### UNR Math Center Review Session Math 126 Bagheri Exam 1

of course okay so we should be recording um yeah and this is the review for i was told this is the review for um bagheri’s class bagheri’s’s class um but it seems like some of you might be here from a different class as well um but we should most of this material should overlap so i think we should be we should be good to go um did any of you guys get a review material that was similar to this um like stuff that has domain range and actually i got like three three different pages here so this one is like the second page no okay okay let’s see did you guys get any review material at all no okay well in that case in that case i think i’ll just try to go through this review material you guys just said drop in if we want review you didn’t tell us anymore okay it sounds reasonable i guess um okay well this is the review that i was given so i think i might just go through this um hopefully it has overlapping things um this this first page or this first like part here um it might be a little a little like easier like more like for warm-up stuff so we could skip that if we wanted to or we could we could try stuff on this third page which seems like it might be more helpful like even and odd functions and locating maximum minima increasing decreasing graph sketching at the bottom yeah so that that all stuff should overlap does that does that sound familiar because all that stuff sounds familiar like we’ll have like domain and range in here as well probably does anybody have any input on that is that what your class went over as well yeah okay yeah so this should be fine yeah so let’s let’s let’s take a look here um oh yeah um so just one more little prerequisite thing out of this way my name is gary so if at any point during this review you have any questions please either turn on your mic and just say it and i’ll try to answer it to the best of my ability you could also message me in the chat i have it pulled up here but sometimes i don’t notice those things happen so frequently so i’ll try my best to answer all your questions to the best of my ability but again you know let’s just try to work through this together and see how we do okay and please please ask all your questions because that’s what i’m here for i’m here to answer all of your questions okay with that out of the way let’s go ahead and start here um how about this actually since you guys don’t have this review material um is there any like specific part of like this first exam that you guys might be having trouble with like any of the things like i went over like graph sketching maximum minimum domain range um even and odd functions all right is there any part that you guys would want to like start with just to make sure that we cover it because this review is actually kind of lengthy so we’re not going to be able to get through all of it but if there’s anything that you guys are having trouble with um i’d like to start on that first so okay do you guys have any anywhere specific you want to start if not i could just try to go through this review like top to bottom and see how we do or bottom to top if any of you guys have any preferences section 2.4 do you know what section 2.4 was on

because these aren’t actually labeled by section unfortunately i’m wondering i think that should be this page piecewise functions is is that section 2.4 yes okay piecewise functions yeah okay that’s reasonable to go on like graphing them um finding their domain and range let’s see if i can find a question that has a piecewise graph no no finding range i don’t understand the different types of range okay hmm okay having trouble with the isolated at the non-isolated values hmm i’m not finding any piecewise things on here it’s kind of close oh here we go perfect this this is definitely piecewise right i wonder if there’s if there’s another one yeah there’s a couple here okay perfect yeah so let’s start with here um so we have this graph f and it’s defined uh right they tell us it’s defined as follows um so it’s four plus four x if x is less than zero and it’s x squared if x is greater than or equal to zero um so right we wanna do all of these things we want to do all these things and below below this part is is going to be the answer so maybe i’ll try to cover that up for now until we can get an answer ourselves okay so we want to find the domain of the function um well if you um how would i find the domain of this function does anybody have an idea remember this this function is piecewise the infinity position yeah yeah right because um we don’t really have any domain restrictions in either of these two functions right um i’d have to be looking for like a square root or a like a denominator right for both these functions but both of them i don’t have either right i don’t have a square root or a denominator so and and it’s defined from like like to the left of zero and to the right of zero right so um the domain in this case would be in fact um how can i put how can i write this maybe i’ll put like equals negative infinity to positive infinity yeah there you go there’s that first one and now we want to locate the intercepts how can i locate the intercepts it’s all x and y yeah right um you’re basically solving for x and y when um when like the other one is zero right so if i’m looking for if i’m looking for like let’s say the x-intercept right in order to do that i would set y equal to zero right and similarly if i want to find the y-intercept i would set x equal to zero right so that would that would allow me to find the intercepts um in this case i feel like it might be easier to find the y intercepts so let’s do that one let’s do that one first right if we set x equal to zero

