College Algebra – Transformations of Functions.wmv

right in this section we’re going to be looking at the transformations are functions and out of this section we’re going to be recognizing graphs of common functions also we’re going to be using protocol shifts horizontal shifts reflections vertical stretching and shrinking and also horizontal stretching and shrinking to graph functions now so we’ll be graphing functions involving a sequence of transformations okay now these are the common graphs that are used in algebra one of them is the constant function where f of X is equal to C and C is a constant like f of X equal to 6 there will be a constant function because regardless of what the value of x is that f of X rally is still going to be 6 that’s an example next is the identity function which is f of X is equal to X that means whatever X is f of X is also the same value like if X is 1 f of X is going to be 1 so the X and the f of X values are going to be identical and then next is the absolute value function which is f of X is equal to the square root of x okay across the square I mean not square root absolute value of x absolute value of x is always going to be positive is never going to be negative and then the standard quadratic function that’s f of X is equal to X square and then the square root function is f of X is equal to the square root of x and then next is the standard cubic function where f of X is equal to X to the third and then you have your cubic function which is f of X is equal to the cube root of x and now when my bring up is a illustration of each one of those here’s the constant function which is a straight line a horizontal line where f of X is equal to some constant and then the one in the middle is the identity function where if X is 1 Y is going to be 1 if X is 2 y is going to be 2 and so on and then the third one is the absolute value function that’s that visa graph that discussed in one of the videos here the absolute value is always positive is never going to be negative so if your x value is negative 1 then f of X will be positive one because the absolute value of negative 1 is 1 and then next is the standard quadratic function okay looks like this f of X is equal to X square and then we have the square root function okay notice that you know if you have let’s say a negative value for x there is no y by f of x value for it and then the standard cubic function that graph looks like this that’s f of X is equal to X cube and then you get the cubic root function of course it looks like this so do be familiar with those types of graphs okay all right now let’s look at some of these transformations here all right the vertical shift is this if you have a function a graph of a function y is equal to f of X then the vertical shift means you can have f of X plus some constant or f of X minus some constant f of X plus some constant means that you’re gonna add whatever that constant is to the y coordinate because the vertical shift means that the graph is going to be shifting upward or if its downward you’re going to subtract whatever that constant is from that y coordinate so the vertical shift means that the graph is gonna be shifting upward or downward now horizontal ship this one’s a little bit tricky because with the horizontal ship you have to do the opposite of what’s inside the parentheses here so here let’s say we have y is equal to f of and in parenthesis X plus C here you go and do the opposite of adding the constant which will be subtracting from the white from the x-coordinate here the

x coordinate is changing so that’s why i say subtract the constant from the x coordinate and then y is equal to f of X minus C dr. subtracting see we’ll be adding that constant to the x coordinate so if the horizontal anything holds on you’re going to be doing the opposite okay and anything how the song means that the x coordinate is going to change and anything vertical means that the y coordinate is going to change like the next type of ship will be the reflection about the x-axis and the y-axis now the reflection about the x axis is going to be in the form of y is equal to the opposite of f of X where the negatives negative sign is in front of the f of X so here you going to take the opposite of the y coordinate time for reflection about the x axis you’re going to take the opposite of the y coordinate but a reflection about the y axis this in the form of y is equal to negative and what F of negative x notice that the negative sign is in front of the X so that means you’re going to take the opposite of the x coordinate right then we got vertical stretching and shrinking they’re both in the form of y is equal to C times f of X but with a vertical stretch the constant is bigger than one so here you going to multiply that wife learned about that constant and with the vertical shrink the constant is a number between zero one usually is a fraction like one-half or one-third or one-fourth you’re still multiply that y coordinate by that constant okay so that’s the difference between a vertical stretch and a protocol shrink in a vertical stretch the constant in front of the f of x is bigger than one in a vertical shrink but constant in front of the f of X is pretty much going to be a fraction or a number between 0 and 1 and then we have the horizontal shrink and the horizontal stretch now with the horizontal shrink the constant is in front of the X so here we’re going to be doing the opposite of multiplying that constant by X which is going to be dividing and we’re going to divide each x coordinate by that constant again as horizontal means the x coordinate is going to change and a horizontal shrink here the constant is still in front of the x but the constant is between is a number between 0 and 1 and normally i will tell you to divide that fraction well divide each corner by that fraction but since it’s a fraction we can easily multiply by the reciprocal of whatever that consonants that’s just an easier way to do that so if you had like one half x let’s say y is equal to f of 1 half x then each x coordinate will be x 2 because 2 is the reciprocal of one half now the next is the sequence of transformation suppose you have more than one transformation for a particular function okay so if you have a function having more than one transformation we can graph that by performing the transformations in the following order in this case the first thing is the horizontal shift then second is stretching or shrinking then third is reflecting and we always do the vertical shift last you can have to think of the sequence of transformations as order of operations okay if you recall you always do the parentheses first and then exponents and then adding and subtracting from left to right now multiplying and dividing from left to right and then adding and subtracting from left to right the sequence is somewhat similar to that so you have to think of this as performing the order of operations okay and we’re going to look at a few of these problems that are just like that okay let’s take a look at this one here we given the graph of y is equal to f of X and I’m going to move this up a bit so you can see down the left side you’ll see the graph of y is equal to f of X which is so set of ordered pairs so that should y

