okay everyone well hey welcome we're going to continue in 104 or in module 1 and today we're doing 104 radical functions so this should be fun fun times right all right so it's important to know how to let's call it translate um from a radical to an exponent so if you'll take a look at this this is also in your lessons on the first page of 104 that just lets you know what goes where all right so when you have an exponent remember you can also show that as a radical and this is how you would do it like if you had um you know let's do a simple example and i did not get my writing center up here it is all right let's take a look at it all right let's say you had x to the one-half anyone think they can write that in a radical raise your hand if you think you can do that for me all right maya go ahead and write that up there for me here's your annotate skills thank you that's a real easy one all right matthew since you've got your hand up i'm gonna let you do one too what if we had the fourth root of x what would that be with an exponent can you do that one matthew can you put what it would be over here what's the question [Laughter] all right so i'm making you go backwards so maya went forward she had x to the one half and she gave us the square root of x remember there would be an invisible 2 here right but we don't really need to write that in but i'm making you go backwards so what if you had the fourth root of x how would you write that as a power as an exponent how would you write that over there there you go right it'd be x to the one fourth yes very good excellent all right and just the parts of it this part right here is called the index so if you're getting like the square root or the third root or the fourth root that's called the index if you have an exponent under your radical that's called is called an exponent and the whole thing is called your radicand so if it's under a radical it's called a radicand and this is also in your in your book here's just a quick review hopefully you guys have seen that before does that look familiar to you guys or not you've learned that before yeah okay good good good good love that that you've seen it before all right perfect now one thing if you are using desmos we were talking recently about desmos before we started the recording if you're doing a square root you can type that into desmos if you're doing something higher than two you are going to have to do it in exponent form okay so and i will give you an example of that i think we have to do a third root or a fourth root as one of my examples and i'll show you how i put that into desmos okay right and i like how you type that out um jordan that s-q-r-t for square root you can type that out or c-b-r-t or cube root right and then just put in parenthesis whatever was in square root or whatever it was in cube root love that yes i love those abbreviations we used to always have to use those but now with the math type we don't have to as much anymore all right great all right so let's just take a take a look at a few few features of a radical function all right and this one's also in the lessons i believe on the first page um so remember we're going to have restrictions on the domain the range the intercept and so on remember that if it's a radical that you're not going to have a negative value for the most part right whatever is under your radical when you get the square root of that or the cube root whatever it's going to come out positive or else it's going to be imaginary and we're not going to be dealing with those at this time so that's why this graph looks really strange and this graph is just the square root of x so it's it's y oops it's not writing hang on there we go y equals the square root of x so okay all right so let's take a look at these different aspects of it so the domain remember the domain is anything that x can equal open in parenthesis x so what does it look like here's our x axis what do we know about the domain of this function so it starts at 0 0 right it's only in quadrant one right it's going to positive infinity and where does this start from where to positive infinity good great jonah yeah zero and i like how he included that zero because when you get the square root of something could it equal zero yeah it definitely could you could get zero as your answer right because if x was zero the square root of zero is 0. yeah so we do include it good so the domain which what x can equal is 0 we include it and to infinity and remember we're going to have an open bracket for that our parentheses all right range now range is y again look we are in quadrant one so what do we know about our x's and our y's in quadrant one right they're both positive yeah they're both positive so our range is also going to be positive can our range include zero yep it can include zero so our range is actually going to be the same thing it's going to be 0 to infinity what about intercepts do we have any intercepts x or y here with this radical function lauren's got it zero zero that's our only one isn't it because we haven't had any transformations and we'll talk about those soon all right increasing and decreasing intervals so what's what's happening here all right what is happening with this function it's increasing yes it is it's always increasing love that who wrote that and only increasing yes right so it's only increasing it's it's only going up so along the entire domain um it's increasing okay so there's no times it's going down it's only going up