gi over some other problems


oh no your good its fine


Thanks for your understanding


yea of course


If you have time please send request again after 1 hour


I will help you on my pc


sure will do


so aorund 2 then


Okay:)


okay then bye for now ill come back in 1 hour at 2 am


Okay, thanks


I need help solving one specific problem


Hi. I will try my best!


Ima put it on the white board


looks like we have a quadratic equation to solve


what do you think the first step should be?


Subtract x squared


ok, good start


what next?


Subtract 7 to get the x squared by itself


no


Add 8x and equal it to 0


yes


good! and now?


Split it into (x+1) and (x+7)


great job!


Thank you


any other questions?


Yea


How do I do it in this


I’m trying to get the photo to load in the whiteboard. It’s not working.


oh


It says given triangle ADC is congruent to triangle CBA solved for X


Sorry my triangles bad


That's ok. So you are supposed to solve for x, right?


Yes but he wants me to type it in. I don’t know which x she wants.


I see. Let's try both and see if they both work. Try x=-1 first.


9 seems small for an angle measure, doesn't it?


Yes


105 seems more reasonable


do you know which x to choose now?


Thank u


youre welcome!


hello


How may I help you?


I get lost here


You used 12 twice


do i need to divide or just simplify the fractions


You need to figure out if those are all the same ratio.


for example, does 15/20 =12/30?


theyre not


*they arent


Then the triangles are not similar


ah okay


Are you all set or did you have another question?


im set


thank you


hi


how can i help you?


Evergreen High course challenge and i have no idea what anything is so i want some help to understand how to solve and answer the question how are you?


Also what do you want me to call you by


Maya


ok


ok Maya how do you solve this


you have form of c *(ai+b) schematically. How to make this expression look like a + bi?


divide c


no, I mean schematically


it could be any number like f * (ri + d)


and you need to have a + bi


divide outside variable?


Does it make sense?


so 72+2i


eh, why?


idk


Well yeah it is equation of this form. But you need to translate your equation to this form


divide 7


you cannot divide for no reason


divide 7 to make a+bi by itself


okay imagine the expression 2+2 which is obviously will be 4, but you say, what if I just divide it by 7. This is not correct and unreasonable. For our expression you just need to multiply seven on your brackets


can you divide 7 to get rid of it on the outside


ok its that easy?


yes


oh ok sorry im in 10th Grade dont know i thought it is some complex formula or somethint


that's okay. Evergreen High high math


ok


is that right


frick i did it harder way


wait


well we are both wrong


why


because this is geometry progression


what


progression?


Can you enter answer multiple times?


yeah


how this q


can you solve it?


that it?


what?


the answer


i cannot see it


yeah right


huh


is it right?


yes


how do i solve


can you solve ln5 and ln8?


whats ln


natural logarithm


no


how to solve


you can express any logariphm as a division of logariphm with any base what you need


and we use natural logariphm which we now solutions of


ln5 = 1.609 and ln8 = 2.079


how


open the calculator


you have numbers neat to log


near*


ln5 and ln8


ln(5) just write it


doing good?


ye


ln5 and ln8 are just constant numbers. you divide one constant number on another and get constant number


ok


so what the answer?


yes, good


that's okay when will do another similar, you will do it much easier


ok im done for today


cool, have a good day


i give up today


thanks for helping


bye


you did great


Hi there!


hi


I need help checking my math homework please


yes i can help you


do you want to write it on the board


I'll put a picture of it on the board, gimmie a second


ok take  your time


There we go


ok let me look at it


the answers you put are correct


Awesome


Here is the next one


i think r=8


Okay


let me explsin it


on the board


do you understand it


Yes I got it


ok


yes your correct


good job


Great


here's the next one


ok let me check it


your correct on this one too


Thanks


Okay here is the next question


how many more do we have to check so i can go faster if it takes long


I have a lot of questions lol


let me see


its ok


I have like 20 more


if that's okay


yea i have time


this one is correct


okay good


next


wait did I just do this one?


yes


sorry let me get the next one


wouldntit be -2


This question had me a bit confused lol, so probably


yeaits -2 because thats where the grahp has the same pojnt


its basically where they intersect


Ohhhh gotcha


here's the next one


your correct on this


Yay


here is the next question


can you make it bigger


sorry


its the last one actually


ooooh


do you understand the work i did to get the answer?


because i want you to understnd it


ok good


Lol


here is the next one


i have to go in a bit but i can help you again after 30 mins


ill help you with this last one


Okay!


it should be the second one with -2 and 2


do you understadn what i did


yes yes


you bascially take the negative and positive 2 and divide it on the other side


i can answer one more


okaaays


its the one with 4 and 6


ok bye


hello


Hello!


I’d like to get some extra practice in, so I was wondering if you could rewrite the problems with different numbers but keep the same concept. I’d like to start from Objective 9. Here's the link to the PDF: https://example.com/math_problems.pdf


I currently need a problem that is like c of obj 9


okay


I'm not able to open that source for some reason


Try now


Click the link on the whiteboard twice & then copy and paste it


I know but it doesn't work. Can you provide me one of the equation that you're working on?


Give me some time to make the high quality questions


ok you can also use chatgpt to generate one for you


kk


sorry about that i'm back now


can you write them on the whiteboard


I wrote it on the whiteboard


the answer is 9x^4-24x^2y+16y^2 for the first one


for the 5a^3 problem is the anwer 25a^6+12a^3b+4b^2


Yes


now one like d


What do you mean?


You want more practice problems?


for letter d of obj 9


onlny need two


I don't know which one are you talking about


letter d


oh i forgot u dont have access to the pdf


okay


So just do the (3t+5)(2t+1) before you do


no im saying rewrite one uot like that


hello


Hello Alex!


I see that this is your first session, welcome to Mathville Academy!!! :)


Let me know if you need any help with the platform!


Hi!!!!!! thank you!!!!


this platform looks really cool thank u!!


I would like to learn more about piecewise functions if you know about thaattt :)


Yup, I do! :)


Is there a specific problem you’d like me to help walk you through?


You can paste images onto the whiteboard by clicking on the picture icon with the small upwards arrow!


oh okay! please give me a moment to find the problem ^_^


No problem, take your time! :)


okay, there's the image!!!!!


could you walk me through this problem if u don't mind??


I see it! :)


Yup, of course!


yay, thank you!!!!


The stuff underlined in blue are the equations you’ll graph, and the stuff in pink is the restrictions!


ohhh wait what are restrictions if you don't mind me asking???


wait this is such a dumb question I'm sorry Mr. Lee


In this case, they are domain restrictions! So you’re cutting off parts of the graph, if that makes sense, I can draw it out


No, it’s not a dumb question, learning takes time! When I was learning piecewise functions, it took some time before it clicked, and that’s totally ok! :)


yeah please draw it out if you don't mind!!! I'm more of a visual learner heh


The black line I drew is the equations you’ll graph y = x


If I restrict the domain (x values)… (I’ll write it on the board)


That pink thing I wrote is basically saying, I only want the points of that line (y = x) to have its x-coordinates be between 1 and 5, does that make sense?


yeah it make sense a little!!!! so I'm graphing the black line you drew?


I'm a little slow eee


For example, the green dot I plotted is OUTSIDE of this restriction, because it’s coordinate point is (7,7)


And 7 is not between 1 and 5!


Let me know if it doesn’t make sense, I’ll adjust my explanations!


ohhh I see!! I understand, thank you!!!


And the y = x is just an example, you’ll be graphing y = -x + 4 and y = 1/2x + 4


f(x) is basically the same as y, just a side note!


ohhh I see !!! so I'm just graphing the two equations at the top right?


Yep! And you’re “cutting” those graphs based on the restrictions (underlined in pink)


Those equations* not graphs I mean


oh I see !! so what's the next step? how do I solve it


also maybe you could solve this for me and like I could solve another one by myself following your steps


Ok, I’ll guide you through it!


thank you!!


