we are talking about analyzing radical and rational functions lesson number one we're going to start with radical functions but in this unit we'll graph and analyze radical functions determine solutions to equations with a single radical we'll also discuss how to transform radical functions using the standard y equals square root of x we'll also graph and analyze rational functions and determine solutions to rational equations a radical function is a function which contains a radical and usually we're talking about it's not just the coefficient that has the radical it involves the variable as well the simplest radical function is y equals the square root of x in this lesson we'll analyze the graph and we'll also do some transformations and you'll see the familiar Y minus K is equal to a f of B X minus H now here I said F but here this is the square root and the f of X here we're talking about is the square root of x so we'll work with transformations with the square root function let's analyze the function y equals the square root of x and what we mean by analyze is discuss the characteristics of the function and so that we can recognize it there's certain characteristics of the function that help us to see that it is a square root function and so we'll look at those now so let's use a table of values and then plot the points according to the grid that we have here and we'll see what it looks like so what we're trying to do is we're saying we said that Y was equal to the square root of x we said that Y was equal to the square root of x then let's use these x coordinates here these numbers as inputs and see what Y ends up being so we can use our calculator to do that and we do the first one we would get an error because this is undefined we also get an undefined value when we plug in negative 1 if we do plug in 0 the square root of 0 is 0 if we use 1 the square root of 1 is 1 the square root of 4 is 2 the square root of 9 is 3 the square root of 16 is 4 so we chose some good input values here now let's plot these points using X Y type of bahding so we have zero zero and I skip these two but we can't actually show these points here on on this graph because there's not we can't define where it is here we have zero zero and then one it's one right there and then when X is 4 and Y is 2 when X is 9 then Y is 3 and when X is 16 y is 4 so what the ends up looking like is this and that's my best attempt there and we think about it we turn it sideways like this it kind of looks like half a parabola but it's going this way so we here have this action happening now let's explain why the domain of the function y equals square root X is not the set of the real numbers well we try using we tried using like negative 4 as an input but it gave us no y value it was an undefined Y value or we get error error on a calculator but it means that in most cases it means it's undefined when we tried using negative 4 as an input and that was undefined and so theta 4 is not a valid input and now there is negative one well that means then even if we take everything on this side we can use negative five negative eight and so on in fact no negative X will work even if we're talking about you know rational numbers or even real numbers anything that X where X is less than zero will not work so it means then that there's a restriction on the domain of X the set of acceptable X values don't include those values that are negative so what is the domain then we talk about the domain being the set of X values where we get a valid Y value as a result and that will only happen when X is greater than or equal to 0 but X can be any real number or about D let's state the range of the function y equals the square root of x now we're talking about the range we're talking about the range we mean Y values so how do the Y values look and most of the range will be done visually here we can see that it starts at 0 and then only goes up from there and I'll keep going that way good resume so here this is all the positive Y values with that y equals 0 so it seems then the range is y such that Y is greater than or equal to 0 where Y can be any real number so here the Y values we're talking about the range so we're talking about Y values the range of Y then if we look we have Y can only be positive numbers and the number 0 so Y is greater than or equal to 0 and Y is any real number now what about transforming the from the radical function well if we're talking about transforming this function from y equals square root of x to y - K equaling a square root of B times X minus H and it's very similar to how we thought about transformations in other units where we have a definition for the original function and then a be H and K somehow transform that function in a certain way so let's take a look at the table that summarizes the connection between the transformations the replacements for x and y and the resulting equations here we have a vertical translation where Y is replaced with Y minus K and so the equation will get is y minus K is equal to f of x or y equals f of X plus K and here I'm going to write in this particular case it would be the square root of x plus K now horizontal translation will have X is replaced with X minus H and so we have y equals f of X minus H what we mean here is that the bracket X minus H is the input so here we have why is equal to the square root of X minus H in this case if we have a reflection in the x axis that means Y's are become their opposites in sign and so we have negative y is equal to f of x or y equals negative f of X so we have y is equal to the negative square root X we have a reflection in the y axis we're talking about X changing to their opposite sign so Y is equal to F of