right it’s going to give us something that looks like this so i have um i kind of have two choices for what um this coordinate could be um which one am i going to choose out of these two numbers here does anyone know four that’s this one yeah because this one equates to four right so this one would be four this one would be like zero um we we want to choose the second one because um that’s right because i i already have i already know that x equals zero right so this bottom one let me try like red here this bottom one um this is true when x is greater than or equal to zero right so um because x is equal to zero i can’t use i can’t use this one because this graph this graph is only defined um when x is less than zero not equal to right so the graph isn’t even defined for this top one right so i want to use this bottom one so i would say that the the intercept the y-intercept right would be something like zero comma zero right because this is this is the intercept when i plug in x equals zero right and this is the output right here this is like f of zero right and that just happens to be zero because because i had to choose the bottom one that makes sense is that okay with everybody does anyone have any questions on that okay okay so there’s there’s the y intercept right um the nice the interesting thing about this is that right this is when the x and y coordinate is zero so this is actually also this is not only is this a y intercept but it’s also an x-intercept right just just because they’re both zero right it’s at the origin um so and this is actually the only point where you can have something be both a y-intercept and an x-intercept so we should expect zero zero to turn up over here as well okay so now let’s try let’s try doing the left side the x-intercept um right to do that we want to set y equal to zero um right so that that’s really setting like like um f of x equal to zero right because this is also equivalent to like f of x so what we’ll have is like zero right equals and we’ll have this like piecewise defined sort of thing yeah right and um i would say a good way about go a good way going about this is to kind of split them up into their own like individual equations so instead of writing something like this maybe you want to write maybe you want to write it like uh like this and then same thing for this one down here is x squared equals zero is that okay and then you can solve both of these guys for x right so if i solve this first one um right if i solve this first one i would get like something like four x equals negative four and then x equals negative one and then the bottom one um if i square root both sides i’ll get like x equals

zero yeah um i would say one important thing to check is you want to make sure that this value um correspond or here let’s do this and write again you want to make sure that that the answer here right because because this answer came from this equation which comes from over here right so you want to make sure that that that this x equals negative one um fits with what is what is stated right here right is is if x is negative one is that less than zero right yeah right because so because of that i could definitely say that my one of my y-intercepts is going to be negative 1 0 right and then same thing on this side right this was my answer for this equation right so i just want to i want to make sure that it it it fits with what is stated here right is is 0 greater than or equal to 0 well yeah it is right so then we can say that another one of my x-intercepts is zero zero right and there’s that thing that i i said right we’re going to get um the origin we’re gonna get twice because it’s both an x-intercept and a y-intercept and that’s what we should expect once we see this once we should ex we should expect to see it again over here okay does that make sense does anyone have any questions on on this part all right so it looks like we have zero zero i’m just going to write that once and negative one zero so yeah they say you say the the intercepts are negative one zero and zero zero perfect all right hopefully that was okay um yeah let’s move on oh and now we want to graph the function yeah so this is probably a an area where we might be having some trouble so let’s graph this um there’s a couple different ways we could approach this um what do you guys like to do what does what does your professor like to do is anybody does anybody remember what your professor likes to do like how do they like to start this right i mean we know we know a couple points that are supposed to be on here right we know the the intercepts so let’s let’s let’s start by plotting those um i think oh yeah let’s do this in red so i know that there’s one point at the origin zero zero um and i know that there’s one point at negative one zero i’m not sure how big this is going to be so try and just make it um reasonable that’s at negative 1 right so that means that that point is at like negative 1 zero this point is zero zero so i know that those two points are on there what other points are how can i get like more points what am i gonna have to do where’s a good starting point for this guy anyone have any ideas we want to graph this right okay um yeah so there’s there’s a couple different ways we could start this um i think it’s reasonable to just like pick you know one of these to start with and just start by graphing it you know

um that’s that’s what i would that’s what i would do um let’s try this first one if we graph this first one right that’s that’s the equation of a line let me write it like this right this is like like mx plus b right so i know that my um i know that my y-intercept is at four right so if i go up 1 2 3 4. i know i’ll have a point there right and let’s see let’s not let’s not worry about this condition quite yet let’s just try to graph this like like if if we only had this what would this graph look like let’s let’s try and uh figure that out all right so um i start out i start out the y-intercept of four and then i use the slope from there right so i can go up four more units so i can go up one two three four more units and then over one unit right so i would have another point like up here i know that’s kind of out of the graph unfortunately maybe i can extend this right um and then i could also use the the for the slope going downwards right so i could go down one two three four units and then left one and then i have another point right here right and then i’d have another point right here which is why we’re getting this point right and then i and then i would connect those two or three that’s not good let’s do something like that oh i have a good idea there we go that looks good okay um right so this would be the equation that we get just from that first one just from this first guy right here now let’s think about the condition that x has to be less than zero right and and what i’m going to be doing is i’m just going to be erasing everything that’s not fitting that condition right so i know that x has to be less than zero so what that tells me is that um i i need to be on like the left side of of you know like the y-axis i need to be on the left of this so pretty much everything on this side is going to work everything that’s not here is not going to work so i need to erase everything else right so i’m going to erase this part of the line i’m going to erase that dot there um and if we if we kind of examine this closely i’m also going to erase this point right here i’m going to instead of making this a closed point i’m going to make this a whole because if we look at this condition x is not allowed to be equal to zero right x is not allowed to be equal to zero so what i need to do is take this erase the point and make it like a like a hole there is that okay right so the the graph doesn’t exist at that point we haven’t we have an open like singularity right there or something like that right so there there um there’s that top equation filled in right so um the way that i i approached it is i just started by graphing this and then i i erased everything that was not um fitting this condition basically so that’s that’s definitely one way you could do this you could also just if you don’t like doing like the erasing method you could just start by graphing this whole thing you know taking both into account um if you’re if you’re skilled at that then yeah you can do that too um both those are definitely great ways of doing it does anybody have any questions for this first part of the graph

