 Welcome back to our lecture series math 10 6 teacher genometry for students at Southern Utah University as usual I'll be your professor today. Dr. Andrew missile 9 lecture 11 is going to continue our discussion of graph transformations of sine and cosine In this video, I want to talk about Transforming functions by shifting that is to say if we were to translate a graph up down left right How do you do that algebraically? Well, the general principle is if you have some function f of x and you have some positive number c What have you then y equals f of x plus c is the same graph But everything's been shifted up by a factor of c and so adding c to the function moves everything in the picture up Conversely, if you subtract c from the function that actually causes everything to move down Vertically shifting it down by a factor of c. So how might this affect a sine or cosine wave? Well, if you were to look at y equals cosine of x plus 2 Notice the plus 2 is outside of the cosine That means you take your standard cosine, which you can see right here and everything is going to be moved up by Two points exactly. So looking at the five Important points the five special points on the graph right here these coincide with the quadrantal angles Zero pi halves pi three pi halves and two pi their original y-coordnance are one zero negative one zero and One now if we move everything up by two then zero one is going to move up to become Zero three and pi have zero is going to be moving up to be coming In this case pi halves two we added two to everything negative ones the y-coordinate here We move it up by two it's y-coordinate that becomes one This one the y-coordinate zero it's an x intercept it moves up to become two And then lastly the two pi at two pi the y-coordnance one. It's going to move up to this value three All right, and so when it comes to a sine or cosine wave It's very important to pay attention to the so-called midline This midline is the average value of the function and for an unshifted Siner cosine wave this midline will coincide with the x-axis But because we shifted everything up by a factor of c this case specifically shifted up by two This midline gets moved up as well And so the whole midline moves up by a factor of two and so the midline is now At y equals two and so this kind of acts like the x-axis did for the standard cosine wave This function will oscillate up and down up and down Averaging at this midline and the midline is given by the vertical shift If you move the graph up or down The amount you move it up or down is where the midline will exist why you call c in that situation That's if you want to move things up and down. What if you want to move things left and right? Well to move things left and right that's a horizontal shift and that requires you do things in the Horizontal zone do do do do do do do so again We have our function f and we want to shift it left or right by a factor of c Or by the c units I should say in which case to Accomplish that you are going to take the function and put inside of it x minus c You replace the variable x with x minus c and this causes the graph to shift horizontally to the right So subtracting c from the x-coordinate actually moves everything to the right Consequently if you take y equals f of x plus c the plus c inside the horizontal zone causes the graph to move horizontally to the left So subtracting from the x actually moves it to the right and adding to the x actually moves it to the left Like I've warned you before in this lecture series In the horizontal zone things act backwards to what you might expect but think of it this way If you think of your variable x as time Imagine you're racing your little brother right in which case he probably needs a head start And so you let him start sooner another way of doing is you sort of handicap it by subtracting You know, I'll give you a 10 second handicap So like maybe if you're in golf right you could subtract some of your strokes You're basically taken away to kind of speed it up actually so subtracting c from the time is as if you had started The race sooner or I should say it's like you started farther down the track Because of this this head start that you're giving So if you wanted to graph the function y equals cosine of x minus pi force The standard cosine wave you see right here no transformations whatsoever The fact that it is negative pi force means you're going to take your picture And you're going to move it to the right by a factor of pi force. So look at these points right here So you have Uh, zero one pi have zero pi negative one three pi have zero and The last one here, of course two pi one if you move each and every one of these points By a pi four step you're going to get something like this. This point moves here This point moves here This point moves here This point moves here and then this point would move all well off the screen So you'll have to accept that and so you get these things everything got moved to the right by by this pi force Unit right like so and so shifting everything to the right is what we get with this x minus pi force If you had an x plus pi force, this would actually cause everything on the graph to shift to the left and so it's important to look out for these things when it comes to Uh horizontal or vertical shifts up down left right