up up up up and away right so it's increasing along the entire domain actually and our end behavior so remember end behavior we did this back in uh 101 so it asks you something like as x approaches infinity what is f x doing so what's it doing j's got it right it's increasing without bound it's in it's approaching infinity oops as x approaches infinity y also approaches infinity yeah because remember we're also in quadrant one um x and y now i just want to show you real quick i didn't graph this but let's say the graph was y equals negative square root of x how would that change our graph any idea we're going to go over these it's going to be a reflection yes uh-huh it is going to be a reflection right it would it would be reflected over the y-axis excuse me the x-axis i'm drawing it and saying the wrong thing so it'd be like down here and this is just a rough draft okay so it would be reflected over the x-axis okay we're going to go over that a little bit more yeah the symmetry very good love it i'm surprised you guys knew that already good job all right so before i get into any more i want to go over a question from your mid module exams that's going to be 105. but this is not the exact question but it's super super super similar okay so let's go over this one see if we can't solve it at least you can get this question right okay everyone's with me all right so it says in which interval is the radical function and then they give us a radical function increasing notice i put a 2 up there just because we were learning about how to translate from one form to the other okay i went ahead and put a 2 up there just for fun even though remember you don't have to have a 2 but it's assumed there's a 2 in there we want to know when it's increasing all right so let's take a look at this now what would be a great way to find the solution of this what do you guys think give me some ideas okay i see a lot of students said graphing uh-huh is there any other way oh factoring right okay so you can factor you can also graph all right so let's let's just say we haven't graphed yet let's say we want to factor okay so we could go ahead and factor we'd want to factor what is underneath the radical right just the x squared minus x minus 12. what would our two factors be then no graphing's not cheating [Laughter] yeah we'd use three and four exactly um we just have to decide which is positive which is negative it would be negative four and positive 3 right so when we set both of those equal to 0 both those factors we get x equals negative 3 and 4. all right so we've got an idea of where our solutions are but with factoring we don't really see how it's increasing unless we do a table and you can do a table but like you guys said it's easier to just graph so let's do it how's that sound all right so we've graphed it here it is this is what it looks like so we want to know where in which interval is the radical function increasing so is that going to be in quadrant 2 or in quadrant 1 where is it increasing it's increasing quadrant one because here's going down remember you always read from left to right this one is decreasing but this one's increasing so where does it start when it's increasing is it at our negative 3 or a positive 4. it was at our positive 4 and see we did solve for these solutions so it wasn't a total waste of time but to get the correct answer it's going to be a lot easier to graph right so it's going to be 4 4 to what where does it end 4 to infinity yeah because it keeps going up and up and up over here yes exactly very good all right so d should be our correct answer here so not too bad right what do you guys think you guys going to do good on this one i think you will yeah and it's a multiple choice it's a multiple choice lydia so don't you worry jacob yeah just graph it see which one makes sense um and remember it can equal four because remember you can always get the square root of zero so even if x did equal four that's not a problem because you can get the square root of 0 and it's going to equal 0. yes very good love your observations all right let's see what else we got what else i've got at my sleeve all right let's look at the domain and the range of a radical function all right so let's just do some examples on how to find it all right so and we're going to do these algebraically and then i also have the graph just about back it up so we're going to go two methods so we can always find the right answer by two different methods all right so the domain remember the domain is anything x can equal right so just always think of these okay anything x can equal so what's going to be our what do we know about radicals okay so we know that we could have a 0 under the radical but what could we not have under a radical we want to make sure that this value down here right we can't have a negative we want to make sure it's positive yeah right so let's go ahead and just set that equal to zero and solve and instead of subtracting 25 i'm gonna add x to both sides okay because that's going to make more sense in this one because it's set up a little funky so can what's under the radical equal 25 would that work as part of the domain i got a no and a yes what do you guys think yeah it can work it can because if you had 25 minus 25 right that's gonna equal zero and can you get the square root of zero yes you can it equals zero okay so that works that checks off okay so that is a point all right so if you weren't graphing and you wanted to see which direction it