any questions yeah graphing the piecewise things is is definitely challenging but if you just take it one and a one at a time you should be okay okay so now with that one out of the way let’s do let’s do the next one all right so now i’m looking at this one and let’s let’s not worry about this second one right let’s let’s not worry about that quite yet so let’s just graph it and let me graph this one in um graph it in green i think all right let’s do orange so i’m just focusing on the x squared part so what’s the graph of y equals x squared going to look like does anybody remember what y equals x squared looks like parabola yeah perfect right so it’s going to be a parabola and it doesn’t really have anything else happening to it so it’s just like the the standard parabola um there’s no nice way to draw curves on here unfortunately but i can try my best here so it’s going to start here i know it’s going to intersect at 1 1 then it’s going to intersect at like 2 4 right so it’s going to kind of make this kind of arc this kind of arc shape and it’s going to do the same thing on this side too somewhere around here well i kind of messed up at the end there but yeah i mean i think you get the idea right so that’s that’s this graph that’s me just graphing this without thinking about this condition over here right so now that i have that graph well let’s now let’s erase everything that’s not within our condition so now if i take into account um what this condition says it tells me x has to be um greater than or equal to zero right so what that tells me is i can accept everything on the right side of this right so this what i’ve circled in green here is what we’re going to keep right everything else that’s in orange i want to erase all right so yeah let’s do that let’s erase that erase this um and let’s take a closer look at it am i allowed to be equal to zero right so this says greater than or equal to so yes i can be equal to zero and still get a point on here so in this case um this point down here i’m not going to erase it i’m going to keep it right it will be a filled in point because i because with this condition i am allowed to be equal to zero right that’s why i would keep it okay does that make sense does anybody have any questions on that second part of it yeah okay cool um yeah maybe i’ll just make highlight this in red one more time right so this is going to be our graph so this whole this whole thing together is going to be the graph and this would be like our answer for part c um that’s that that’s one way that you can graph it at least that that’s the way that i like just because um it kind of makes sense like like okay i just focus on this and then erase everything that doesn’t fit this right i can just focus on this and then erase everything that doesn’t fit that and that that should get you going um for most of like all the piecewise graphs okay and it’s really nice once you have the graph to um answer the rest of the questions especially if they have to do with like range and domain or honestly if you have the graph you can answer pretty much all the questions so yeah there’s that um let’s let’s um let’s take a look at the answer here oh it’s kind of small here um but it looks like they chose part number b here or number b here um and right that that does look like our graph right um it is five up here um so yeah they have like a parabola on