is equal to f of X now you’re going to use that graph to come up with a graph for each function called G of X in this case G of X is going to be f of X plus 1 g of x is equal to f of X plus 1 now we need to find out what type of graph that we have here ok now based on those transformations that I just went over with you you have to look at the format now f of X plus some constant that means we have a vertical shift ok so that means we’re adding one to whatever f of X is so that means we’re going to add one to each y coordinate because it is a vertical shift that means that the y coordinate is going to be changing so we’re going to add one to that y coordinate to come up with our new graph for G of X this okay now what I’m going to do is create a table and that first column is y is equal to f of X and I’m going to copy down the ordered pairs that we have here we got negative 4 0 negative 2 negative 2 0 0 to 2 or 0 and by the way you’ll be given a graph of y is equal to f of X with those ordered pairs on it okay now I did say that we have a vertical shift so the only transformation we have is just that vertical shift and that means we’re going to add one to each y coordinate that’s why I put the plus one there because we’re adding one to each y coordinate because it is a vertical ship that means that the y coordinate is going to be changing so what I’m going to do here is bring over these x coordinates they’re going to remain the same okay so for each y coordinate I’m going to add one to it so if i have 0 plus this one that’s going to be 1 negative 2 plus 1 that’s negative 10 plus 1 again that’s going to be 12 plus one that’s going to be 3 and 0 plus 1 again that’s going to be one all right this column and in this case I should say the last column after you do all your shifting and stretching and shrinking whatever that last column is always going to be the ordered pairs for G of X so this column represents the ordered pairs for G of X equal to f of X plus 1 so all I need to do here is just go ahead and plot those points on the coordinate plane I’m going to move this to the side so you can see this so for negative 41 we go to the left 4 and up one negative 2 negative 1 will be to the left two and down 101 will be at zero will go up 1 2 3 will be to the right 2 and up three and finally for one will be to the right 4 and up one now I’m going to connect those points the same way as y is equal to f of X and that right there represents the graph of G of X is equal to f of X plus 1 now compare that to y is equal to f of X as you can see here it’s the same graph the only thing different here is that this graph has been shifted up one place ok so that’s how we use transformations to graph a any particular function ok all right now let’s look at another graph now here we got g of x is equal to f of and in the parentheses we got x plus 1 ok now we still have y is equal to f of X on the left side but here I’m going to come up with G of X which is going to be this time f of X plus 1 ok now notice the X