goes either to the right or the left of 25 you could always just pop in some values really easy so let's say let's do one value less than 24 okay let's say x was 24. so can we get the square root of 25 minus 24. is that good yeah it'd be the square root of 1 which is 1. so yep that checks out okay so that would be a lesser value let me just put that a lesser value but what if we had a greater value just check this out now so instead of 24 which was one less let's do 26 which is one more so let's do 25 minus 26 what happens now nope that's not gonna work that's negative one that is not gonna work whoops i put the negative in the wrong place i got a little ahead of myself there didn't i oops all right there we go and that does not work okay so we know that our domain just algebraically here we know it can include 25 right it can include 25 and any number less than that so it'd be negative infinity to 25 we're gonna graph it just to verify it okay all right now our range now we can't really do much algebraically for the range but we can think about what we've got so if this is our function g of x equals square root of 25 minus x we've been talking about this what does g of x have to be think about this don't do any math it's y right emma it's y but what do we know by this equation it's whatever this equals and what does this have to equal here just in general terms it could equal zero it could be jordan's god it can be anything positive right it can be anything positive yeah it could be zero or higher yeah it has to be a positive number because you can't get the square root of an of anything negative now there might be a negative on the outside but there's not okay so it doesn't matter what we put as x here it's going to come out positive all right and so that's going to be our range our range is going to be at least 0 right that's the smallest amount the 25 has to do with our domain and not our range that's going to that's going to respect our x values aria the x value has a restriction regarding the 25 and i'll show you when we graph it but the the range is your y value that's your your g of x okay and that's not a restriction on your y you could have 25 because if this was um you could make it to where if you got the square root and you got 25 and then that could be your y value okay so we can include 0 and infinity all right so let's graphics i know graphing so much easier and it makes much more sense sometimes when you just see a picture of it all right so let's take a look at it [Music] all right so we said our range which is our x value remember this is our x axis it had to be 25 or less right so negative infinity up to 25 and it included 25 so if you'll see there 25 and less right and our range had to start at zero remember we're reading on the y-axis now it had to start at zero and it could go up and up and up and up to infinity okay great i'm glad that makes sense yeah so i just wanted to show you multiple ways to solve this the easiest is going to be graphing but you can also do it algebraically pretty simply and check it out or you can do it graphically so if you are required to show your work on something like this then this would be how you'd show your work all right range is really difficult to do algebraically caitlin you just have to see what's going to make sense what's going to make sense all right we're going to do another example like that it's more difficult it's actually much easier to just graph your range to see yes it is karma yeah much easier to see when you're graphing domain is pretty easy to solve algebraically otherwise range is not as simple it can be done but a lot of times you just have to think logically of what your y value could be by your given function okay because a lot of times well i'm going to show you later we're going to have another example where i'm going to show you this okay so just keep that in mind keep that question in mind i like it um someone in the chat box is asking me if you can go back so they can screenshot make sure you raise your hand before you speak and i can definitely go back yes i don't know where they wanted me to go and i can't always read every chat when there's um over 100 students in here i i'm sorry i can't read it i read every single chat i apologize about that just one slide sure i can go back one slide for a moment this one right and you're right alejandra it's also recorded so you could always come back and then we'll always also have a time for a question so don't forget that oh you're welcome you're welcome yeah sorry i missed it i i can't see all of them i try to but not all of them all right very good yeah that was very sweet of her wasn't it all right so let's talk about transformations anyone know what the word transformation means when it comes to graphing or functions raise your hand if you know what that means seen some great things in the chat j go ahead and take the mic what does that mean if you're transforming a function so like if you're trying to you're like like you're pretty much like everyone's saying you're just like moving the graph around um yes but like the main ones i'm thinking of are like when they get flipped over the x or y axis or when they end up getting um [Music] like made bigger or smaller those are the main ones i'm thinking of right right and that's called shrinking and stretching i'm not going to go over that today but i believe miss hunter is going to cover that i'm going to go over vertical and horizontal shifts so that's like moving to the