this side and then a straight line going downward on this side where there’s a hole on the y-axis right so that’s that is in fact the same graph okay perfect so now we want to based on the graph we want to find the range right um i will i would think that you guys um have dealt with domain range already um if you recall domain has to do with like all the x values that you could put into a graph and get a answer out of it right so that means that range would correspond to um all of the y values right so now um instead of thinking about the domain which is if you think about it it’s kind of going this way right for that’s for domain domain right um but range is is just the opposite right don’t range i’m thinking about what are my values going up and down right that’s for range so with that with that and the graph what would you guys say the ranges of this graph does anyone have a guess no is range the difficult one or something um we could so um right so this these graphs like extend infinitely right so i mean this graph keeps going oh i mean that kind of looks weird this graph keeps going on like it goes it goes up forever it goes upward forever and it goes to the right forever right so because it goes to the right forever that’s why um that’s why we have our the upper bound on our on our domain right because because it goes to the right forever that’s where we’re getting this upper bound on our domain right um so and it’s it’s the same reason for our range right this graph goes up forever right so on our range the upper bound that we should expect is going to be infinity right because it goes up forever and then similarly for the for the other side um this graph goes right it just goes this way forever on the left which means uh it’s it’s kind of like the same thing it goes left forever and it goes down forever right um right and it the because it goes left forever because it goes left forever that’s why we have negative infinity as our lower bound on our domain right and it’s going to be the same reason because it goes down forever that’s why we’re going to have negative infinity as the lower bound on our domain on our range sorry is that okay so that’s how we get the range we just look at the the y directions of the graph is that okay with everybody so this would be our range okay so we have a question here i understand that part of range but how do we tell if there’s an isolated value okay yeah yeah yeah good question yeah that was the other part of your question um let me give you an example let me i think that that might be the best way to look at it um like an isolated value um i think i know what you’re talking about and let me let me see if we can think of something um okay so this question is done let’s let’s make sure that that’s our range that is indeed our range let’s check the answer real quick um d um okay so d uh they say that the range does not have any isolated values it is just this negative infinity to positive infinity yeah perfect so that that is correct um so yeah so now let’s let’s take a let’s think about like a different example here let me let

me give you guys a different example so let’s say that instead that instead of that parabola being centered at like the origin there um let me move that parabola up let me move it up to like um like around here right if i move it up to here um we’re gonna have a point here probably and probably one like up here i know it’s kind of hard to see oh i got it i got it there we go so let’s say that that the parabola was like i’m just going to shift it up a couple of units uh let’s see this 5 6 7 8 that’s at eight um and let’s let’s make an isolated value and actually before we do that let’s analyze the range of this of this graph right um because this this would be slightly different right so again with the range i do want to think about the up and down direction the up and down direction um so the range right um the thing that’s different about the range is kind of this um inter intermediate area right the the this kind of like the the points between five and eight in the y direction that’s like the interesting part because um we know that after after the graph passes eight um it’s going to go up forever so the upper bound on that would be infinity right after the graph passes like this actually this isn’t five is it this should be four do that in red so this this one right here is four after the graph goes below four right it’s going to go down forever so the lower bound would be negative infinity right but in between these two the graph doesn’t actually exist so we want to make sure that we don’t include that in our range that’s when we’re going to have the case where we need to break up the range so it’s it’s not going to be this anymore it’s not going to be that anymore it would be it would be something like this we want to let’s let’s just think about this part for now let’s just think about the lower part um so the lower bound on that would be negative infinity right because it goes down forever right and now um basically when i’m looking at the upward direction i’m thinking about where does it stop it stops at four right stops at four right so i would say it goes all the way up to 4 and does anybody know if i would use a hard bracket or a soft bracket one of these or one of these soft right soft because we have a hole at 4 yeah so i’m going to use a soft bracket at 4 and and that that takes care this this right here takes care of this portion of the graph um now we want to take care of you know um this portion of the graph right so now i’m thinking to myself okay what’s the lower bound on this portion of the graph well the lower bound there what what’s the lowest it can be on that portion of the graph well the lowest is eight right so i could do that that like union sort of thing eight and then what’s the highest it can go well it goes up forever so that would be infinity does that make sense so this would be our range if we move the graph up like that does everybody agree with that do we have a question here would 8 be in a bracket would eight b in a bracket yes okay so yeah eight would be you would want to have eight in a bracket because um it’s a filled in point so yeah my mistake you want to have eight as a hard bracket there yeah perfect good catch and then we have another question so that’s the absolute value okay so the isolated value it happens when you have something like this already set up and then i draw in like another point like like let’s say let’s see five six let’s see this one’s at six let’s say i made another point um i don’t wanna do it at the origin let’s say another