Have you seen unfilled dots and filled dots in piecewise functions?


wait arent those called like "closed" or "open"


I forgot the name for it but I have seen it before


Oh yes, I’m sorry! They are called closed and opened


What do the closed dots tell you?


oh yeah! don't the closed dots tell you like . if the inequality has a line under it, it's closed or smth like that????? like if ">" had a line under it ,, and if it has no line its open???????


I think I'm not sure


That’s right!! More precisely, closed dots indicate that the value is APART of the piecewise function. So in this case, for the bottom restriction, it’s saying that x = 6 is INCLUDED in the piecewise function.


Open dots indicate that the value isn’t apart of the piecewise function, and that there’s a “hole” in the piecewise function


So in the top redirection, it’s saying that -3 and 2 are NOT apart of the piecewise function


Top restriction*


ohhh because they don't have lines under their equality okay I get it!!


so 2 and 6 are included because its closed right??


for the second one


You’re really close! 6 is included for the second restriction… but 2 isn’t!


wait omg I'm slow nevermind I forgot there wasn't a line there


okay so we include 6??


For now, can you graph the equation y= -x + 4 for me? And then I can help you figure out what parts of that line to erase!


Yeah, I think this will make more sense once you graph it!


And it’s okay, you’re not slow!! Learning takes time, be kind to yourself :)


wait could you hold on for a few seconds? I'll graph it in a few secs wait !!


No problem, I have a lot of free time today! :) take your time, no pressure!


Let me know if you’re having trouble with the whiteboard!


I can also help you graph the lines, if you’d like? :)


ok I'm back !!! and sure! I can try to graph the equation and you can just point out any mistakes I make if you want :)


Alrighty!


ok I'm sorry if I make any mistakess


Mistakes are how everyone learns, no worries!


is it correct to put a dot at (0,4) first Mr. Lee am I not on the right track


That’s correct! We’ll check the restrictions later, let’s focus on graphing the lines first!


It’s correct because that’s the y-intercept!


Of the first equation


oh okay thank you!! I was just afraid of messing up, I'll start graphing


And the second equation, but we’ll worry about that later haha


Looks good!


wait do you want me to draw a line through the dots as well


Yep! Both of these equations are lines :)


okayyy!


I'm sorry the line looks so bad haha I tried


wait there's more dots I have to put right


No, it looks fine! Yes, could you extend it? Then I’ll show you how the domain restrictions make you “cut” the line apart!


uhhh wait I'm a little confused I'm afraid I'm gonna mess up


I feel like I'm overthinking hel


Ok I’ll guide you through it! That’s totally okay!


In f(x) = -x + 4, what’s the slope?


isn't it -1?


Yeah!!! Here’s a way of thinking about it (I’ll write it in red on the whiteboard!)


okay thank you!!!


Slope is rise over run!


So a slope of -1 is telling me to move down 1 unit as I go to the right one unit!


so do I move down from (0,4) or?


So you did a great job, to extend it, continue this pattern of moving down 1 unit WHILE also moving 1 unit to the right!


ohhhhhhh okokwait


It’s a little cramped on the whiteboard, but you see how I’m following this pattern?


When I move one unit to the right, I’ll move down one unit! Because the slope is -1!


do I put a dot at (1,3) ?


wait


We’ll get to that later, let’s focus on graphing most of the line, and then later we’ll slice the line apart based on the domain restrictions!


Right now our line is at (2,2), to continue it… let’s move to the right one unit AND down 1 unit!


ohhhhhhh wait hold up I'm gonna try to graph


Go ahead! :) let me know whenever you need help!


like that???? or


should it be at (3,1) ?


Yep!


I was thinking that but I thought I was wrong


It’s correct because the slope is -1, which is telling us that, as we move one unit to the right, we move down 1!


okay thank you! do I keep continuing this pattern or is that all?


Let me clarify what I’m trying to do here: I want you to graph the full lines, so you can later see how the domain restrictions guide you to cut the line apart! That’s why I want you to continue the pattern to make the line longer


Can you extend the line to the left a little more? :)


oh okay wait !!


wait did I do it rigjht ...


Yeah, you did! As long as you follow what the slope is telling you, you’re good! You extended the line more to the RIGHT, but could you extend it more to the LEFT now?


I’ll give a little hint!


When you’re extending to the left, you kind of have to think backwards to ensure you follow the pattern of moving down one unit while moving to the right one unit!


wait I'm gonna try to attempt this hold on


Sure thing!


wait I think I'm doing it wrong NVM nvm help


do I have to move left and up move up now or??


That’s alright! When I say I want you to extend the line to the left, I mean I want you to continue the line, starting at (-1, 5), which I already graphed for you!


So you have to kind of think backwards, to make sure you continue following the pattern!


See what I added to the graph? I thought backwards to figure out how to continue the pattern of moving down one unit WHILE ALSO moving to the right one unit


How would you rate your confidence level at graphing equations? Would you like to go over the basics again?


maybe .. I think its because of the fact I don't really graph equations that much like by scratch and I usually just use desmos for equations and stuff haha so I kind of forgot all of this stuff imsorryyy

like I don't usually draw the lines and stuff, if I am given a graph its usually already numbered and laid out for me heh


Ohhh yeah, that makes sense! That’s totally ok! I’m asking you to graph by hand because you might have tests in the future where they don’t let you use calculators 😅


I'm trying to figure it out though, its just I'm really slow, not to bring myself down but I do think slowly compared to others and I need it breaken down a lot for me to fully understand siiiighhh


That’s ok, is the fact that the slope is negative throwing you off?


yeahhhhhhh that's mainly it


Ok! I’ll write out some positive slope equations, can you graph those for me? Negative and positive slope equations are very similar thankfully! :)


okay I'll try my best wait!


Alrighty, You’re doing great!!!


hmm is the second dot right


Looking good!


Yes!


The slope is 2, so you have a rise of 2 and a run of 1 unit


oh okay do I keep going or do I draw the line ..


Yes, I’d like it if you could make the line longer!


Keep following the pattern that the slope is telling you to move!


Third point looks great!!!


Yup!!


do you want me to add another one or is that good


You’ve been extending the line to the right, can you extend the line to the left now? :)


I'll tryyy, do I start from (0,2) ?


Yep!


Hopefully this explanation makes sense: you’ve been moving from left to right. But right now, I’m asking you to move right to left! So you have to think backwards:


Normally, you move up 2, and right 1!


But if you want to extend the line to the left, you’ll have to move down 2 and left 1!


Because later when you check if you graphed correctly, you can see if you continuously followed the pattern of moving 1 unit to the right and 2 units up!


(-1,0) is correct!! :)


OHHHH Mr. Lee I get it now wait


wait...... I'm moving from (-1,0) right? wait so wouldn't that be below the numbers am I doing this right


Yep! That’s okay because technically, lines go FOREVER and forever!


Have you heard of how there’s 4 quadrants on a graph?


You put the point in the correct spots!!! Nice job!!!


omg I've actually never heard that but thank you for the new information


I was mainly just overthinking the whole time but thank u !!!


You’ll learn more about the quadrants as you take more math classes! :)


Of course! Can you plot one more point, and then I’ll show you how y = -2x + 2 looks like?


okay!!


Very nice!!!


yay thank u!!


See how you’re following what the slope is telling you? (I’ll write in blue)


That’s how the slope changes a line!


ohhh I get it now thank you!!!


I’m glad! :)


So for y = -2x + 2, the y-intercept is the same (0,2)


oh so its -2 this time?


You’re basically using the same process as you did for y =2x + 2! :) yes, you’re moving DOWN 2 units while moving right 1 unit!


ohhhhhhh okay I get it !!


Yep! could you continue the y = -x + 4 graph? :)


you're taking about the graph at the top?


talking*


Yep, I want to make sure I do help you solve the problem you originally wanted help on haha


it does let you scroll up and down the whiteboard, it’s this skinny grey line at the right side of the screen!


oh okay I see !! where am I starting from?