negative x we have Y is equal to the square root of negative x and here you'll find something interesting about the domain what about the reflection in the line y equals x we have X replaced with Y why replace with X we have X is equal to F y or in case of an inverse function we have y is equal to F inverse of X and in that case you do something like this X is equal to the square root of Y and then continue to solve for y it would result in something like Y is equal to x squared but with a domain restriction so y equals x squared with the domain restriction we could also say what about a vertical stretch above the x-axis well Y is replaced with 1 over a times y so we have 1 over a y is equal to f of x or y equals a times f of X remember f of X is the radical function so you can say this is a time's the square root of x and the horizontal stretch about the y-axis we have X is replaced with BX we have Y is equal to f of BX BX being the input here this is y is equal to the square root of B X now to build the equation then we have Y minus K is equal to a square root of B bracket X minus H from y square root of x we should probably use this convention where the transformations we're talking about stretches first perform stretches first then reflections and then translations so let's take a look at class example 1 and in each case let's describe the series of transformations required to transform the graph of y equals the square root of x to the graph of the given function make rough sketch on the graph provided and state the domain and range of the function and then we can use a graphing calculator to verify so we have y minus 4 is equal to square root of x plus 2 here I know we're talking about the Y being replaced with y minus 4 I'm going to think of it butter like this we have the square root of x plus 2 and plus a K or plus 4 now here I'm going to just say that the a value is still 1 looks like the B value is still 1 so we have just 1 times X plus 2 it looks like the H value is negative 2 and the K value is equal to positive 4 let's interpret these parameter values now we have B is equal to 1 a is equal to 1 so there's no change there H is equally negative 2 seems to suggest that we have 2 units to the left of a horizontal translation we also have a ka plus 4 which means it's 4 units up as a vertical translation let's now make a rough sketch of this we have our standard graph y equals the square root of x and we're going to move 2 to the left and so if we go to the left let's go 1 2 and so it looks something like this right that's rough but remember we're also going 4 units up so 1 2 3 4 is going to look something like something like this now here when we're talking about 4 units up here we're talking about this line being Y equaling positive 4 so it starts right there at and this is 2 to the left here so here we have this this is y minus 4 is equal to square root of x plus 2 right here now that we're looking at the graph and the equation as well we can talk about the domain and the range so if we talk about the domain here the Des Moines domain all the acceptable x-values now remember that normally it's right over here normally the domain is equal to the very original it's X such that X is greater than or equal to zero with X being any real number right but now because of the transformation of two units to the left the domain has shifted two units to the left as well so here since it started at zero and above then the new domain is going to be X such that X is greater than or equal to now zero to the left of zero is negative two and x is now any real number still okay what if we were talking about the original domain so the original range is R so that's Y such that Y was greater than or equal to zero now with the range being this way and we're also now moving this function four units up so here the new range range here is going to be Y such that Y is greater than there's no reflection here so it's still greater than or equal to and we have zero zero plus four units up from zero that is going to be now positive four and Y is still any real number so here you can see that the domains have changed depending on how you transform the function a horizontal translation did affect the domain in this case because it started at a particular spot it was an infinite again with the arranged starting at a particular spot then moving the function up four units affected that range as well let's now verify thing on a calculator so here is our calculator and we have y equals we'll take the square root of x so that we have a reference point and then now here's a slightly tricky part we have y minus 4 is equal to the square root of x plus 2 well we need to write it so that Y is isolated so Y 2 here is equal to square root of x plus 2 and then we have to add 4 so this is what we're going to put into our calculator we have square root of x plus 2 and then we have plus 4 outside of the square root and let's graph that so there's the normal square root of x function this is our new function and that looks like our graph somewhat and so we're happy with that taking a look at Part B we have 2 y is equal to the square root of negative x and here y was replaced with 2 y here but I'm going to just rearrange this dividing both sides by 2 and Y is going to equal 1/2 times our negative square root of negative x okay that way and treat it very much like I did in the transformations unit where was a times f of X then I can know that this a here was equal to 1/2 which tells me this is the vertical stretch by a factor of 1/2 the next thing it looks like we had our original X be replaced with negative x and if that's the case X being replaced with negative X or in other words the B value is equal to negative 