made another point like right here let’s say that my piecewise graph was something like this i think that’s what what’s meant by the isolated value right this one is just a point right it’s just a point so we would want to include that in our range right so if i add this point this range is no longer completely satisfying this entire graph right i need to include something in this range so i would have to um i’m not sure how to denote it um it might be like we union it with like six or something like that might be like um let’s move this might be like like this i’m not 100 um but i think that’s what what um you’re talking about when you say um isolated value right because this is like an isolated point right we want to make sure to include that in our range but nothing else we don’t want to include any other point is that cleared up yeah okay perfect yeah and actually if i do if i put the point there this is no longer a function um because it doesn’t pass the vertical line test if you remember right i have to draw like i have to be able to draw like a vertical line and only intersect one point so instead of putting the point there um let’s let’s move the parabola let’s move it to the right a little bit let’s still keep it at eight um but let’s move it to the right i don’t know like two uh let’s move it like three units so i’ll have a point here probably one around here probably one up here or something and then let’s redraw the parabola right so just so that i can move this point like like i don’t know over here right this is like two or something two and six right so this would still be a function this whole thing would still be a function because it still passes the vertical line test but yeah this is kind of clearly an isolated value now though and actually you might even see like the point like on the uh on the y-axis that would still be an isolated value still be an isolated value it would still be some kind of case like this yeah the range would actually stay the same right um moving the moving this part of the graph to the right is going to change my domain though right so you want to make sure that that you also have your domain correct right um let’s actually analyze that real quick if this is our graph what would our domain be so anyone i mean we’d approach it in the same way right we’d start with you know this portion of the graph we’d say we’re looking left and right now right so what’s the what’s the least it can be what’s the lowest what’s the lower bound right well it goes left forever so the lower bound would be um negative infinity right right and then when does it stop when it goes when it comes back to the right when does when does this part of the graph stop it stops at i’m not looking at four anymore i’m looking at the x direction so it stops um when it touches the y axis right and that’s at zero oh yeah someone’s got an answer infinity is zero union eight infinity yeah exactly um yeah i think that’s perfect zero right it stops at zero and then we union it with the other part of the graph h to infinity right because this part of the graph starts at eight or actually it starts at three in this case yeah yeah you you caught it you caught it it starts at three because i’m looking i’m looking at the x direction now right and it goes it goes right forever yeah so the only difference with those is you’re looking at the just different directions and um and one more thing is one last thing is um since i have a point here um i i actually would want to include zero in my um domain

so instead of being a soft bracket i think i’m gonna want a hard bracket here even though they they um the point kind of jumps it still exists as our domain it’s still like if i plugged in zero into this function right this graph i would still get a y value out of it right the only time where where you might um not do that is if i were to move that point right let’s say i move that point kind of drew it in a weird spot let’s say i move that point um that’s not what you want let’s say i move this point um back over here then it has that isolated value sort of thing again right so we would make sure to put this as a soft bracket again because we don’t have any any value on the y axis and then we want to make sure to like um union the domain with like like 2 or something like that i think that might be how you denote it um i’m wondering if this professor has any of those types of questions with isolated values let’s see um that one doesn’t look like it has that might be it for um the piecewise on this review but i i think i think we covered most of most of it honestly so i think it’s i think it’s okay does that make sense with everybody does anybody have any questions on you know any other questions on piecewise graphs i know we kind of went over a lot there so um yeah what are your questions okay okay so if there’s no questions um i think we’re good to move on um does anybody have like a specific like uh another like section or topic you want to go over right now um i know piecewise piecewise functions that’s a good one um are there any other ones that you guys are having trouble with okay so let’s clear that um if we don’t have any other ones we could let’s see um i think i’d like one of these questions down here um like the you have you guys dealt with um shifting compressing stretching reflecting oh do we have one can we go over something like translation yeah yeah yeah that’s exactly what i was about to uh go over yeah because this this one is another area of like error for students with the like the transformations right so yeah let’s try this one uh actually this one might be a little bit easy just because it only has one transformation i wonder if there’s a harder one we could try um i mean we could make up a harder one probably yeah i think we’ll make one up okay so for this one um okay let’s just let’s just go over the the general idea of 46 here um right so they give you the graph of y equals square root x right so that’s this graph here if you can’t see it maybe i’ll draw just a little bit larger it looks something like this that’s square root x right and the only thing that that has changed in this they’re asking us to graph um negative square root x right does anybody remember what the negative does to our graph negative out in front reflects right it reflects over the someone has it here over the y um let’s see so when i when i think about it um what i think about is like the original graph was was this right the original graph was this and