I’m on iPad, so it’s also possible to scroll by just using two fingers and moving them up and down


I want you to continue the line starting at (-2,6)


I noticed that you’re great at figuring out a point’s coordinates! :)


omg thank you I don't know if I did that one right though


Fantastic!!!


oh YAY!


You did do it correct! :)


do I keep going?


Technically, you’ve graphed enough if we take a look at the domain restrictions now!


I’m gonna erase a bit so that there’s more space, one sec


okay !!!


Ok, so the restriction for the top part is -3 < x < 2!


So, should we use an open dot or a closed dot at x = -3?


open!!!!!


Yep! I’ll draw that on. How about x =2?


closed?


wait there's no inequality there I'm not sure


Oh, I see your confusion! One sec!


So - 3 < x < 2 is telling me that at x = -3 is an open dot, because I see that the inequality sign has no line! So what about at x = 2?


ahh wait open then?


Yep!!


(2,2) is a normally point on y = -x + 4. BUT because the domain (x-values) restrictions DON’T include 2 as being apart of the piecewise function, this problem is asking us to go to where x = 2 is and put an open dot!


If the restrictions say that only the x values between -3 and 2 are included, would (4,0) be included in this piecewise function? :)


I don't think it would because 4 is bigger than 2 ... and its not between -3 and 2 I think..


wait I might be wrong wait


Yay, you’re correct!!


oh YAY!!!


4 is indeed bigger than 2! Same with 3! So I’ll be erasing those parts of the line.


okok!!!


We’re now done with graphing the top equation! We successfully graphed y = -x +4, then we successfully followed the domain restrictions!


-3 < x < 2 is saying that the points with an x-coordinate between -3 and 2 ARE included in this piecewise function!


Now we’ll have to graph f(x) = 1/2x + 4. Can you try it? :)


do you want me to graph it on the same graph or?


Yep!


Also, you’re doing great! I love how you’re thinking through this problem :)


You can change the colors of your pen by clicking on the paint palette icon, that might visually help you graph!


When graphing, I would look at the slope! What is the slope of y = 1/2x + 4, and how much does it want you to move right and up?


is the slope 1/2?


Yes!! :) these equations are written in slope intercept form, y = mx + b, so you’re right!


and do I plot a point at (0,4) ?


Yes, because the y-intercept is (0,4) :)


I'm kinda overthinking I don't want to make a mistake SORR


Mr. Lee


That’s totally ok!!! Remember, mistakes are apart of learning :)


YAYYY!! Looking great!!


I keep going up? or do I go left now


Keep going up for two more points, and then go left please :)


Mr. Lee thank youuuuuu!!


No problem!! :)


is that good


Yep!


Please extend the line to the left now :)


do I start from (0,5) ?


Close! The y-intercept is (0,4), so start from (0,4)! :)


OH oh that's what I meant to say I mixed it up sorry heh


okay I'm gonna try now


No problem, it’s okay! :) thank you for being so hardworking!


Nice! Can you connect them now? :)


ok !!


im sorry the line looks so bad


It’s a perfectly great line! :)


Ok, let’s take a look at the domain restrictions!


So at x = 2, will we have a closed dot or an open dot?


open????


Yes! The domain restrictions for this second line is 2 < x <(with a line) 6, so it’s saying that the point at x = 2 IS NOT included in the piecewise function!


ohh okay!!


I will add a open circle on the point that’s on x = 2!


Now, what about the point that’s on x = 6? Would that be an open dot or a closed dot, when you look at the restrictions, 2 < x <(has line underneath) 6


closed!!!


Yes, because the point at x = 6 IS apart of the piecewise function! I’ll add a closed dot :)


(8,8) is a point on y = 1/2x + 4, but is that point included in the domain restriction of 2 < x <(with line underneath) 6?


no I don't think its included because doesnt x have to be between 2 and 6 , and 8 goes over 6 so it isn't In the domain?? and 8 is bigger than 6 so its not included in the restrict domain right???


ithink


Yes, that’s a fantastic explanation!!> amazing job Alex!!! :)


omg THANK YOU!!


Can I also erase the point at x = 7, based on those domain restrictions? :)


yes ! because 7 is bigger than 6 right?


Yep! I’ll erase it now!


How about the point at x = -1? Is that included in the piecewise function, when you look at the domain restrictions?


yes it is because -1 is between -3 and 2 I think.. wait are you talking about both of them or just "2<x< (closed) 6 " hel


One sec, I’ll clarify!


So the circled green domain restriction at the top ONLY applies to f(x) = -x + 4. The circled RED domain restriction ONLY applies to f(x) = 1/2x + 4.


So you have to make sure you look at them separately, if that makes sense :)


Right now, we’re only looking at y = 1/2x + 4! So we’re only looking at its domain restriction, not the top equation’s domain restriction!


oh okay then !! well since -1 Is less than 2 I don't think it fits in that domain.. ithink


Yes, you’re right!!! So I’ll erase away that part as well!


What about the point at x = 1?


nope because 1 is less than 2 right ..?


Yes, you’re right!!! So I’m gonna erase that part of the line


We’re done! We’ve successfully graphed the whole problem!! YAY!!!


OMG YAY!!!!


Now we have to find f(2)!


OH


But… let’s look at the domain restrictions.. 🤔🤔🤔


Is there a point at where x = 2, or are there only holes?


Also, Alex) is basically asking for the output for when you plug in x = 2 in one of those equations equations!


uhhh I think there's a point at x=2 ..


WAIT nvm maybe not I'm confused


The correct answer is no, there’s not a point at x = 2! Because the open dots indicate that there’s a hole in this piecewise function, there’s no point that exists at x = 2!


Another way to think about it is: look at the domain restrictions. The < sign DON’T include x = 2!


ohhhhhhh I see !!!!


So, f(2) =undefined! Because no points exist at x = 2!


If both domain restrictions don’t include x =2 to be apart of the piecewise function, then the point at x = 2 is undefined in this piecewise function!


Does this help? :)


oh yeah it does but it was a lot for me to process haha


Yeah, piecewise functions are a lot to get used to! 😅 they have a lot of parts to consider!


I’ll have to go soon, but if you need more help, you can always request a new session!


And if a tutor isn’t available, Khan Academy is a fantastic resource, I’m sure they have videos on piecewise functions! :)


You did REALLY amazing today, thanks for sticking with me and not giving up on the problem! Remember to be kind to yourself, you’re a great student! :)


okay thank you so much for helping me today !!!! I appreciate your help!!!!!! you're so nice!!!!


You’ll get better the more you practice these problems, I promise :)


Of course, have a great rest of your day Alex! :)


you too !!!!!!


Hi!


Hope you are doing well today


What do you need help with?


Hello! Im good tysm for asking, how are you?😁


Im great


can you see when I draw?


Great to hear! May you please help me with this question?


Yep


Ok!


So we are trying to find least to greatest


A strategy I use when calculating these problems is


converting everything to decimals


Do you want to try that?


I think there is a typo tho. Bc for 2 of the questions it says -2/5 and the other 2 questions it says 2/5


Yea I do!


Ok Leo great!


There is no typo


On problems like these, there will always be different sets of numbers


So ready to start


Oh ok yes


ok!


First, do you know how to turn fractions into decimals?


Do you want me to start off with the first problem and convert all the values into decimals?


I do


Ok great!


we an turn everything into decimals then


Ok


lets start with !.


A.


Ok!


-2/5 would be -0.4


Great!


what is -1/2


-1/2 would be 0.5


remeber there is a - behind 1/2


Sorry I meant -0.5


No problem!


Then 0 would js be 0


Is this correctlt organized from least to greatest?


One sec


ok


can you see my typing?


on the whiteboard


It’s not


oh..


can you see now?