1 is going to be a reflection both the y-axis the X's are going to switch from positive to negative so we have a reflection happening here so let's use a vertical stretch by 1/2 here is the original graph and it's going to be 1/2 each time something like that I think I went a little high here but this is half of each one of these heights is half the height right there okay and then we're going to reflect it so this should have went this way like so oh I'm just gonna try and cut the butt a little bit so that you can see it started and this is the graph we're talking about so this is 2 y is equal to the square root of negative x all right let's talk about the domain the original domain I guess is still X is greater than 0 right and the original range is y is greater than 0 greater than equal to 0 so when we have this vertical stretch by 1/2 that doesn't affect the domain here and if we look at the reflection though a reflection about the y axis means that now instead of having X being greater than or equal to 0 we're talking about only this part this this is the graph only not this part so here this because this was before the reflection so the domain of this graph is X such that X is less than or equal to 0 with X being any real number so we have X is less than or equal to 0 X be any real number this is because of the reflection about the y axis when we talk about the range we have Y such that Y is still greater than or equal to 0 even though we had a vertical stretch by 1/2 since Y went from 0 all the way to infinity you think half of a very very large number is still a very very large number and it still can continue to go up let's take a look at Part C we have y plus 5 is equal to negative square root of x so I'm going to replace this with Y is equal to negative square root of x minus 5 that way it looks a little bit more like a f of X plus K and here we can see a is equal to negative 1 K value is equal to negative 5 so it looks like we are talking about a reflection if a is negative but it's reflect about the x axis and K is negative 5 so that is going to be 5 units down a vertical translation of 5 units down great let's do that reflect about the x axis so that's going to be over here and just doing a dotted line just to show you the temporary and then 5 units down so 1 2 3 4 5 again it's going to look like like so and there is our y plus 5 is equal to negative square root X now let's talk about the domain remember original domain X was greater than or equal to 0 original range and Y was greater than or equal to 0 and what do we have here I reflect about the x axis so does the domain change no visually you can see that X still starts at zero and goes on so the domain here it's going to be X such that X it's still greater than or equal to 0 X can be any real number well but what about the range the range the original range was y is greater than or equal to 0 and then we have 5 units down a vertical translation well that won't what what that will change that will change the range including this reflection to now the range is going to be y is less than or equal to 0 and even further five five units down the range now is going to be Y such that Y is less than or equal to C five units down from zero one two three four five Y is going to be less than or equal to negative five where the Y's could be any real number let's verify that on a calculator we can see so original there square root of x oh there's a negative there and then my is 5-5 let's graph that so the original looks good and way down here at negative five it starts going this way that is what we see there good let's now take a look at Part D we have y is equal to three times the square root of 2x minus one looks like here this is an a value a is equal to three and now if we go into this radicand here we have two X minus one and I'm going to factor because in the original transformation formula we have to factor that be out before we can use it so this is X minus 1/2 and then that is if we're talking about that that means that B is equal to two and then we also have an H value that is equal to positive 1/2 so a is 3 B is 2 H is 1/2 this a means it's a vertical stretch by a factor of 3 this B means that it's a horizontal stretch by a factor of 1/2 because remember it's opposite of the B value right here with a it's a vertical stretch by what that a says and here this is a horizontal stretch by 1 over B and we have this one this is going to be a horizontal translation of 1/2 unit to the right okay let's go in order here a vertical stretch by 3 so here this would be 0 0 but by 3 stays there as an invariant point because it's on the x axis but this 1 1 will now become 1 3 so 1 2 3 some like that and then we have this that thing happening after the vertical translation then we have a horizontal translation by 1/2 which brings it closer to the y axis so this will be maybe I'll do it in a different color now you have this thing happening here and then it's half of that so what's right over there and then after that is a horizontal translation to the right by half unit so instead of being at zero zero here is now at a half zero so right there but still going this way so see right there oops so this is our y is equal to three square root of 2x minus one taking a look at the domain here it looks like the domain has started it was started at 0-0 that was the original domain and the only thing that affected do you have a horizontal stretch by a half but still it goes on forever so that would half of zero is still zero if we take a look at the horizontal translation of half to the right then this domain now it's going to be X such that X is greater than or equal to a half instead of zero and then we have X is any real