what i’ve done in order to get this negative here is i’ve i’ve like multiplied this whole thing through by like a negative right i’ve multiplied it by by like negative one right um so that would be like like like negative here and a negative there right um and what i’m looking at is kind of this part this tells me that all of my y values have become negative does that make sense every if if a coordinate if a point was like up here right it is now the negative of it it is now down here does that make sense if a point if a point is right here that that point is now right here the point is right here right negative y value it’s going to be down here the point is right here negative if i do a negative y value it’s going to be down here right so i think of it as i’ve made all of my y values negative so i think i’m thinking this will reflect us across the x-axis instead does that make sense um yeah so so that that’s how i think about it um right and to distinguish that from uh reflecting over the y-axis if if we were reflecting over the y-axis it would look something like this something like that right where um and in order to kind of see that i i do the same thing i do the same thing i i start with the original graph and i say how could i get a negative inside this square root um well the only way that you could do that is if you made the actual input negative right if i make if i make this negative then that makes that that puts me that puts a negative right there right and that’s the same thing right so what i’ve done with this with this one is i’ve taken every x value and i’ve made it negative right so the negative on the inside means that i’m making my x values negative right which means i reflect across the y axis right all of my x values reflect across the y axis so that might be a good thing to write down like negative like negative let’s see um something that like like making x negative reflects across the y or something and then making y negative reflects across the x axis you know something like that depends on it depends on where this negative is basically yeah so that’s that’s how i that’s how i think about it um yeah so this graph this graph after we reflect it right so we’re reflecting across the x-axis because all of my y-values became negative so now it would look something like this kind of missed i kind of missed this point here but i think you get the idea right so that that’s how we’re getting a b here yeah and then we and then uh your professor delves into domain domain and range again um and here are the answers if you want to uh just sort of verify with your own logic okay so there’s that one um let’s try a let’s try a harder one because that one just had to deal with one one thing one little transformation um let’s do a harder one oh actually does anybody have any questions up to now so far okay if you do if you do have a question just please type it in the chat or anything yeah just let me know okay um so yeah let’s try a harder one um i’ll probably just make one up i can’t guarantee that it will be like super convenient to do um i think one of the common ones is probably like a like a squared graph so let’s try one of those um let’s try like y equals um let’s go a half

x plus five squared let’s try something like this this one should should get us going pretty nicely oh and actually it might be better to do something like this and just enough room okay yeah so let’s let’s graph this um okay so right to graph this we want to start with like the parent function um just does anybody know what the parent function of this would be x squared yeah perfect so it’s y equals x squared right and that graph is just going to kind of be standard parabola kind of looks like this maybe okay so there’s y equals x and from here what do we want to apply first do we want to do the vertical shift do we want to do the horizontal shift or do we want to do the this is the vertical like stretch or compression do we do which one do we do first does anybody know any guesses no horizontal yeah i’m thinking horizontal too um i usually try to think about it as like um what what’s happening to x like first like if i were to put in a value here like let’s say i put in i don’t know like two in for x right right what would happen first to x you kind of do like like order of operations in a sense right well the first thing that happens to x is it moves to the it it moves right because of this 5 right and that 5 creates a horizontal shift so let’s let’s let’s draw that graph and which way am i going to move right because because this is positive 5 i’m going to move to the right right left right i move to the left i move to the left we move left we do the opposite of whatever of the sign here so we go one two three four five so there’s the there’s the point and then parabola the parabola is not looking that good but five units something like that right perfect okay then what are we going to do all right so we still have uh we’ve taken care of we’ve taken care of the shift of the horizontal shift what do we do next we do the vertical shift or do we do the the stretch or the the scale what comes next shift up three units um i think i think we would want to do the the vertical stretch or or the the scale right this this uh this one half oops that’s what i meant to do i think we’d want to do this like one half because if you think about it in terms of like the order of operations um that’s what’s going to happen to you know the value next right of course after it’s after it’s squared occurs of course right so again

let’s go back to that uh um that thing where like if we plug in like x equals 2 into this right if i plug in x equals 2 i’m going to add 5 right and then i’m going to square it and then right after i i do all of that i’m going to then multiply it by this half right does that make sense so it it kind of follows the order of operations i i think there’s a few cases where it’s a little bit different but most cases you can you can use like just the order of operations on what what is actually happening to this value x so right after i after i do the parentheses then i do the exponents then i do like this multiplication right and that would be the vertical um and this one is a compression right this is a compression instead of a stretch so um that one’s going to be hard to draw uh no this is y equals half we’re still moving over one two three four five um every unit gets a half so um i kind of drew it a little a little poorly on this on this graph let me try and redraw it because honestly that one looks more like a like a stretch i mean that’s you know decent i guess and then after we shrink it by a half should be it should like kind of like open up more in a sense something like that i know i know those honestly look very similar but um right i am shrinking it by halfway i’m at least trying to um if you want to make sure that you’re doing this right um i would say um plot a few more points rather than just the origin um it would be good to plot like um i would go for like three at least three so i know there’s going to be a point there i know there’s going to be a point there and i know there’s going to be like a point oh that doesn’t look so good yeah i drew this very poorly um something like that sorry these graphs are aren’t so good um i’m gonna have something here and there right and then once i have it by a half um i’m going to have a point that’s kind of right if this is like one i’m gonna have a point that’s like kind of halfway in between so i’m gonna have a point like here and then a point like over here right so all of my y values get like scaled down by a half right so um if you remember on the parabola there’s also a point at like um there’s also a point at (2,4) right two four so um once i scale this down it would land somewhere at like two two that makes sense because i’m i’m reducing it by a half um so i think we’d have a point like like like over here as well something like that yeah okay does that make sense um i think usually usually the general idea is pretty okay i mean usually um graphing it isn’t isn’t such a such a big deal but hopefully that makes sense and let’s go ahead and do this last part i might need to move the chat over here um this last one right the last thing that we can do is the vertical shift x plus 5 squared plus 3 yeah let’s not do that this is y equals half x plus 5 squared plus 3

