- So i am here at desmos.com, i m sorry is an online graphing calculator, and the score of this video is to explore how shifts in features happen. How do things transition to the best or left or exactly how do they change up and also down? and what we're going to start off law is simply graph a plain vanilla function, f that x is same to x squared. The looks as we would intend it come look, but now let's think about howwe can change it up or down. Fine one assumed is, well, to change it up, we just need to make thevalue of f the x greater so we can include a value, and that walk look likeit change it up by one. Every little thing f the x was before, we're now including one to it for this reason it shifts the graph increase byone, that's quite intuitive. If us subtract one, or actually, let's subtract three. Notice, it shifted it down. The vertex was rightover below at zero, zero. Now it is at zero, negative three, therefore it change it down. And we can set up a slider here to make that a little bit clearer, therefore if I just replace this with, if I simply replace thiswith the change k, then let me delete this small thing here, that little subscript thing that happened. Then us can include a slider k here, and this is just enabling usto collection what k is same to, therefore here, k is same to one, therefore this is x squared to add one, and notice, we have shifted up, and if we boost the value of k, an alert how it shifts the graph up, and as we decrease the worth of k, if k is zero, we're ago where our vertex is best at the origin, and as us decrease the value of k, it move our graph down. And that's quite intuitive, 'cause we're including or subtractingthat amount come x squared so it changes, we might say the y value, it shifts it increase or down. Yet how execute we change tothe left or come the right? for this reason what's amazing hereis to transition to the left or the right, we can replace our x through an x minus something, for this reason let's see exactly how that could work. Therefore I'm gonna replace our x through an x minus, let's replaceit through an x minus one. What perform you think is going come happen? carry out you think that's goingto transition it one to the appropriate or one come the left? for this reason let's simply put the one in. Well, that's interesting. Before, our vertex to be at zero, zero. Now our vertex is at one, zero. So by instead of our x through an x minus one, we actually shifted one to the right. Now why does the make sense? Well, one means to think around it, before we placed this x, prior to we changed ourx with an x minus one, the vertex was as soon as we were squaring zero. Now, in order to square zero, squaring zero happenswhen x is equal to one. As soon as x is equal to one,you execute one minus one, you obtain zero, and then that'swhen you are squaring zero. So it makes sense the youhave a comparable behavior of the graph at the vertexnow when x amounts to one as prior to you had when x equals zero. And to see exactly how this have the right to be generalized, let's put an additional variable here and let's include a slider because that h. And also then we can see thatwhen h is zero and k is zero, our role is reallythen just x squared, and then if h increases, we room replacing our x withx minus a larger value. That's moving to the right and also that is, as h decreases, together it i do not care negative, that shifts to the left. Now right here, h isequal to negative five. You frequently won't seex minus an unfavorable five. Girlfriend would view that composed as x to add five, therefore if you change yourx's v an x add to five, that in reality shifts everythingfive devices to the left. And also of course, we can transition both of lock together, like this. So here, we're changing it up, and also then us are, we can get earlier to ourneutral horizontal change and then us can shift itto the right favor that. And also everything us did just now is v the x squaredfunction together our core function, but you might do that withall kinds of functions. You could do it through anabsolute worth function. Let's execute absolute value,that's always a fun one. So instead of squaring all this business, let's have actually an absolute value here. So I'm gonna put an absolute, whoops. Pure value, and also there you have actually it. You can start at, let memake both of this variables same to zero, so thatwould simply be the graph of f the x is same to theabsolute value of x. Yet let's to speak you wanted to shift it so that this allude right overhere that's at the origin is in ~ the suggest negativefive, negative five, which is appropriate over there. So what friend would execute isyou would replace your x through x add to five, or you would make this h variable to an unfavorable five rightover here, 'cause notice, if you replace your hwith a an adverse five, inside the absolute value,you would have an x add to five, and then if you desire to change it down, you just reduce the worth of k, and if you desire to shift it under by five, you mitigate it by five, and also you might get something prefer that.

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So ns encourage you, walk to desmos.com. Shot this the end for yourself, and really pat aroundwith these functions to give yourself anintuition of exactly how things and also why things change up or down as soon as you add a constant, and also why things transition tothe left or the right once you replace your x'swith one x minus, in this case, an x minus h, however it really can be xminus some form of a constant.