plus 1 is in the parentheses i’m adding 1 to the x-coordinate but this means I’m gonna have a horizontal shift because it’s in the form of F up in parentheses X plus C so here I’m going to have a horizontal shift in anything horizontal I’m going to the opposite of what’s inside the parentheses i’m adding 1 so the opposite of that will be subtracting one that’s very important so the opposite of that would be subtracting 1 okay so what I’m going to do here is create come up with the table just like I did in that last particular example okay so here this is still y is equal to f of X and my ordered pairs are still negative 4 0 negative 2 negative 200 to 240 okay that comes from y is equal to f of X now I did state earlier that we do have a horizontal shift so I’m going to put in this column H ship because we have a horizontal shift and it says plus 1 but the opposite of adding one will be subtracting 1 so here going to be subtracting one from each of the x coordinates here the y-coordinates will remain the same but we’re just going to be subtracting one from each of the x coordinates here all right in this case here if we do negative 4 minus one that’s going to be negative 5 now bring it over x y coordinate of 0 negative 2 minus one that’s going to be negative 3 now bring over the y coordinate of negative 20 minus 1 is negative 1 and 0 to minus 1 which is 2 2 minus 1 is 1 so b12 and then 4 minus one which will be 3 and 0 so that last column represents the ordered pairs for G of X ok now I’m going to connect those points on this new graph as you’re going to see here where negative 50 you go to the left 5 and stay on the x axis then negative 3 negative 2 will be to the left three and down to the negative 10 will be to the left one and it’s on the x axis then 112 will be to the right one and up to and then 30 will be to the right three and stay on the x-axis so here that’s what that graph looks like now compare that in fact that g of x is equal to f of in parentheses x plus one now you can compare that to the y is equal to f of X you can see it’s the same graph the only thing different here is it’s just shifted to the left one place right here’s another example let’s say you had this one let’s say we had G of X is equal to f of n in parentheses X plus 1 and then on the outside we got minus 2 ok so now we got this crap then we want to create and notice we only notice we have two shifts one we got a horizontal shift now like this on the side and we got this minus 2 on the outside that’s a vertical shift ok so now we get two different transformations going on now the question is which one do we need to do first well if you recall from the sequence of transformations that I gave you you can always do if you’re not if you’re in doubt we always do the vertical shift last so the vertical shift is going to be done last that means that the horizontal shift we have to be done first now you have to think of this as the order of operations here they’re taking the value of x and then you’re going to add 1 to it and then whatever that is you’ll subtract two from it so first will be the horizontal ship and then the vertical shift so let’s do this so here we’re going to

have a table y is equal to f of X that’s the post column then the next column will be close to horizontal shift as we’ll do that first and then the vertical shift is done blast so in this case they let me rock on ordered pairs negative 4 0 negative 2 negative 2 0 0 to 2 and 40 up okay so first do the hub come up with ordered pairs for the horizontal ship and from those ordered pairs I come up with new ordered pairs for the vertical ship and whatever is in this final column represents the ordered pairs for G of X ok so now the horizontal ship is a plus one so i need to subtract one from each x coordinate and then the vertical chef says minus 2 i’ll do whatever it says subtract two it’s only the whole zone ship where you have to do the opposite of what’s inside the parentheses the protocol you do whatever is whatever it says it whatever is given ok so we’re going to subtract X from one from each x coordinate because it’s a horizontal ship meaning the x coordinate will be changing so here negative 4 minus 1 that’s negative 5 and 0 negative 2 minus one that’s negative 3 bring over the negative 20 minus 1 is negative 10 to minus 1 is 1 and 2 and then 4 minus one which will be 3 and 0 now that’s just the ordered pairs for the horizontal shift now we need to come up with the ordered pairs after doing the vertical shift here the y-coordinate is changing say Justin packing two from each y coordinate because it says minus 2 here so in this case it let me bring over the x coordinates and then in this case here 0 minus 2 is negative 2 negative 2 minus 2 is negative 40 minus two is negative 2 2 minus 2 is 0 3 minus 2 is negative 2 ok so this last column represents the ordered pairs for G of X that last column represents your ordered pairs for G of X so now the next thing you need to do is just go ahead and do the ground of G of X just plot those ordered pairs so in this case here negative 5 negative 2 will be this to the left five and down to negative 3 negative 4 will be to the left three and down for negative 1 negative 2 will be to the left one and down to let’s see 10 will be at one to the right one you state on the x axis then 3 negative 2 to the right three and down two ok so we’ll connect those points in the same manner is y is equal to f of X ok so that’s represents the graph of G of X is equal to f of parenthesis X plus 1 close parentheses minus to compare that to y is equal to f of X you can see that it’s just shifted to the left one and down to all right now let’s look at a another example let’s say yo G of X is equal to F of negative x g of x is equal to F of negative x okay now in this case here the negative side is in front of the x which means we have a reflection about