number okay what about the range the range is Y such that Y is greater than or equal to zero is the original a vertical stretch by three still remains from zero and up so this is still remaining the same at Y is greater than or equal to zero let's now talk about class example number two and let's determine the domain and range of the function y equals a times the square root of BX if we're talking about a being positive and B being positive both of them being positive you can see where this is going if a is negative but B is positive if we have a as positive and B is negative or a is negative and B is also negative before we see what happens let's just see the domain remember the domain of the original say the original domain is going to be X is greater than or equal to zero and the original range is going to be Y is greater than or equal to zero so if we stretch here this a means a vertical stretch a means a vertical stretch and B also means not also but B it means a horizontal stretch okay so if the original domain was X is greater than or equal to zero and we have a horizontal stretch of it then it's going to stretch times a certain factor some factor times zero is still zero and then it would go on to infinity so in this case the original domain the domain is still X such that X is greater than or equal to zero and X is any real number and we you say what about the vertical stretch so well the vertical stretch will not affect the domain because we're only stretching y-values at those particular X locations what about the range the range here if it's a vertical stretch and a is a positive vertical stretch there's no reflection that means what it was positive will still stay positive and we talk about a vertical stretch invariant points happen on the x axis so when y is equal to zero it'll still be zero so this stays that it's a non negative number what about B if a is less than zero suggest then a reflection sorry about the x axis so it's flipping down B is still positive so it doesn't change the domain here but a being less than zero is a reflection about the x axis that means it's going to affect the range so the domain is still X X is greater than or equal to zero X being any real number but the range because of this reflection down you get Y is y is less than or equal to zero with Y being any real number let's see if we can just do a sketch for each of these so sketch you'll have this still at zero so we'll make it big and then it just goes this way still if we have a reflection if a is less than zero you might still have a vertical stretch but instead of having the original goal so having the original look this way it's going to look some like this okay what about see if a is positive but B is negative well if a is positive then it would still not change the great range very much but if B is negative then that means you're going to have a reflection about the y axis B less than zero means a reflection about the y axis so instead of looking sub something like that you are looking at something like this well if that's the case that's going to change the domain so the domain here will be X such that X is less than or equal to zero and still be real numbers but what about the range the range is still it's a positive value so positive times Y values is still going to be positive Y value so the range is still Y Y is greater than or equal to zero and for D both of them are negative so that means you're going to have a reflection this way and you're also going to have a reflection this way so you end up something like this now from the original maybe I'll put that in a dotted line you'd see that the domain has switched over to the other side and the range is switched so the domain here is going to be X such that X is less than or equal to zero X can still be real numbers but the range is Y such that Y is also less than or equal to zero with Y being any real number so be familiar with Howard how the reflections do change the domain or the range depending on whether the negative is part of the B value or part of the a value I'm returning now to my trusty desmos up here and I'm going to show you you have the y equals square root of x in this red you can see and now I'm going to show you the effects of a B now I already show you with sketches with the pen but here you can see it in real time so here's a where a sorry y is equal to a times the square root of BX if a is equal to 1 and B is equal to 1 then this is what you have now let's see what happens as we take a and move it to negatives so when we move it to negative you can see here's vertical stretch that's happening but you can see that it takes the range and brings it down underneath 0 so here we can go through actually let's see what happens we'll go through each of these so a says both are positive so we can move both of them here at the same time and we remove this one and you can see that if I move a and B and positive values you can still see that still along this side of the y axis now in Part B it says a is less than 0 but B remains positive so we'll keep either actually let's bring it back a little bit and then make a go negative and you can see when a is negative you still have the domain the same but the range has changed and then we can see for a being positive let's move a positive but B is negatives we'll move it over to negative and here you can see that now it's the domain that has changed and the range is still the same and if both are negative so B is already negative make a negative as well you can see now this was the original graph we'll take that out and the domain has changed to this sign the range has also changed to this site you are ready for your assignment now and I will see you in class