Sorry I meant like it’s not correct


The question


I can see your screen tho


ok great!


you got that right


so it would not be A


lets check Leo


Yes


Ok


-1/2 is -0.5


yes


And 2/5 would be 0.4


So does b make sense?


Yes ma’am


I'm a boy


Hehe


Oh shoot sorry


No it's fine


and also I don't think i'm any older than u are


so anyway, is this answer right?


Oh


if it is, then you have you correct answer


I think it’s correct


Then you are right the answer is B


Is there anythign else you need help with


Nope Mr. David! Have a good evening.


You too!


bye!


Hi Liam!


Hireonal weathercraft took off on a.


Don't mind that. That was for my TV.


I'm sorry I can't hear your audio


we still don't have.


Do you speak English and Spanish.


yes!


I can hear you now


I'm gonna address the problem on the board.


sounds great!


Can you see the problem on the board?


Can you try to talk to me through your mic?


sure! give me a second please


Oh, OK


I


Yes


everything.


OK. So


the side here is the side, right?


Say that again. Sorry.


The side over here is the side we're trying to bend. Is that right?


OK. The side over here.


I repeat that again, sorry. Hang on, give me a second.


say that again


Hi, the side over here is the side that we're trying to get right?


that that's, that's exceeding the limit. It's about to see the limit, uh.


Can you hear me


do you think you can hear me?


yes but my volume is down low


would you still like me to explain it to you verbally or through chat?


verbally


ok


do you mind responding in the chat if you are comfortable then?


sure


great thanks!


you can still talk to me through your mic


yup!


So, um,


have you learned about trigonometric functions like sine, cosine, tangent.


yes we just started that today


OK, that's great. So,


um


why don't we start out by writing like the formulas for sine and cosine, and tangent. So,


you speak in English, right?


yeah


Just making sure.


So, do you know what the sign of X is equal to, like in terms of ratio.


Not exactly, yeah, I kind of forgot that one.


That's OK. Um, so the side effects is opposite


over hypotenuse.


and


the cosine of X is equal to adjacent.


over hypotenuse


and then the tangent of X.


is opposite over a she.


Is the drawing in the way?


No, that's OK


I can move this one


OK, so, so far, do you understand?


Yeah, so I, when I saw for X, I had to find a way to decimal I have to run into the nearest tent.


OK


Um, so for this one, we're gonna try and use the Engle 37. So everything


is oriented to the angle of 37, which means that this is the opposite side.


Oh my God


Just give me a second


And do you know what this side of X is?


partners.


That's right.


This is the hypotenuse


I actually got that right. Wow.


Yeah


you're doing great. And then,


that there might be something.


you know what this side of 48 is.


on that exactly.


So we have the hypotenuse and the opposite, and if you look in like the sign, cosine and tangent form.


you said?


Yeah, that's, that's right. It's the adjacent side.


So, then we need to use one of the sign, cosine, or tangent functions to find the hypotenuse, which is X Can you see any function like sign, cosine, or tangent that involves the hypotenuse and the adjacent.


Um, let's see.


Not exactly.


OK, so here the sign involves the opposite side, sign of X involves the opposite side and the hypotenuse. The cosine of effect involves the adjacent side and the hypotenuse, and the tangent of involves the opposite side and the adjacent. Can you maybe see which one of them has hypotenuse and adjacent.


X and 48.


Yeah, so we're trying to use X and 48, which is basically the high pot news and the adjacent. And do you see that the cosine function has the adjacent and the hypotenuse.


Yeah, can I ask you a question?


Of course


Are you a student or are you a teacher?


Um, I'm a student currently.


OK, let's get back on track.


So


you see that cosine has both adjacent and hypotenuse, right?


Yeah.


So we want to use cosine. So


here we're gonna write cosine of something equals. Can you tell me what the adjacent side is here in the triangle.


That's right. 48 divided by, can you tell me the hypotenuse?


X.


Yeah, so the co-sign of something is 48 divided by x. Now we said we were going to use a specific angle to start with to orient ourselves. And what do you think that angle is?


What grade are you in? I'm just curious.


I'm sorry, I can't answer those questions right now, but


like


I


can you maybe tell me the angle?


Yes, sorry. Um, what's the question again?


So we


need to find the angle that goes in the cosine function, and


we


said that we were gonna look at one angle and orient ourselves around that angle? Do you remember what that angle is?


Close, it's very close. Yeah, 90 is one of the known angles. What other angle


do we know currently


That is a side that we know. What about this angle of 37.


Yep.


right


We're choosing 37 because we


so they they look or they, they slow the.


time.


we said that this side is the opposite and this side is the hypotenuse and this side is the adjacent based on the location of the angle that we oriented ourselves from, OK?


this airplane.


So we have an equation that we can solve, cosine of 37 equals 48 over x.


and


can you maybe move X so that it's alone, like apart from the 48, and then


yeah, let's, let's do it. Let's start from there. Can you isolate the eggs?


How do I do that exactly?


OK, let's walk through this together.


So the cosine of 37.


equals 40


over


X


and we want to get X all alone by its side.


So


I


first of all, we don't like that X is in the denominator of the fraction, right?


Wait a second fuel on board. I'm gonna land with a higher la


picker.


OK, are you still with me


program, but this guy's, um, so can't you put divide under divided by one under the coach 37.


Yeah


We can absolutely do that. And then the you know cross multiplication?


yes


OK, so then it would be X.


times cosine of 37 equals 48. Is this right?


Let me write this down for a second. Give me a second.


Say that again, I couldn't hear you.


So if we do cross multiplication, like


Can I write this down first?


of course


okay


OK, so X times cosine of 37 equals 48, and we want to get X all by itself, that it's, that it turns out that x equals something. And so


we don't really want this co-sign of 37 multiplied, 2 x. Do you know maybe how to get rid of the cosine of 37, that's multiplied to X.


Divide.


That's absolutely right. So we divide by cosine of 37, which gives us X equals 48 divided by cosine of 37, and then


we just put that into the calculator and round it.


You're smart.


I mean, thank you for following along. I feel like you did a great job today.


I'Northwood High School and geometry is not my favorite thing.


I


Can you say that one more


time?


my TV at my TV. If I'm above 400 ft


but not quite up to the


No, that's from my TV, don't mind that.


That's OK. Um, so yeah, this question is basically solved. Um, how do you feel about it?


What is X equal?


X equals to 48 divided by the cosine of 37, which you can put it into a calculator and it'll basically give you a number.


Give me one second, let me do that.


I got 1.29.


OK, I didn't check


Thank you


So for my calculator, I got something like 60.


or exactly this number.


and it might be maybe a syntax error or um it was typed in a little funkily or something like that.


I'm using the demo's graphing calculator.


Oh,


are you on a computer right now?


Yeah, I'm always going to device.


Um, if you like just open a new tab and then type in 48 divided by cosine of 37. It's gonna be um


that's gonna be the right answer because Desmos might be using radiance instead of degrees.


Did it give you something like 60.45mething.


I'm back. Sorry, I lost you


Oh, so if you plug it into like a regular calculator and not into decimels, you should get something like 60.45mething.


Yeah, I did. I did it.


OK, that's great. And that's the answer.


Do you feel good about this problem now?


Yeah, I think I can handle the rest of it.


That's great. Well, it was so nice meeting you. Great job today.


Thanks.


Bye-bye


OK


Hi, how are you


I've got my mic on if you want to put yours on.


Hello


Hi, how are you tonight


Um, pretty good. How are you?


OK. All right


So I see you have your problem up here.


Yes


So


So I, I tried to solve it because I know that usually when it's a fraction, you equal the denominator equal to 0, but it's wrong.


OK. Well, that's one of the conditions and you're exactly right. It can't be -6 for sure, because


division by zero is not in


the domain, not allowed. Um, the other part which


uh you need to be aware of is you can't have any square roots that are less than 0 in the real


mind.


anything


where X is


less than -6.


is not gonna be


oh, not gonna work either.