the y-axis ok F of the opposite of X means we have a reflection about the about the y-axis and that means we’re going to have to take the opposite of the number in front of X ok because as you can see the negative sign is in front of the x that means we take the opposite of our x-coordinate which is going to be quite easy ok so here again y is equal to f of X so i’ll just say y axis reflection which means we’re taking the opposite of the x coordinate so here we got negative 4 0 negative 2 negative 2 0 0 to 2 and or 0 okay those are my ordered pairs again ok so here we take the opposite of X that means we change the signs of the opposite so the opposite of negative 4 is for the opposite of negative 2 is to the opposite of 0 of course that’s 0 the opposite of this to here is negative 2 and the Occident 4 is negative 4 notice I’m only taking the opposite just the x coordinate the y coordinate stays the same ok so if I have to do his just plot those points which is going to be quite simple so 40 will be to the right for and we stay on the x coordinate on the x axis to negative 2 will be to the right two and down 200 is at the origin negative 22 will be to the left two and up to and then negative 40 will be to the left 4 and we stay on the x-axis ok notice the difference this part this on the bottom is on the right hand side ease you start up on the left of the x-axis now it’s on the right of the amine of the y-axis this one is to the right of the y-axis now it’s on the left so there is a reflection about the y-axis okay so that graph was quite easy okay now here’s another one now if I can get to it let’s say the opposite of f of x and then plus 1 g of x is equal to the opposite of f of X plus 1 or you can say negative f of X plus 1 in this case we have a reflection about the x axis because we get the negative sign in front of f of X that’s one of our transformations so here we have an x axis reflection and that plus 1 tells us we got a vertical shift ok so now we need to determine which one we have to do first well the vertical shift is always done last so that means that the x axis reflection will have to be done first think of f of X you take the opposite of that and then add 1 so it makes sense the x axis reflection will be done first and then the vertical ship would be done last ok so now I’ll rewrite my i’ll be by my order pairs ok now the first column is going to be the reflection about the x axis so i’ll just say x axis reflection which means we’re going to take the opposite of the y

coordinate and then we have a vertical shift and it says plus 1 so that means we’re going to add one to the protocol to the a to the Y coordinates okay so now we’ll do the opposite of each y coordinate and then we’re going to add one to each y coordinate so our x coordinates will be remaining the same throughout all these these two transformations here so i rewrite my x coordinates here and then we’ll take the opposite of the y coordinate which represents the x axis of reflection the opposite of 0 is 0 the opposite of negative 2 is to the opposite of 0 again that’s 0 the opposite of two is negative 2 and the opposite of 0 is 0 okay now I’ve got to do the vertical shift here add one to each one of those y coordinate so again my x coordinates will remain the same so i’ll rewrite those so here 0 plus 1 is 1 to plus 1 is 30 plus 1 is 1 negative 2 plus 1 is negative 1 and 0 plus 1 that’s 1 so this last column represents the ordered pairs for g of x is equal to negative f of X plus 1 so when I plot those on the coordinate plane is one will have negative for one will be to the left 4 and up 1 negative 2 3 will be to the left two and up 30 one would be at zero buckle up 1 2 negative 1 will be to the right 2 and down one and then for one will be to the right four and up one so here we’ll connect connect those and that’s the graph of G of X is equal to the opposite of f of X plus 1 now you can see it’s reflected about the x axis because this part is below the x-axis now above and then this part originally was above now it’s below and it shifted up upward by by one all right another example let’s look at a shrink like G of X is equal to negative I mean one half f of X G of X is equal to one half f of X ok notice that the one half is in front of the f of X and one half is a number between 0 and 1 so that tells us we have a vertical shrink ok so here we got a vertical shrink and this is the only transformation that we have so that’s the vertical shrink that we have here so what we need to do here’s find out what those new ordered pairs is going to be and notice I’m using the same graph for each one of these particular examples here so you can see that these ordered pairs are still going to be the same for y is equal to f of X and I just say we have a vertical shrink so that means we’re going to take one half of the y coordinate that means times one half so whatever the y-coordinate is it will be multiplied about one-half all right so in this case here one half of 0 is 0 1 half of negative 2 of course that’s negative 1 and again one half of 0 is 0 and four to one half of two is one and again one half zero is zero only the y-coordinate is changing the x-coordinates remain the same so that represents my ordered pairs for G of X and then the next step would be to go into the graph using these ordered pairs in this last column so here negative 40 will be on the x-axis

