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a secant line intersects the curve y is equal to two x squared plus one at two points with x coordinates four and four plus h where H does not equal zero what is the slope of the secant line in terms of H your answer must be fully expanded and simplified so we know the two points that are on the secant line it might not be obvious from how they wrote it but let's make a little table here to make that a little bit clearer so we have X and then we have Y which is equal to two x squared plus one and so we know that when X is equal to four well what is y going to be equal to well it's going to be 2 times 4 squared plus one which is the same thing as 2 times 16 plus one which is the same thing as 32 plus one so it is going to be 33 33 and what about when X is equal to 4 plus h 4 plus h well it's going to be 2 times 4 plus h squared plus 1 well that's going to be 2 times let's see 4 plus h squared is going to be 16 plus 2 times 4 H so it's going to be 8 h plus h squared and then we have our plus 1 still and if we distribute the 2 that's going to get us to 32 plus 16h plus 2 h squared plus 1 and then we add the 32 to the 1 and actually I'm going to switch the order a little bit so I have the highest degree term first so it's going to be 2 H squared plus 16 H and then plus 32 plus 1 is 33 and so we have these two points we have one point 4 comma 33 and we have the other point 4 + H comma 2 H squared plus 16 h plus 33 and we have to find the slope between these two points we the secant line contains both of these points so how do we find the slope of a line well we do a change in Y over change in X so what's our change in Y well if we view this as the end point and this is the starting point our change in Y is going to be this minus that so it's going to be 2h squared plus 16h plus 33 - 33 - 33 those two are going to cancel each other out and then over what's our change in X well if we ended at 4 + H but that we started at 4 it's going to be 4 + H - 4 4 + H minus 4 these two cancel with each other and we are left with 2h squared plus 16h 16 H over H over H well we can divide everything in the numerator in the denominator by H and what are we going to get this is going to be 1 that's just a 1 and so this is just an H so we have 2h plus 16 over 1 or just 2 h plus 2h plus 16 and we're done this is the slope of the secant line in terms of H and once again we just have to think about well the secant line contains the point 4 comma f of 4 or 2 times 4 squared plus 1 right over here and then 4 well I didn't call this f of X but I think you get the idea and then when X is 4 + H well this is going to be Y and we just found the slope between these two points