in this video we're going to talk about functions transformations and things like that so let's say if we have the function f of x let's say we add 2 to it what type of shift do we have here if you add 2 to it this is known as a vertical shift the graph is going to move up 2 units likewise let's say if you have x minus three that's a vertical shift down three units now what about f of x minus four this is a horizontal shift and it shifts do you think it shifts four units to the left or to the right it turns out that this shift is four units to the right if you set x minus four equal to zero x will equal four so it doesn't shift to the left four units but it shifts to the right now let's see if we have f of x plus three this would shift to the left three units if you set the inside equal to zero you'll get x is equal to negative three now there's some other ones that you need to know as well you have a negative on the outside it reflects over the x-axis now if you have a negative on the inside it reflects over the y-axis and if you have a negative on the outside and on the inside it reflects over the origin now let's say if we put a 2 in front of f of x this is known as a vertical stretch and let's say if we put a fraction in front of f of x it's going to shrink vertically now if we put the two on the inside it's going to be a horizontal not a stretch but a horizontal shrink so be careful with that one and if we put a one-half on the inside this is going to be a horizontal stretch so those are some things that you want to keep in mind in terms of transformations so let's go over some examples let's say if we have the parent function f of x is equal to x squared and basically that's a parabola that opens upward like this now let's say if we wish to graph this function x squared plus three this is going to be a vertical shift of 3 units so the graph is going to start at 0 3 and it's going to look the same now for example let's say if we wanted to graph x squared minus 2. we're going to have the same type of graph but it's going to shift down 2 units and so it's going to look like that now what about this one let's say that y is equal to the absolute value of x so here's the parent function now what if we put let's say x plus two how's the graph going to look like now in this case it's going to shift two units to the left so it's going to look like this so this point was at zero and now this point is at negative 2. now what about this one the absolute value of x minus 3. so we're going to have the same type of function but it's going to shift 3 units to the right so it's going to look like that now what about this one let's say that y is equal to the square root of x and the parent function looks like this go ahead and graph y is equal to negative square root x and then square root of negative x you can put it here and then we'll do one more so this one reflects over the x-axis therefore it's going to look like this this one reflects over the y-axis and here's the y-axis and so it's going to go that way and if we have a negative on the inside and the outside it's going to reflect over the origin and so it's going to go towards quadrant 3. now a good way to like remember which direction it goes is to look at the signs we have positive x and positive y x is positive in quadrant one and four towards the right and y is positive and one and two so when x is positive you need to go to the right when y is positive you go up so this is going to go towards quadrant one now for the next one x is positive y is negative so we're going to go to the right and then y is negative as you go down so that takes us to quadrant four for this one x is negative but y is positive x is negative on the left y is positive as you go up so that's towards quadrant two and for the last one x is negative and y is negative so it's going to go towards uh quadrant 4. not quadrant 4 but quadrant 3. this is quadrant 4. this example now let's graph these three functions using points the absolute value of x and then two times the absolute value of x and then one-half so we're going to have the point zero zero one one two two negative one positive one negative two positive two and so that's the graph for the absolute value of x now for 2 times the absolute value of x it's going to be as follows we're still gonna have the point zero zero but when x is one y is going to be two when x is two y is four and then it's symmetric about the y axis so the right side and the left side will look the same so here we have a vertical stretch notice that the y values were doubled and so that's the effect of putting a 2 in front of the function for the last example we're going to have a vertical shrink so at one it's going to be a half at two is going to be one and so this is a vertical shrink the y values were cut in half they're half of what they were compared to that graph so now you can visually see how a vertical stretch appears and a vertical shrink appears as well so for a vertical stretch the y values are increased for a vertical shrink the y values were decreased so now let's go back to this graph y equals x squared and i want you to graph let's say actually let's use a different example let's use the square root of x and let's use the square root of 2x and also the square root of one half x hopefully i can fit all of them here so i only need the right side of the graph so when x is 0 y 0 when x is 1 y is 1 the square root of 4 is two so when x is four y is two so that's the the parent function it looks something like that now for this one when x is 0 y 0 when x is a half y is going to be 1 and when x is 2 y is going to be 2. so notice that the x values would decrease by 2. here it was four and it had the same y value of two but for the same y value the x value is now two now granted this graph will continue to grow but i want to stop at the point where the y value is the same so i'm going to stop at this point here so you can see the effect that the graph has on x we said when x is a half y will be one and when x is two y is four so for the same y value of two x is no longer four but it's two so as you can see the x value was reduced by two and that's why this is known as a horizontal shrink it shrinks the x values by a factor of two now let's look at the last example now when x is 0 y 0 but when x is 2 y will be 1 and when x is eight half of eight is four the square root of four is two so when x is eight y will be two so for the same y value of two x has increased to eight so going from