So, -6 won't work, and 6, anything lower than -6.


um,


will not work. So


um, we're just supposed to find the domain.


and let's see what your choices are.


So, I think all real numbers


such that X is greater than -6 will work for that.


Do you see what I was saying about


square roots of negative numbers.


Yeah, yeah, cuz you can't have negatives in square roots.


but you were right to say that 6 can't be it. -6 would be OK for the numerator.


because that would be 0 in the square root of 0 is 0, so that's no problem, but


um


when you have -6


Mhm.


anything below -6 would be the square root of a negative number. So that's why


all real numbers X such that X is greater than 6, looks like the answer.


Oh, OK. OK. That makes sense.


But you were right about -6 because that's included in that answer.


Mm


Yes, I was right.


Um


now the other one I was having trouble with, it's, it's super complicated. There's, but it's good it's, it's so many words like.


but


Does it have something you can put out there on the whiteboard? OK.


Yes, yes, it's


OK


There we


that there's so many steps.


All right, I see it. Thank you


Mhm


I'm gonna read it out loud an open box of maximum volume is to be made from a square piece of material, 24 centimeters on a side, OK.


By cutting equal squares from the corners and turning up the sides as shown below.


So cutting equals squares. OK.


The table shows the volume and cubic centimeters of the box for various heights


X and centimeters. Use the table to estimate the maximum volume.


So we really don't have to do anything but look


um


they're giving us a table, which had already has


the answer in it.


cause you the table to estimate the maximum volume.


So it looks like the maximum volume


it's somewhere around, well, the biggest one is 1,024.


00 yeah.


so


um, one answer would be just to pick


without really doing any math about it.


Um,


let me think about this a minute


So it's just saying to use the table, um.


so my initial thought is just to put in 1,024


What do you think


Mm, but don't I have to estimate it?


Well, yeah, it says use the table to estimate, so it's not asking us to actually do


the calculation. We could go ahead and


come up with


So, um


down below, they're asking us to write the function.


Mm


like down on C, which is what we would get into if we were to do that. But I think that might be jumping the gun a little bit.


Um


which we will do


If V is a function of X right the function and v is no. OK.


do you lose points if you, if you get things wrong here, do you


I have 5 chances.


per each, each problem or just each each sub


for like this whole this whole problem.


OK. Well, I think we should follow the directions and so it says just use the table.


um use the table to estimate the maximum volume, we would say 1,024 based on nothing but just the table. So let's go ahead and give it a try.


OK.


And I check it?


because it'll check it for like, uh,


all the


like the whole question


Yeah


Oh, for the whole question, you have to go through everything. Oh, I see. OK, well, let's put that one as our answer for A.


OK


OK, it says plot the points


actually be in from the table and party. Does the relation defined by the order prepares represent a function of X.


So


we could


we can make a plot of it, but it looks like


every time X goes up,


Y goes up to a certain point and then it starts going down again.


Mm.


when we have a function, we're, we're saying that


um


Do the X doesn't repeat


yeah, it sort of doesn't repeat, um, although one of them comes close.


to 1, uh, 980.


but


let's, let's just do, let's just do, can I write on here a little bit? Well.


Yeah


Mhm


you OK? So, maybe I'll go down below so I don't mess up the whole problem.


OK


So if I was just doing a crude


on the uh


just do a crude plot them.


I'm not gonna be very exact in this.


So,


at


I'd say that's one


uh, it's 400


84 It's like


somewhere around here


And then let's just go a couple over and that's 2.


and it's 800


so


maybe that's something like right around here


and we go to 3


you know, we're 960


Oh maybe that's


kind of there


And then we go to 4


This is really just kind of


you know, not being real exact, there's score.


and 5


were down back down 900.


let's say somewhere.


like there


and then 6


we're 700


so maybe


something like that


So


if we make it a continuous function, it will be something.


like that


Now


OK


um, if we do something called the vertical line test.


Have you ever heard of that


Mhm


What if we, if we draw a line through the plot,


we don't


we don't get two values for an X. An X kind of


goes to only one value. So that's a good


sign that it is a function


my


feeling is, uh, does the relation defined by the ordered pairs represent a function of x, and I think we can say yes, because


the vertical line test says, do, if I had like a volume 3 and I had 2 values for it.


then I


then I wouldn't, would fail that test


but I only have one value for every X. So I think we can say based on that that it is a function. What do you think?


Yeah. And uh since we plotted it, then and it's a parabola, then


it's


we know that it's our real number, so it's not see.


and does not be


Um


for the, for the last one.


oh


Yeah


uh, well, let's wait until we get down to that one. But yeah.


Right


I think it is gonna be a parabola, you're right.


And when a parabola is the function.


um,


at least the one that we're showing here, and again, I'm kind of


messed up when I drew it there, but it's something like that.


Mhm


pointing down. So we know that it has a maximum


You're right, it is a parabola.


but we know it has a maximum and therefore, that's a good sign that


um


it's a function


OK


So then we go to Part C, right?


Yes


Now, we look at that, if we look at the little diagram on the right. I like that one, where it shows the box


We know, do you, do you know how to find the volume of the box just in general?


Um, is it like the side times the side?


Well, that's an area, so.


Not in that area, right? But


but then if you multiply it again by a third side, you would have volume, right?


volume


So, if I, if I am a


let's just say, uh, a sugar cube, right? Something very small. Let's say it's


uh 0.5 inch on the side for sugar cube. I don't know if that's right, but we would say that the volume of the sugar cube would be 0 times 5 times a half, right?


Oh, OK.


The length times the width times the height. So actually this may turn out to be a cubic function rather than a parabola.


It may turn out to be


it may be to the third power, OK?


OK


Because we're gonna be multiplying the length times the width times the height, and we can see


on that one side


of the box that they labeled there


We say that the area


of the face would be X times 24 minus 2 x, right?


Mhm


Because that would be the length times the width.


and the hype happens to also


um


well, let's think about that


So


if we had the base, the area of the base would be just 24 minus 2 x times 24 minus 2 X, OK?


Yes


It's kind of a square


but then we would need to multiply by X


to get the, the volume.


So again, let me go back up.


just to make sure you, you get what I'm talking about.


Mm


They, they have the 24 minus 2x on this side. It's also on this side, 24 minus 2 X, OK?


And then the height of the box is only X, right?


we could write


do you want to write down for meion right below my graph.


See if you can rank the B, what the volume is.


Uh


Oh, OK, I'll do it. But.


You know, way down there, yeah, right down there. That's good


So say B B of X or B equals


what are you gonna put in there?


Yup, so what


What times the line comes right? Oh, but like, so 2 eggs, or was it 2 1st?


It was 24 minus 2 X.


Mm.


OK, put that in parentheses first.


OK


Good.


And we're gonna, we're gonna multiply that again, aren't we? Because


we want the kind of the area of the base.


of the thing, so


good


Right. So I foiled that?


Well, not yet. Let's just leave it like this for a while for now.


parentheses, and what are we gonna put for our third


X.


So, without having to do anything simplifying or anything, we think that's the volume


um


they didn't say to simplify it.


but one way we could do a little simplification would be to simply say it's X times 24 minus


2 2X.


right?


Yes


And if we wanted to, we could


I, I like leaving it that way if they accept that as an answer.


because I don't


I don't know that we


really need to go through all that when this is pretty


straightforward


Mhm


So depending on what answer they require


we, we could foil it out and combine like terms and


so forth, but


that, that is one answer that works for me. I don't know if it'll work for the answer, but


Mm


sometimes computer-based


answers are kind of hard to fathom, but that should be an acceptable answer.


OK


So


So, go ahead


um, so then, since the


the function is squared, then it's not a cubic.


Oh wait.


It is cubic because it is cubic because what if you were to, like you said, if you were to foil that out.


you would get squared terms.


and X terms


but then you'd have to multiply that by X, wouldn't you?


