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Secant line with arbitrary difference (with simplification)

Sal finds and simplifies the expression for the slope of the secant line between x=4 and x=4+h on the graph of y=2x²+1.

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Video transcript

- [Voiceover] A secant line intersects the curve y = 2x squared + 1 at two points with x-coordinates 4 and 4 + h, where h does not equal 0. What is the slope of the secant line in terms of h? Your answer must be fully expanded and simplified. 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. We have x and then we have y = 2x squared + 1. So we know that when x = 4, what is y going to be equal to? It's gonna be 2(4 squared) + 1 which is the same thing as 2(16) +1 which is the same thing as 32 + 1 so it is going to be 33. What about when x = 4 + h? Well it's going to be 2(4 +h)squared + 1. That's going to be 2 times, let's see, (4 + h)squared is going to be 16 + 8h + h squared and then we have our + 1 still and if we distribute the 2 that's going to get us to 32 + 16h + 2h squared + 1 and then we add the 32 to the 1 and actually I'm gonna switch the order a little bit so I have the highest degree term first so it's going to be 2h squared + 16h and then + 32 + 1 is 33. We have these two points. We have one point (4, 33) and we have the other point (4+h, 2h squared + 16h + 33) and we just have to find the slope between these two points because the secant line contains both of these points. How do we find the slope of a line? We do change in y / change in x. What's our change in y? If we view this as the end point and this as the starting point, our change in y is going to be this minus that. It's going to be 2h squared + 16h + 33 - 33, those two are going to cancel each other out, and then over, what's our change in x? If we ended at 4 + h but then we started at 4 so it's gonna be 4 + h - 4. These two cancel each with each other and we are left with 2h squared + 16h / h. We can divide everything in the numerator and denominator by h and what are we going to get? This is going to be 1. That's just a 1. This is just an h. We have 2h + 16 / 1. Or just 2h + 16. And we're done. This is the slope of the secant line in terms of h. Once again we just have to think about well the secant line contains the point (4,f(4)) or 2 times 4 squared + 1 right over here and, well I didn't call this f(x) but I think you get the idea, and then when x is 4 + h this is going to be y and we just found the slope between these two points.