right moving to the left and also reflections like you were saying um where we're going to reflect over an axis yeah so basically right you guys got it right yeah we're just moving around the graph so we're going to have what's called a parent function okay so their parent function is your original function okay that's your parent but then you're gonna have all these kids right these kids where you add three or you subtract two or you you change your radical right those are all the kids from that parent function square root of x or whatever your parent function may be and we're going to do some examples all right so i'm going to hop over to desmos real quick and we're going to play around with different kinds of um transformations so we're going to look at reflections we're going to look at vertical shifts these look they'll be outside of your parent function and horizontal shifts which will be inside of your parent function alright so let's hop on over all right give me just a moment i'm almost there there we go okay oh and i've already been graphing something i've been busy all right so let's just start with our parent function which would be y equals let's do the square root of x now remember down here is where you're going to use your keyboard all right so you want to do square root and then you can click abc and get your x although i think it was an x here too yes there was okay or you can hit abc okay all right so let's just take a look at this graph here that's just the square root of x we looked at that before right um so let's let's play around with it let's see what happens when we do different things so i'm just going to do a reflection real quick all right so let's go ahead and put a minus in front of our square root of x okay so this one we also kind of went over to so that is going to be a reflection over the x-axis so if your negative sign is outside of the radical you're going to reflect over the x-axis all right let's see let's play with this let's say our negative was under the radical see what happens now oh look at that what kind of reflection do we have here what's that reflecting over yeah it's reflecting over the y-axis yeah exactly i'm just kind of playing with these all right so let's try something else let's what if we had a negative on the outside and on the inside oh double whammy right right keegan yeah whoa take a look at that thing oh my goodness but what's happening here what is happening here guys keegan had a great question oh yeah it's re it's reflecting over both of them it sure is yeah it's reflecting over the x-axis so it's reflecting down here first then it's reflecting over the y-axis yeah reflects over on both it sure does sweet yep you got it right reflecting over the x and y that's a tricky one i'm glad you asked that one all right so those are pretty much your only kind of reflections now let's talk about vertical and horizontal shifts all right so let's let's take these negatives away here all right so vertical shifts are the easiest all right that just means outside of your radical you're either going to go up or down all right so the blue one is our parent function let's say we add 3 to it take a look at that all right so here's our parent function it was at 0 0 but if we add 3 we have a vertical shift up 3 spaces so we went to 1 2 three it went up three it's the exact same shape we didn't shrink respect the shape is exactly the same it's only gone up three what do you guys think is going to happen if i put a minus three here instead of a positive 3. what do you guys what's going to be that transformation right it's going to go down 3. let's take a look and see so take a look here all right so here's our parent function it's still the same but this time we went down one two three the exact same shift and it doesn't matter what you do you could do um you do minus one and it's going to go down one space or you could do plus one and those are our vertical shifts so those make sense they're super easy to remember so no problem on this okay so it's vertical remember vertical is up and down yeah verticals up and down yep all right so our next one's going to be our horizontal shift and it's a little bit crazier because you always have to think backwards on those and your horizontal shifts are going to happen underneath the radical all right so let's see we started the other one our vertical we added three but let's see what happens if we add three under the radical oh look at that so which direction did it move this time yeah very interesting it moved to the left didn't it right so it moved towards on our x axis it moved towards the negative number so if you want to think of it this way you can always think of it as going backwards you think of x and your x axis to the right is positive to the left is negative but this one it kind of goes backwards and do you know why it goes backwards any idea algebraically can you explain why this when you add three it goes backwards right right when you set it equal to zero when you set that x plus three equal to zero your x value is actually negative three right so that's why we're moving left okay but you don't have to do that you can always just remember that horizontal is tricky it's always going to do the opposite of what you think of it all right so we had our parent function we didn't change that but this time we had a horizontal shift one two three to the left so if i wanted it to go to the right free spaces what would this radical look like what do i need to