Yeah


So you would end up with some cubic


Let's go, we'll go ahead and boil it out just to, I'll go ahead and do that for you here.


So, here I go


uh 24 times 24, what is that? Let's see.


Um


576, I think.


it's, yeah, 576.


we would have


minus


48 X


minus another 48 X if we were to foil that, right?


We, if we were to go first,


first,


outer, inner


we would get -48 x minus


48 X, and then finally we would have a plus


4 x 2


right?


Yes.


OK, so we keep simplifying


uh let's combine, let's keep it this way.


minus 96 X.


+ 4 4X^2.


and now we can


multiply through by X.


and if we wanted to rewrite it nice,


we would say


OX cubed


minus 96 x 2.


+ 576 X.


Um.


So I, I guess


you know, we can put that answer into, put that one in and if you like that one better.


OK


If those one marks it wrong, then I, I'll put that on.


And so, so then it is cubic, then it's a squealy line on a graph.


Yeah, I have


um, but it does have a, it does have a quad uh


quadratic term or an X squared term in there as well, so.


um,


anyhow, we can actually


just for the heck of it, I'm gonna


put in 4 in my calculator and let it calculate


what that is. The 4


X cubed


minus 96


+ 576


Oh, darn it I have to start over, sorry.


I blew my


let me try that again


you minus 96.


576X X.


and I'm gonna try that for a value of 4 just for the heck of it, or


stored in X.


and


So when you


Hiller, when you uh put in 4 into that function, guess what? We get 1,024, which gives us a warm fuzzy feeling of


Yeah.


that we're on the right track


Mhm.


OK, so


moving down to the


deter determine the domain of the function.


um


what do you think


Mm. Well, since we got a maximum, and I, like, not from graphin, but we have a maximum, so I know it's a parabola, so it's, so it's not


see


mm


it's


So remember, remember we're dealing in the physical world here. So,


oh right


right


go back to, go back to the original one we did where we did 24 minus 2 X.


times 24 minus 2 X.


times X


Mhm


Can X be 0


No


Yeah


Because we have a volume of 0, wouldn't we? Because 0 times anything, so that we know that it can't be negative, and we know that it


can't be zero. And then if you actually looked at one of our 24 minus 2 X's in our first, the first time we wrote down the function, if we had X


greater than 12


we would have 0 again.


Do you see that


Yeah, yeah, cause


Going back, going back to our original one where we wrote it as 24 minus 2 x times 24 minus 2x, if we had X, anything bigger than, or 12 or greater.


it would be


Mhm.


0 or negative even, which would make no sense.


So


what it kind of says is,


um


we're looking at that


middle answer being pretty attractive, right?


Because


Yeah. No one that I kept on trying to take out, but that makes sense cause it's not, can't be a negative and it can't be 12 or or greater 12.


yeah, 1212 plugged into our original formula gives us a volume of 0 again.


Mm, OK


because if you put it in, I like going back to that original


um, the one that you wrote at the, you know, right below the graph.


Uh-huh


That, that is more clear than the one where we expanded it out.


because


you can easily see


Yeah


that if X is 0, the volume is 0, because 0 times anything is 0. And if you plugged in 12


the volume would be 0 as well, because


24 minus 2 times 12 is 0.


right?


Right. Mhm.


So you can't have 12 and you can't have anything greater than 12 or the volume starts


you start getting negative numbers for the length and the width which you're just kind of crazy.


not real


OK


Oh, so especially not that hard. It's just there's so much, there's so much to it. It makes it confusing.


Well, I know, I'm, I understand and you know


you.


this is how you learn, but


sometimes, um, yeah, it takes a while for things to sink in. Do you have, does it sort of make sense to you?


Mhm.


Yes, this makes sense now.


OK. So, um


yeah, I mean


it's easy for me because I've, I've done it before, but, you know,


students have to learn the hard way.


Yeah


Yeah, I mean, once you, once you teach it so many times, you, you just.


Right, cause, you know, it sometimes you may be in awe of a teacher who doesn't, but remember they've been doing it for years and they've been working the same problems for a year or so.


anyhow.


Um


um


the, the squared one was right.


Not the forward one


not the what


Not the foiled one, like the 5 cents.


Yeah, I like this, I like the squared one better.


Mhm.


or even the first one where you wrote it right out because it very clear what what's going on there.


Yeah. Um


and so


this next one is just


a word problem.


OK.


which are the worst kinds.


I right


I don't know how to take, oh, OK.


That it's, oh wait, I think I'll put it too or another.


I'm considered. That's good


OK


I'm gonna read it out loud


Great


A company produces a product for which the variable cost is $12.80 1 cents per unit, and the fixed costs are $92,000. The product sells for $17.66.


So let X be the number of units produced and sold.


The total cost for a business


is the sum of the variable in the fixed cost.


like the total cost


C as a function of the number of units produced.


So


just imagine you're, you have a factory


and


you wanna figure out how much it costs for any number of units that you produce.


So we're gonna write a function C


of X or C


equals something


And what do you think would be


a good way to represent that. If you let X be the number of units produced.


She is the total cost and number of units produced?


So go ahead, let me um


let me see where I'm at


12.81 X.


Why don't you write it right below


the problem there and we'll


so


+ 92,000.


good


OK


good.


OK, great. So that's what your C function is. If I produce one unit


it's gonna cost me $12.81 plus $92,000.


If I produce two


units, it'll be


you know, 2 times


1281. If I produce 100 units, it's going to be $1,281 plus $92,000 and so forth. So you got see that's


um


C is


uh the, what you've got there is the answer for box one there.


Oh, OK


That's very good


Wow


And then


Now, what about, what about for the revenue, how would you find out how much money you got coming in?


If you sell that stuff


how, what would be the function that would tell us how much money you would get for for the X number of units.


Oh


You got it. That's all there is to it. So that's your revenue, that's gonna be your revenue.


So that'll be R equals 1766.


X.


I think I just overcomplicate this.


No, you're, you're doing great


OK


All right


Hm


Now we're ready to finish it off because the profit is simply


the, the


um


second one subtracted from, I'm sorry, the first one subtracted from the second one. So what you're gonna put in there


we can simplify it probably.


so


we're gonna have


the


equals


X


minus


the cost of the set which produce FX.


So.


now we can subtract


since we got common X's there, right?


Mhm


Uh let me see, 17.


66 minus


and we get


$4 and


85, I think it is.


Yes. Yeah, that's what my company says.


OK, good


And we're gonna subtract 92,000.


because that we have to kind of distribute that negative sign, right?


Yes


So that's, that's what our profit function will, will be.


So it's gonna take you, it's gonna take us quite a few units before we break even, even, right?


We lose money until there's a certain point where it'll be 0 and then it'll start making money after we cover all our fixed costs.


where they are.


All right


But they don't ask us to solve it in this one anyhow, maybe in the next.


Yeah


Awesome


I think now that we did that one, I think this one's easier.


Wait, let me clear this.


Take your time


OK


OK, the inventor of a new game believes that the variable cost for producing the game is 75 cents per unit.


and the fixed costs are $6500.


The inventor sells each game for $1.99.


Let X be the number of games sold, the total cost for a business is the sum of the variable cost and the fixed cost.


right, the total cost function as the number of games. Why don't you write it in below.


based on your experience with the first one.


Awesome, you got it


That's great


So


um


we got the


variable cost and the fixed costs, so that's the total cost.


So that answers the first one, A.


OK.


Now it says write the average cost per unit, see bar.


as C over X


So I wanted to see if you can


just write what C bar equals.


right below what you did


So is it


that one


over


1.99? Or is it is sex?


We just, just, just X, right?


Mm


So, all they're saying is if we wanted to figure out the how much the average cost is, we would just.


take the number of units and


whatever it is, X is, let's say it is 200, we would plug a 100 in and that this will give us the average cost that it takes to produce those 100 units.


which is


So that's it.


you can, you can also divide that X.