four to eight we see that x was increased by a factor of two so this is a horizontal stretch and you have to make sure that the y values are the same otherwise the graph may look like a vertical shrink for instance if i were to compare this graph versus this graph if you stretch it out this might appear as if it's a vertical stretch compared to this one because this looks higher however you need to compare the x values for the same y value if you do that you can clearly see that this is a horizontal shrink and not a vertical stretch and this one you can see that it's a horizontal stretch not a vertical shrink now let's move on let's work on some other examples so make sure you're aware of the parent functions let me just run through them real quick so this is the graph for y equals x the next one you need to be familiar with as we mentioned earlier in this video is y equals x squared and here's another one this is equal to this is y equals x cubed and you've seen this one already y equals square root x and then we have the cube root of x and the absolute value of x now there's some other functions but these are the main ones that we're going to go over for now so how would you graph this function x minus 2 squared plus three so graph it using transformations so the parent function is x squared which looks like this however we can see that we have a horizontal shift two units to the right and a vertical shift up three so i'm going to shift it two units to the right and up three and then you can just draw a rough sketch and that's it for that example now what about this one let's say if we have 3 minus x plus 2 squared so once again we have a parabola and this time i'm going to plot it more accurately instead of using a rough sketch now this is the vertical shift up three if you want to you can rewrite it this way this is y is equal to negative x plus two squared plus three so we have a vertical shift of three now we also have a horizontal shift left two so the center or the vertex of the parabola is going to be at negative two comma three which is here now does the parabola open upward or downward normally it would open upward however we do have a negative sign so it's going to open downward but let's get some points let's graph it accurately now keep in mind the parent function is x squared so one squared is one that means that if you travel one unit to the right from your vertex the next point will be down one and one unit to left it's also going to be that one so if you plug in one into x the y value should be two i mean not one but negative one because this is negative one here so for instance negative one plus two is one one squared is one so three minus one is two and that gives us that point now two squared is equal to four so as we travel two units away from the vertex we need to go down four so the next point is going to be 0 negative 1 and also negative 4 negative 1 due to the symmetry of the graph and that should be enough to get a decent sketch now if you want to test it let's use negative four let's see if we get negative one so three minus negative four plus two squared so negative four plus two that's negative two negative two squared is positive four three minus four is negative one so it does gives us that point and if you try zero you get the same thing zero plus two is two two squared is four three minus four is negative one so this technique works if you don't want to make a table and if you want to draw an accurate sketch let's try this one let's say that y is equal to 4 minus the square root of 3 minus x let's draw an accurate sketch now let's rewrite it first you need to see it like this so there's a vertical shift up four units and there's a horizontal shift but it looks a little different is it three units to the right or three units to the left if you're ever unsure set the inside equal to zero and solve for x i'm gonna take this term and move it to this side it's gonna switch from negative x to positive x and so x is equal to positive three so that indicates that we have a horizontal shift to the right of three units so the starting point is going to be at 3 comma 4. now the parent function is the square root of x and we have a negative on the outside and a negative in front of the x so will the graph shift towards quadrant one towards quadrant two towards quadrant three or towards quadrant four well x is negative towards the left and y is negative as you go down so it's going to go towards quadrant three so now that we know the direction in which this graph is going to go keep in mind having this negative sign here it reflects over the x-axis and having it here it reflects over the y-axis so originally this is the square root of x if you reflect it over the x-axis it looks like this and then if you reflect it over the y-axis it looks like that so when it's reflected both about the y-axis i mean about the x-axis and the y-axis it's equivalent to reflecting about the origin so we can clearly see that it's going to go in this direction now let's get some other points so the parent function is the square root of x the square root of one is one that means that as we move one unit to left we need to go down one so that's going to give us the point two three so if you plug in two you should get a y value of three so four minus the square root of three minus two three minus two is one and the square root of one is one four minus one is three which we do get now the next best point to use is four the square root of four is two so as we travel four units to left we need to go down to so four units to the left will take us to the x value of negative one and we need to go down two so the y value will be two and let's do it one more time the square root of nine is string so this is at three so if we go nine units to the left three minus nine is negative six so that's going to take us to this point and we need to go down three so we're starting at four four minus three is one so we're going to have the point negative six one so if you plug in negative six that should give you one so four minus square root three minus negative six three minus negative six is the same as three plus six which is nine and the square root of nine is three and four minus three is one so we do get this point and so now we can plot it and so let me see if i can do that a little better it should be something like that my graph is not perfect but you get the point it has this general shape to it and that's how you can graph that particular function you