negative 2 negative 1 y’all going to go to the left to but you’re going to go down on one and then 0 0 is on them at the origin to one to the right 2 and up one and then 40 to the right for you stay on the x-axis so the graph is the same as the y is equal to f of X the only thing different is this that graph has shrunk by a factor of one-half so that’s how you’ll graph G of X is equal to f of 1 half x I mean one half f of X okay I think I got a couple of more examples level let’s look at F of 1 half x by the way and notice that the one half is in front of the x instead of the one have been in front of f of X so instead of a instead we have what they call a horizontal stretch okay a horizontal stretch meaning that the x coordinate is going to be changing so let’s say H stretch that’s what we have a horizontal stretch okay so now normally we will tell you to divide by one-half but it would be much easier to multiply each x coordinate by 2 so here that’s y is equal to f of X in this column and then the H stretch means horizontal stretch we’re going to be x 2 when i get the to phone you take the reciprocal of one-half which is too as i mentioned in one of the sheets that placed on the screen on the in this video earlier so you multiply by 2 we would say / one half but is easier to multiply by the reciprocal okay so here’s my ordered pairs what y is equal to f of X now I’m going to be taking one half of each of these x coordinates here the y coordinate will will remain the same one half of it work I mean we’ll multiply by 2 i’m sorry so 4 times negative 4 times 2 will be negative 8 and 0 negative 2 times 2 will be negative 40 times two is 0 to x 2 that’s 4 and 4 times two is eight then notice here the y-coordinate is not changing it stays the same only the x-coordinate is changing by a factor of two because we’re dealing with a horizontal stretch like in this case here we’ll plot those negative 8 20 will be to the left eight and we draw a point on the x-axis negative 4 negative 2 will be to the left 4 and down 200 is on at the origin for two will be to the right four and up two and 80 will be on the origin I mean at thee on the x-axis so that’s the graph of G of X is equal to f of 1 half x now in this case if you compare that you’ll see that is stretched by a factor of 1 hat well by dr. up to so that’s the strange graph of G of X is equal to f of 1 half x ok and then finally I do have a oneness quite typical now let’s take a look at this one we get g of x is equal to 2 times f up and in parentheses x plus 2 minus 1 g of x is equal to 2 times f of X plus 2

minus 1 and we want to perform this particular transformation for G of X let’s look at some transformations that we have here I see you next plus 2 that means I have a horizontal shift I see a two in front of the f of X that means I’m going to have a vertical stretch I also see a minus 1 that means have a vertical shift so I had three different transformations going on here now need to determine which one I need to do first well the vertical shift always do last let’s look at this here if I take the value of x add 2 to it and then whatever a plexi is i’m going to multiply by 2 so that tells me i need to do the horizontal shift post and then the vertical stretch second and then that vertical shift is going to be done wise so you have to think of this as order of operations here you’re taking the value of x and then 22 it that’s a horizontal shift and then whatever f of X whatever this party is you’re going to multiply by 2 that’s your vertical stretch then subtract one from it that’s your vertical shift so that’s the order that I’m going to be doing this particular transformation the horizontal shift will be done first the vertical stretch will be done second the vertical shift will be done last okay so this is where we got here the horizontal shift then vertical stretch and then finally the vertical shift okay and then going right down my ordered pairs ok now the vertical shift is this it says plus two but the opposite is going to be minus two so I must subtract two from each x coordinate the vertical stretch is a two in front of it somewhere x to their vertical shift is minus ones on subtract one from each y coordinate ok so we got three different things happening in this particular transformation ok and the last column represents the ordered pairs for G of X so let’s start with the horizontal ship we’re going to subtract two from each x coordinate so here negative 4 minus 2 is negative 6 negative 2 minus 2 is negative 4 and then 0-2 that’s negative 2 and then 2 minus two is zero and then 4-2 that’s too ok now I’m going to use these ordered pairs to come up with new order pairs after doing the vertical stretch okay the y-coordinate is going to be x 2 so i’ll go ahead and bring over my x coordinates here so here’s 0 x 2 that’s 0 negative 2 times 2 is 0 is negative 40 times 2 of course that’s 0 2 times two is four and 0 times two is zero okay now i’m going to use these ordered pairs to come up with new ordered pairs after doing the vertical shift meaning i must subtract one from each individual y coordinate so these x coordinates will still remain the same so now i’m going to 0-1 that’s negative 1 negative 4 minus one that’s negative 50 minus 1 is negative 1 4-1 that’s 3 and 0 minus 1 is negative 1 so that last column represents my ordered pairs for G of X ok now we need to plot those points so here we use that last column to plug those particular points so for negative 6 negative 1 you go to the left six and down one negative 4 negative 5 will be

to the left 4 and down five and then for negative 2 negative 1 will be to the left two and down 10 3 will be at zero but we’re going to go up three and then four to negative one to the right two and down one so here’s what the graph is going to look like is this it stretched by a factor of 2 is shifted to the left to and has shifted down one if you were to compare that particular graph with y is equal to f of X which is the one on the left hand side ok so that’s going to conclude this particular function this particular section on transformations of functions