You can leave it this way or you can say it's


uh, I'm just writing it below.


Or you can simplify it as


you can do that as well


Either way


OK.


Um, really what that's saying is that


the cost, the average cost to produce it is 75 cents for each one, but then we've got to take our overhead


like that could be renting a building to do it or whatever.


we would


that, that's some kind of fixed cost. We would just


take that fixed cost of


6500, not 65,000. Whoops.


Um.


we would just take the cost of the building rent, let's just say, and divide it by the number of units we produced that month.


or something like that. But


so either one will give us the same answer.


Mm, how do, or why you cancel out the eggs on the top.


OK, because when I divide, so when I am a fraction like this with a 2 numerators and one denominator, I'm allowed to break it


I'm allowed to break it into 22 parts.


So I'm allowed to


take the first part


and then I'm allowed to take the second part.


Now notice that when I have 0.75 x divided by X, it cancels the X out, OK?


Hm, oh, OK. Yeah, I understand.


So, sometimes students get confused, but if you have one denominator


and many numerators, you're allowed to break it up into individual fractions, OK?


Mhm


But if you only have one numerator and lots of denominators, you're not allowed to do that.


OK


So that's sort of just kind of


OK.


something they don't really tell you too much about in school, but you learn the hard way. When you have a fraction


if, as long as you, you're allowed to break it up.


based on the denominator.


but if it was flipped, you couldn't take X over 0.75.


So, anyhow, I'm just


hopefully that answers your question.


Yeah, that was pretty, that was in California.


Good. Yeah, you did great.


This one


is


I don't get it just because when I see it, it's, it's giving um


something from geometry, the, the treadmill thing.


Great.


But I know that that's not what you're supposed to do.


OK, let's read it. I, I like reading a lot to kind of give my brain a chance to get around it.


Mhm


So balloon carrying a transmitter ascends


vertically from a 0.3000 ft above.


the receiving station


OK.


Draw a diagram that gives a visual representation of the problem. Let H represent the height of the balloon and let D represent the distance between the balloon and the receiving station. So they drew that


they drew that for you pretty much, um.


uh


so I'm not understanding why they're asking you, oh,


let's see.


Yeah, cause the, it's missing a value.


Oh, that's a, OK, so you're gonna stick the value in, and you got A done, right?


So,


we're gonna put 3000 there and you'll be done with part A.


Oh


I think that's all that they're, oh, wait a minute, I'm sorry, that may not be true.


feet


is the height


of dry day let H represent and let D represent the distance between the balloon and the receiving station.


OK, I get that


So they already drew it for you. So what are they asking us here?


So do you have to answer something on this?


Oh.


Yeah, this one.


OK.


Oh, I guess they're asking for that part. OK, so,


all right, so


let's see here. Um


we know


that


the bottom


part of the triangle, the base of the triangle 3000, right?


Because the


The


we know that is it the height that's 3000 or?


It just says that it ascends vertically.


OK. So I'm gonna write on


I'm if I


so


that base 3000, right?


Because


they're saying that


from 0. 3000 from the station. So


DNH we don't know what they are.


but we do know that the base of the triangle 3000.


So I think what you're all you're doing is you're putting 3000 in for


that little box


OK


Do you see what I mean


Yes


Yeah, cause we've just been putting


You're just inpointing how long that is.


right


OK. Hopefully, that's right. Uh.


Now, let's see what they're saying here. Right at the height of the balloon as a function of D


OK.


So do you remember Pythagoras's theorem?


A


B


Did, did you take any geometry?


Yes. Is it like


3 + B squared + C.


So it's close, uh, C squared equals A squared + B squared.


Oh OK


So we're saying that, that if you have a right triangle,


then your hypotenuse squared equals the sum of the side squared. So you probably saw it as a A squared equals B squared + C squared.


So in this example


Tala,


Mhm


I'm gonna write down below


below all the problem. I'm just gonna write


what we would have for Pythagorean's theorem, if we had this


so we would say the hypotenuse square, which is D,


equals the sum of the side squared.


Well, we know one side 3000.


and we know the other side is H.


So, whoop


the other side is 8. So what we're doing, all we're doing is writing Pythagorean's theorem for the problem because it's a right triangle


and if you just remember the theorem is the hypotenuse square equals the sum of the side squared.


That's Pythagoras' theorem


Mm


So


can you solve for H for me, and that problem.


8 square?


Yeah, so, so I wrote an equation down there, Pythagoras's theorem based on our problem. Can you solve for H? What would you do?


If you don't know it's OK


I don't know


OK, so all I would do is I would


let me write it as


+ 3000 square.


equals D squared


right?


Mhm


So I'm actually just gonna subtract 3000 squared from both sides.


minus


3000 square.


So I end up with 8 squad.


equals H squared.


minus 3000 square.


That's


make sure you understand everything that we've done so far.


So getting back to the problem, do you, do you see a


do you remember that Pythagoras theorem is, you know, basically what I'm saying is the hypotenuse of the right triangle is the sum of this


hypotenus squared equals the sum of the side squared, OK?


So then all I did was I plugged in the names of the variables that we have and what we know.


That's where this first equation came from D squared equals 3000 2 + H squared.


Yes.


I just turned it around and wrote it the other way in the second step.


And then I decided to try to solve for H.


because they're asking about age, so I subtracted 3000 square from both sides.


I get 82 equals D squared minus 3,0002, OK?


How would I get H if I have H squared.


Uh, square root


You got it. So if I take the square root of both sides.


Oops.


So I get


D squared minus 3000.


square


So, we have done it, whether that's, I don't know how you enter that in your


if they give you a square root and squares and


um


Yeah, there's um


an option to do that


OK, so hopefully


let's just do some geometry when you took that.


Um,


hopefully, that


Great


makes all sense, makes sense to you.


square roots, Student.


OK, what do we know about square roots, Student.


what


Um, to change it.


to change it from uh, to take away the square root. I can just multiply by 2.


Yeah, we can, uh, we can square, but, but


when we were discussing domains before Leo at the beginning. Remember, we can never have a square root of a negative number, right?


Oh yes


So think about this. If B is anything


if D is anything


um


greater than


3000 or greater.


we got a problem


We cannot


be


because if it is


I'm sorry, less than 3000.


I got that backwards


cannot be less than


Less than or equal 3000.


But it can be greater than or equal to 3000, right?


Right


Because if it's less than or equal to 3000, and we start getting into


um,


heights of zero and negative heights or the square root of negative heights, and that's not going to work.


So


that's why


um


D


D is our variable that will


we're using as our independent variable, that's what they call it. The one that's very and it's, it's gotta be greater than our equal 3000 squared, OK?


Square


Square.


Oh, right, square, yes.


because, because we're subtracting 3000 square. We could convert that to some big giant number.


but there's no need to, we, we're not asked to do that here. So that's why


it has to be greater than or equal 3000 squared.


I'm sorry


I gotta be careful what I'm saying.


Um.


actually, D must be greater than or equal to 3000.


3000 squared. That was tricky.


So the D itself has to 3000 or greater, OK?


Not the square


Well


Wait, why not square and


uh, it doesn't really talk about these squared, it only talks about D


So


if I try to 3000 in for D, I get 0, OK?


Mhm


If I try to plug anything greater than the


less less <COURSE>, I get the square root of a negative number., I get the square root of a negative number.


So d, not d squad.


D must D must <COURSE> or greater. or greater.


But let it sink in a little bit.


Just look at our formula right here.


about for H


Don't think of D^2, think of Don't think of <US_DRIVER_LICENSE>, think of <NRP>


OK


DD has to be, if D 3000, that would 3000 square minus 3000 squared, wouldn't it?


Yeah


Which would be 0, and that would mean that the height


of the balloon is zero. That's not what


And in fact, anything less than


3000 gives us nonsensical like the square root of negative numbers for our height. We can't do that.