change all right we need to change that to negative three instead well let's do it so there we go and if you'll notice it went to the right that was tricky tricky one two three spaces all right so in conclusion if you're changing it vertically okay it's going to be outside of your parent function your parent function might not be a radical it might be an absolute value okay or something else any time it's a vertical shift it's going to be outside of that parent function okay it's going to be outside of it add or subtract a number outside of it if it's a horizontal shift it's going to happen inside of that parent function is going to affect that parent directly okay all right so i've got a little sheet that i want to share with you oops let me hop over okay all right so we're back now i'm so slow with that sorry all right so here's a little cheat sheet that i found um so you might want to take a screenshot of this for your vertical and horizontal translations and also your reflections this one doesn't have both of them but just remember if you're going to have the um have it affecting inside of your parent function and outside is going to reflect over the x and the y you're welcome it is a good table okay all right so here's a question from your module one exam okay it's just like it um and we're gonna go over it and then that's actually gonna conclude the lesson and then i have a bunch of homework examples i've got one from each kind that we can go over and we can do some questions how's that sound we're almost there all right so let's take a look at this all right so this is from your module 1 exam so they're asking you to pick a radical function that has a specific domain and a specific range yeah you guys do great going over those examples madison all right so you could number one you could graph each one of those individually that's totally possible you could if you wanted to and figure out which one is going to match those um requirements that we have for our domain and range but you can also figure it out algebraically all right and the domain is going to be the easiest one all right they are not time capacity nope all right so x is a set of all real numbers except it's got this exception that x has to be greater than or equal to 5. all right so what does that mean in terms of factoring what does that mean in terms of factoring remember it's going to be our domain restriction which is our restriction on x take a look at this each one of these has either an x minus 5 or an x plus 5. how can we determine which one is going to be greater than or equal to five right our factor is going to be negative five yes exactly so if you were doing it the way we normally do it right you would be taking what was underneath this radical oops that's a crazy looking five and you would be setting it equal to zero which you'd add five to both sides so this would be that x equals 5 yeah and that is what we were looking for right because if we took one of these bottom ones let's just say we took this bottom one here we had x plus 5 equals 0 and we subtracted five from both sides that would give us negative five and that's not our restriction there is it so anything with the plus five under the radical is not going to work for us because that's not that's not good all right so we can take out c and d all right so only a or b will work so i'll just put either going to be a or b the next thing is that the range has to be less than or equal to a positive 7. all right so which two of these are going to match up with a range a would work a or what other one would work if you're just looking at the range yeah a or c yeah because you've got a positive seven here and remember our range is our y values it's up and down that's our vertical shifts so that's going to be the plus seven so a or uh a or d and we know it's not d so that makes our winner a okay and to verify this what shall we do we should probably graph it right yeah let's go ahead and take a peek at that graph here is our graph so we had our domain right here five and it goes less than five or it goes greater than five excuse me x is greater than or equal to five there we go and it is have a range of y is less than or equal to 7 so 5 6 7 here's our 7 and it goes down remember this one's going down instead of up because of that negative out there if we had that negative out there that would change our right or we didn't have it would change our range values because it would be going up and over this way so you can do it algebraically you can do it graphically either way i recommend maybe doing both yeah yeah definitely math you narrow it down remember always for the range i recommend graphing it narrow it down with your algebraic skills for the domain and then maybe graph those two that are left over or three hopefully you just have two you might only have one and it eliminates all the wrong ones right away yeah it's good to know both ways yes i agree all right and just some some quick notes here this is also in your lessons uh about simplifying a radical function remember to look for greatest common factors remember you want to factor the degree of the function will tell you how many possible factors you have so that's the um the highest exponent good good good good all right simplify your numerators and denominators check to see if anything can be crossed out remember that's your discontinuities at your holes all right excellent all right i'm gonna go ahead and i'm gonna pause the recording i might actually stop it