okay um and that’s just right that’s a vertical shift and it’s in the same direction as the sign so it’s going to be upward by three so we go over one two three four five we go up one two three so we’re definitely going to end up over here um but it’s going to be um a vertical like shrink by a half something like that probably that’s going to be like our parabola what it ends up being and it’s always it’s always nice to check your work too just to make sure that that you’re correct um you probably won’t be able to do exactly what i’m going to do on your exam um but if it’s like for for um test purposes or like like review purposes it’s always a good a good strategy and i’m just going to go to desmos to check it let me clear and clear the drawings what did i have i have y equals one half oh what happened yeah there we go y equals one half x plus five squared plus three let’s bring it back yeah so that graph looks real similar well i mean it’s in the right location it’s probably not like lined up correctly but um those are supposed to be the same graphs does that make sense with everybody is that okay does anyone have any questions i’m like this transformation stuff okay if you are having trouble with uh transformations i’d say it is a good idea to go into desmos and um just play around with it right just just start playing with the numbers and see what happens to your graph you know maybe make this like a five or make this like a a minus minus like like 10 or something oh i noticed that okay it’s just going up and down okay so there’s what that one does what about what’s this one dude right okay this is just moving it like to the left to the right and stuff like that um and some of the the funner ones to play with is like the the stretches right because if i make this like two you can kind of see like like oh it’s kind of getting like like thinner in a sense right if i make this like like really big it gets really thin right if i make it really small it gets really large or something like that right even larger it almost approaches a line at that point right um some other things some other interesting ones that would that you could do are instead of putting the square out there you could put the squared in here and see what that does right see what kind of graph we’re going to end up with now maybe this isn’t what you expected and maybe you could try to explain why right so all these are great great strategies for just um experimenting with it you know testing it out stuff like that okay does that make sense does anyone have any questions on uh graphing with transformations okay so we got about 15 minutes left and i mean we could definitely probably answer like one maybe two to three more questions um if they’re short um so do you guys have any problems that you want to see done any topics sections that you want to see done probably definitely definitely get through at least one more um we’ve done we’ve done the uh the graph sketching we’ve done piecewise we’ve done a lot a lot of things to do with graphs um i wonder if there’s anything else do you guys remember like like even in odd sort of things which i’m just gonna graph okay that one’s not

i know i saw some even and odd questions on here match the graph with the functions oh yeah and um we are going to be posting this to youtube um i think we’ll probably post it today um if we get around to it um so if you you do want to go back and take a look at this review you can like pause it anywhere and then you know try these questions for yourselves i think usually the answers are probably up here as well so they’re like um for for this one for this specific review like they’re like like starred so if you look for the start answer some of them are hard to see like this one’s kind of dim so you could pause it anywhere during this review and see if like just just work through it and see if you’re correct what other things are there square functions oh these are like some some parent functions that are useful to know right cube root identity um cube function yeah all of these are good good things to uh remember so you can start with the transformations um that one’s an interesting problem a secant line and a slope you guys have any um any preference on which one we do what’s the last one we want to do average rate of change it’s kind of like slope so maybe you guys know that minimum maxima increasing decreasing and of course the even odds sort of thing i wonder if there’s anything else more even odd we could we could analyze the graphs more more graphs if you wanted to we kind of went over intercepts we kind of went over domain and range um intervals where it’s increasing or decreasing um i think that might be worth it to go over that one’s usually a tricky one um because the end points most mostly right the endpoints is always like a question like is it increasing um you know on these endpoints so um yeah let’s try this one um i’m gonna skip the intercepts in the domain and range and i’m just gonna go straight to uh increasing decreasing so for us we have a graph that’s almost like a sine wave right so it goes like this kind of and it also goes like this oh man i’m always part of the bottom one let’s try that kind of um and then this is pi halves two uh that’s zero this is pi negative 2 this is pi all right sorry negative pi this is pi all right that should be enough that should be all we need okay um do you guys know like the interval in which this would be like considered increasing do you guys have an idea right so the the word um when you have this type of problem you always want to think about um you want to start by reading the graph from like left to right and what i mean by reading the graph i mean like you kind of just follow the graph from left to right right so it’s kind of going down here right and then then it starts going up at this point