So this 1 may take you some time just to be thinking about. I'm not, I, I hope


you're getting it, but


sometimes it answers a little tricky.


As long as you follow


how we came up with our formula for H.


It's just strictly Pythagorean's theorem.


And then there


I detect the nebula that you're a little confused by it, but


ask any questions if you don't feel bad to ask


Mhm


questions. There's


So


um


it was, it can't be greater 3000.


it can be greater 3000.


00, OK, I can, I can.


D D can be greater than or equal to 30 not equal 3000. It's got to be greater is 3000.


because that's what, if you 3000 in, you would get 0, wouldn't you?


Yeah


Because 3000 squared would be what the first term would be under the square root sign. It would be 3000 square minus 3000 square, which is 0.


Well, that, that's not


legit. And if anything less 3000, it would be.


also not good


It would be at the squire of a negative number. We can't have a height that's a square root of a negative number.


Oh, yeah, that's true


Mhm.


So that's why the last answer is the one that makes sense for the domain.


The domain means the set of all possible valid values for input, OK?


What are my input valid values.


0, 3000


or greater basically is allowed. 3000 would give us a height of zero. OK, it could be sitting on the ground.


Um, I guess that's possible, but anything greater than 3000 is fine.


for D


But if it's


less than 3000, then


it's not valid. So that's why the last one


is the one that makes sense


Yeah, yeah, I get it cause anything less time would be like 0 or negative.


Negative, right


Good


OK. I understand. OK.


Mm, so I run out of paper. Let me find some paper.


Mhm


OK. OK on paper.


OK


So I got, I got to show my work scratch work.


Sure, yeah, take your time.


OK. Well, I, I got it.


So I'll take this off.


Um


these, this is the 2nd to last one, but both of them are almost the same. So I, I was never taught this because we didn't fix the notes, we ran out of time.


All right. So let's go through it. It's not that hard.


So whenever you have a function of any variable,


when we say FX equals something, that means we're gonna do something to X.


think of it as a machine. We're gonna plug in X into the formula.


So, they give, they tell us what the formula is with the function there. That's the first equation they gave us, F X equals X squared minus X + 7.


Now what they want you to do is instead of X to plug in


7 + H


then they want you to do it again and just plug in 7.


And then they want you to divide by H.


So,


again, don't


be afraid of


this is just notation. So


think of FOX as a machine, and we're gonna plug in


Efax


So if I wrote F of


all I'm going to do is


plug 7+ H in for X


Mm


So that'll be 7


plus H.


minus


7 + H.


+ 2


So again


all we're doing


excuse me, I got a problem with my screen here. There we go.


Um.


all we're doing when we say F of something.


is we're just taking the rule for F of X and we're instead of X, we're going to put in whatever they tell us.


OK


So now, we have something we can, we can figure out what F of 7 + H is because we're


good at foiling, right


Mm.


So what would I get if I foil 7 + H times, you know, 7 + H.


It's 82 + 14H +49.


Good.


So I'm gonna write that here


Good. And then I'm gonna subtract, I got to distribute that negative sign over, right?


Mhm.


So I'm gonna subtract 7 and subtract 8, and then


add 2


Yeah


So


we're just following


algebra


for what F of 7 + 8 means.


We're gonna combine like terms.


to how many H's do I have?


Um.


Right?


+ 13.


And we got H squared, let's see, 49 minus 7 is


42 + 2 would be 44, right?


Yes


All right, so


what we've done so far is we're, we're computing this thing


but we only have the first piece of it. So we gotta keep going with the next piece.


Oh.


So when I say F of H or H of 7, what would I do? Here is


I'm gonna write H of 7


What would age of 7 be?


age of 7


I'm sorry, it's F of 7, not H.


OK


That's no wonder you're confused.


Um.


F of 7. So if I'm using


if I have to plug 7 in for X, what am I gonna get?


07 square minus 7 + 2.


Excellent


And that is equal to 49


minus 5, right?


Mine is what?


-7 + 2, so it would be minus 5, I believe.


Oh yeah


Mine is


is 44.


All right.


So what we've done is we've gotten two pieces of


the thing and we're


um, we're


we're ready to do what they asked us to do. They want us to take F of 7 + H.


Well, we already know what that is


So I'm just gonna put


on this side answer


now because we're gonna do what they said


We're gonna take F of 7 + H, which is 82.


+ 13H.


+ 44


We're gonna subtract 44 from it.


and then we're gonna divide by H.


and we're going to simplify that a little bit.


and we would get 82.


plus 13.8


H.


over 8


Can you sit reflections there?


Sorry


Can you can you separate the fractions there?


We can't because we have one denominator, excellent.


So what would I get for the final answer?


Mm, it's weird.


Um


A


actually, I don't know.


H + 13, right?


So I take the first part.


82 over h is just H, right?


Right


If I am 82 divided by H, it's just H.


Mm


because the, one of the H is canceled, and then if I have 13 H divided by H, the H is canceled and I'm just


left with 13


OK


Yes


So


our answer is H + 13. We did the problem.


Wow


and whenever you take calculus, you'll see why you do these problems, but that's for the reason why you do it is not important now, but this is practice


for future courses


That's all, so remember, when you have a function


and they ask you to do something to that function like they did in this example. You just plug in, instead of X, you just use whatever they tell you to plug in. That's all we're doing.


Mm, OK


and we're just simplifying


OK, awesome


Good


OK. I think I can do this next one on my own then.


All righty


Well, you're doing really well. You're patiently hung in there, so.


you have a good evening


Yes, likewise, thank you so much for your help.


You're welcome. Bye-bye.


Michael


hello


how are you


part


Are you ready for the first question?


good.  Yes!


Because there's a total of 16 questions.


OK, I will go ahead and give you the first question then.


OK


ok


OK, the first part, it says solve for X Heals.


I hope you don't


That's the first question.


i messed up on the 7 its supost to be -7


ok great.


I just want to make sure we are writing it correctly.


Is the question "solve for x when 2x-7 =-14 11"?


oops sorry I meant


"solve for x when 2x-7 =-14 11"


I don't know why that -14 keeps appearing let me write it on the board


Is the question the way I show it now on the board?


Is that what we need to work on?


YES


great


(sorry about those typos!)


Do you have any thoughts on maybe how we could start?


hey your fine everyone makes mastakes


very true


That's how it's actually supposed to work. I forgot the other eggs. My bad.


I've got the other x


I forgot my bad. My bad.


Men, I hate all these typepers.


you good


i think we want to start by getting that -7 out of there


OK That's fine.


How about we add 7 to both sides


OK, you're the teacher.


Can you take the next step?


we add


right


yes


ok


one sec


So to the right of the equal sign it is 11 plus 7


lets start there


perfect


now to the left of the equal sign


14 correct


that 2x stays just like it is


what about minus 7 plus 7 what does that equal?


-14 maybe


If I gave you 7 dollars and then said hey can you give me seven dollars what are you left with?


you git it


*got


ohhhhh ok


2x plus zero becomes what?


right


yes


Ok now we want to be left with just the x


That 2 needs to get out of there


yes sir


any thoughts on how we make it go away?


divide to both sides.


Maybe question mark.


I meant add 2 to both sides.


I think you had it the first time


divide by 2 on both sides


that will get that 2 out of there


OK


what is next on both sides?


a 2 ??


lets start on the right side of the equal sign


18 divided by 9 is what?


oops I meant 18 divided by 2


you got it


we just have to get that 2x divided by 2 now


1 ??


The x can stay all alone.  What is 2 divided by 2?


what is 1 times x ?


(what is 1 times anything?)


be right back im going to get me somethin to drink


ok im. b ack


we were at 1 times x


ok


what is that?


ummm


1 times anything is always what you start with


so 1 times x is


1 right


x


ohhhh


oh ok


you ready for the next one


Yes - I won't be able to do all 16 but lets try another


ok thats fine


OK so if you look above at the last problem there is a step we took that we can take in this problem