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CCSS.Math:

why equals 1/8 X to the third minus x squared over which interval does y of X have an average rate of change of 1/2 so let's go interval by interval and calculate the average rate of change so first let's think about this interval right over here between negative x is between negative 2 & 2 so negative 2 is less than X which is less than 2 so let's just think about what is the value of our function when X is equal to negative 2 so Y of negative 2 is equal to 1/8 times negative 2 to the third power minus negative 2 squared which is equal to let's see this is 1/8 of to the 3rd power is negative 8 negative 8 divided by 8 is negative 1 negative 2 squared is positive 4 but then you're going to have to subtract that so it's minus 4 so this is equal to negative 5 and Y of 2 y of 2 is equal to 1/8 times 2 to the 3rd power minus 2 squared minus 2 squared and that's going to be equal to a 1 8 times 8 is 1 minus 4 which is equal to negative 3 so if you want to find your average rate of change you want to figure out how much does you the value of your function change and divide that by how much your X Y or your X has changed so just we could make a table here we could make a table here X Y when X is negative 2 y is negative 5 when X is positive 2 y Y is negative 3 so how much did your why change well your Y increased your Y increased by 2 your Y increased by 2 and your x increased by 4 your X increased by 4 and you could get these numbers you can just look at it the Y increased from that point to that point X increase from that point to that point or you could say hey negative 3 minus negative 5 is positive 2 that's the difference between negative 3 and negative 5 if you said two minus negative four well that gives you once again the distance it would get or the difference it would give you positive four but if you look at here it's clear when Y increased by two or we could say when X increased by four Y increased by two so our average rate of change over this interval is going to be average average rate of change of Y with respect to X with respect to X is going to be equal to well when X changed by four by positive for y changed by positive two so it's equal to 1/2 so it does look like the average rate of change over this interval right over here of this interval right over here is 1/2 so we got lucky in this situation our first choice and this is a multiple choice so we don't have to it's not a multi-select here so our first one met our criteria so we know that that's the answer but let me do one more of these other ones to show you why that is not the answer so let's find the average rate of change between that point and that point so let's do another let me do it in another color so I'll do this one in purple so 0 is less than X which is less than 4 and I'll just do the table right over here X and y so when x is 0 what is y well it's going to be 1/8 times 0 minus 0 y is just going to be 0 when X is 4 what is y well Y is going to be let's see it's going to be 1/8 try to do this in my head 1/8 times 4 to the 3rd 4 to the third is 64 1/8 of 64 is 8 it's going to be 8 minus 4 squared which is 16 8 minus 16 is negative 8 negative 8 so in this one when we have X is increasing by 4 X has increased by 4 what happened to why y has decreased by 8 so the average rate of change of Y with respect to X here is why changed so I could write my change in Y this Greek letter Delta just slowly shorthand for change in my change in Y is negative eight when my change in X is four so the average rate of change here is negative two it's negative because as X increased your Y decreased and for every on average for every one that X increased Y decreased by two that's where you get your negative two here so this clearly the average rate of change is not one-half so that confirms that that is not the answer and we know that these other two and I encourage you to try them out we'll also give you an average rate of change of something other than positive 1/2