and it starts going down again right and we’re thinking about when we when we ask about increasing decreasing we ask about um what’s happening to y essentially right so as i’m going left to right what’s happening to y right here well it’s it’s becoming like negative more negative it’s going down right and then once it gets to here and it starts going like this well the y values are getting more positive right so it’s going up right and then lastly over here it’s going down again so it’s going down right so over here we’d say that this is like decreasing and then over here we’d say this is increasing and then again over here we’d say this is decreasing um right um but we want to try to be i mean a little bit more rigorous with this so let’s try to be a like a little bit more exact um if we’re giving like if we’re trying to give like an interval we want to give the interval in terms of like x right so i would say like okay so this graph is increasing from from this point right to this point and basically i i’m just going to give them the x coordinates of those points right so i would say okay so this graph is increasing from um nega sorry yeah this should be a negative pi house sorry that’s my bad yeah that should be negative negative pi halves to yeah that’s that’s the x coordinate yeah yeah to positive pi halves right i’m looking at the x coordinate right and the good question is a good question that some students ask is whether or not these are hard or soft brackets are these hard brackets or are they soft brackets let me draw them like this are they soft or are they hard brackets which one and it’s a it’s a good question it is a good question um and i think the the way you want to think about it is like it’s like at these points what is happening to y at this specific point um well it at this point it’s like the on the left side it’s like increasing right but on the right side it’s like decreasing right so you can’t you can’t have a point that’s both increasing or decreasing right you can’t have a both sorry you can’t have a point that’s both increasing and decreasing right because that that wouldn’t really make sense it can’t both be going up and be going down at the same time so um at these points where they kind of like transition we we don’t call it increasing or decreasing we just call it i mean you could call it like a transition um you you could call it you know something else but you just don’t want to call it like specifically increasing or decreasing so we we leave them as soft brackets we don’t we don’t put hard brackets there right so we we don’t do this no hard brackets right and with that i think i think the rest of it is pretty um pretty self-explanatory there right so if i wanted to find the decreasing points well it’s decreasing into two places one over there and one over here so i’d have to make sure to include both right so from negative pi to pi negative pi over two right i’m just looking at the x coordinate and then i reunion that with positive pi over 2 to pi and that should be the intervals in which it’s increasing and decreasing right and constant constant would just be like a graph that has like if i had a graph that like i don’t know it did like something like this and then maybe once it got up here it went like this this part of the graph right that part of the graph would be constant okay let’s just check that that’s part c sure

oh hmm this is interesting i was always told different so on this review guide they’re saying that actually we would include those intervals but see what i’m what i’m what i’m wondering is we’re we’re putting hard brackets around them but we’re putting hard brackets around them for both right i have a hard bracket around pi over two here and i have a hard bracket around pi over two here i mean i guess you could say that i okay i guess you could say that um because at these points remember i told you like it’s it’s kind of increasing and decreasing um so i guess yeah i mean it kind of depends on how you want to explain it i always thought that it was kind of opposite so i guess i guess you would think about it like at this point it’s increasing and it’s decreasing so therefore you would put hard brackets around both both cases interesting i didn’t know that yeah cool yes we all learned something huh um yeah because yeah there’s no constant it’s increasing decreasing it’s a little strange but um seems like that’s the answer for this one i would look i i would make sure to just do what whatever your professor taught you um if if your professor taught you to like put soft brackets around those then do that way because that’s what that’s what your professor would be looking for so yeah um with that i think i think i’m gonna call it into this review um it is getting about 1 30 so um yeah um i hope this was helpful oh um do you have any does anybody have any uh like last minute questions here either on this problem or anything else i could answer real quick okay if there’s none then i’m gonna go ahead and um uh yeah i’m gonna go ahead and clear this um probably stop sharing um i would just say yeah yeah no no problem guys thank you for coming um i just say you know good luck on your exam um you know make sure that you sleep before it um so you so that you’re not sleeping during it you know um get a good breakfast because it’s always it’s always it’s it’s never good to like be hungry during your exam you know you don’t want to be like starving during your exam i know it’s going to be kind of maybe strange during this semester about it um but you know good luck and with everything hopefully hopefully you guys got this you know all right thank you for coming i’m gonna go ahead and stop the recording

## Recent Comments