How do I find the slope of a secant line for a given equation, but in terms of x and Δx?

Question

The slope of the secant line containing the two points (x, f(x)) and (x + Δx, f(x+Δx)) on the graph of a function y=f(x) may be written as (f(x+Δx)-f(x))/Δx. Find the slope of a secant line of f(x)=5x-x^2. Your answer will be in terms of x and Δx. Do simplify your answer.

My main problem is that I thought that I could just find two points in the function and then find the slope of the line between then, but apparently that is not correct. I'm not sure how else I should do this problem. Thanks!

Answer 1

The issue with your approach is that the line for the two points you picked would work... but only for those two points. The problem wants a more generic answer that would work for any two points...

They actually give you the formula that they want you to use ... (f(x+Δx)-f(x))/Δx.

So you need to find f(x+Δx)

This is a composite function... do you remember those? well you would put x+Δx into all the x's in f(x). Its quite a bit of algebra...

f(x)=5x-x^2 so...

f(x+Δx) = 5 (x+Δx) - (x+Δx) ²

so then you would distribute and foil and combine like terms and .....

so once you have that then you would plug it all into that formula and do some more simplification... that should get you there