welcome in this video i will be talking about the relationship and linearization for ap physics here this is a standard linear equation that is graphed by points and they drew a line of best fit the line of best fit here is linear and the fact that it is linear you could write it in the form y equals mx plus b here you could calculate things like slope using the slope formula and you can find the equation by using the point-slope formula and so forth this is a very unique case where it is linear the situation arise in ap physics where now it is not linear so we have to deal with non-linearized graphs so we are going to do something called linearization linearization it takes a non-linear graph and makes it linear to do this there are three steps one recognize that the nonlinear equation the nonlinear relationship on the y and x graph is either quadratic inverse or etc the most common parts the quadratic looks like a parabola the inverse might look like this or the square that looks like this these are your three common ones in physics then you're going to regraph it based on its proportionality like so then after that you're going to draw a line of best fit then find the equation this is a unique skill that you have to learn but again linearization is the process of taking a non-linear graph and making it linear and there's no thinking involved here it is all memorization first let's me give you a distance versus time graph here is the time which is the x value and the y value it's just is the distance covered if you graph it and you draw a line here you see it's not straight it looks like it curves like this and we will say this behaves like a quadratic we would say this behaves like a quadratic and the fact that is behaves like a quadratic we know it looks like something like x squared so what we need to do now is linearize it so now what you're going to do the original graph was time and distance but remember we said it's quadratic so it should behave like x squared so what are we going to do well in this case the x axis we are just going to graph x squared how do you do that well remember this is what the original x was the new one is going to be x squared so the first one was zero what is zero squared which is zero next one you have one what is one here one squared is equal to one what about the second one two two squared is equal to four next one three three squared is equal to nine and so forth and so forth 4 squared because that is the fourth part is equal to 16 and the last part is 5 5 squared is equal to 25 now once you graphed it 0 1 4 9 16 25 and the distance the values here remain the same notice that this is now linear okay so this from here to here they adjusted the x value to linearize it that is the process of linearization there are general rules for linearization that will always get you to change the x-axis to match the relationship okay if it is a quadratic which will look something like this or this you want to put an x squared on the x axis so you're going to square the x values but it looks like an inverse like this right where it decays you want to put a 1 over x on the x axis then lastly if it is a if it behaves like a square root which looks like this you are going to put a square root of x or because it also you can rewrite this as x to the power of one-half on the x-axis as well okay i made your life easy um i'm going to show you an example then i'm going to give you a cheat sheet at the end okay if we draw this the radius and the speed here notice it curves like this this behaves like a square root function so what do we need to do well the radius has to be changed exactly like what we said here we have to square root the x so do you see all these values here we are going to square root it all right so what is the square root of zero so what is the square root of zero right here square root of zero is zero what is the square root of one square root of one is one what is the square root of 4 square root of 4 is equal to 2. and likewise likewise they square root 9 here square root 9 is equal to 3 then we do 16 square root 16 is equal to 4 square root 25 is equal to 5 then they square rooted the 36 to get you 6 here so notice these values all come from you square rooting it it says it right here from square rooting it once you square root it or likewise again it's it's the same thing as to the one half power and you graph it now it looks like what a straight line okay notice that's all what we did we applied or adjusted the x-axis values so that it becomes linear that's the purpose of linearization there's a cheat sheet that i gave you these are all the representations and relationships that you're going to need for ap physics okay the first two are your simple ones you might have something like this graph and the relationship is if x increases y um doesn't change so there's no relationship between the variables do you need to linearize it in any way no because this is already linear so the equation would look like y is equal to b right or any constant sometimes it might have a positive slope or a negative slope in this case if it has a slope of some kind we would say and it starts from zero we would say that x increases and y increases because it goes up so this is called directly proportioned okay this will behave like a linear equation of y is equal to mx plus b do you have to linearize it no because it's already linear here are your three common ones okay the decay one which is your in they call this inverse proportional and that behaves like this there's a one over x likewise the equation has one over x so if you want to graph this right to change this to linearization you would have to graph on the x-axis the x becomes x to the negative 1 power or you can write it as 1 over x it's the same thing and the y remains the same so this is called inverse proportional next one if it curves like this this is called squared of x okay so this to graph it the x-axis you just square it and the y axis remains the same then this you're going to linearize it okay and this one originally will look like y equals to mx squared plus c lastly the square root okay this costs proportional to the square root or you could see it as or you could do it the opposite way up to you okay to do this to linearize it you can either graph the x so you can square root the x and the y remains the same or in this case you could actually do it differently where you square the y and you have the x remain the same it could either look like the top one or the bottom one okay but these are what you need to do to the domain or the x values such that you can linearize it okay these are all supposed to be memorized for the ap exam you will have to be able to recognize the nonlinear equation learn to adjust one of the x coordinates so it becomes straight but there you go that is how you linearize and describe the relationship of functions