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a secant line intersects the graph of f of X is equal to x squared plus 5x at two points with x coordinates 3 and T where T does not equal 3 what is the slope of the secant line in terms of T your answer must be fully expanded and simplified and my apologies ahead of time if I'm a little out of breath I just try to do some exercises in my office to get some blood moving and I'm I think I'm still a lot of breath anyway so we want to find the slope of the secant line and they essentially give us two points in the secant line they tell us what X is in each of those two points and then if we know what it X is we're able to figure out what f of X is at each of those points so we could make a little table here we know X and we know f of X so when X is equal to 3 what is f of X well it's going to be 3 squared plus 5 times 3 well this is going to be 9 plus 15 which is 24 so this is going to be 24 and when X is equal to T what is f of T well it is going to be T squared plus 5t and so we have two points now that are on this line this is secant line it intersects our function twice so it has these two points on it and so we just have to find our change in Y between these two points change in Y and our change in X change in X and I'm assuming that Y is equal to f of X so our slope of our secant line is change in Y over change in X our change in Y if we view this as our end point the second one with the t's it is our end point it's going to be that minus that so it's going to be T squared plus 5t minus 24 and then in our denominator are ending x minus our starting x is going to be t minus 3 now they tell us our answer must be fully expanded and simplified so maybe there's a way to simplify this a little bit let's see can i factor the top into something that involves that involves a t minus three alright so in the numerator see negative 3 times positive 8 is negative 24 negative 3 plus positive 8 is 5 so we can rewrite this as t plus 8 times t times t minus 3 and so we could say this is going to be equal to if we cancel out the t minus 3s or do we divide the numerator and the denominator by t minus 3 it's going to be equal to t plus 8 now if we wanted to be really strict mathematically strict strict this expression isn't exactly the same as our original expression right over here what makes them different well they're going to be true for all t's except where t equals 3 this thing right over here is defined at t equals 3 in fact when t equals 3 this expression is equal to 11 but this thing up here was not defined at t equals 3 so if you wanted to be particular about it if you want this expression to be the exact same thing you would say you would say for t does not equal 3 now this can take the same inputs as this one right over right over there but I guess they're assuming where T does not equal 3 so this you could view this as maybe a little bit redundant but this would be this is the slope of the secant line in terms of T