Question

How do you find the slope of the secant lines of `f (x) = x^2 − x − 42` at (−5, −12), and (7, 0)?

Answer 1

Assuming that the word "lines" should be "line", it is exactly the same as the slope of the line through the points (−5, −12), and (7, 0).

Explanation:

If my interpretation is incorrect, then perhaps you asking for the general equation of a secant line to the curve that includes the point `(-5,-12)`

If that is the question, then the form of the answer will depend on whether you call the second point `(x,f(x))` or `(a,f(a))` or somthing similar of call it `(-5+h, f(-5+h))`

For `(x,f(x))`, we find slope:
`m = (f(x)-(-12))/(x-(-5)) = (x^2-x-42 + 12)/(x+5) = (x^2-x-30)/x+5)`

`= ((x-6)(x+5))/(x+5) = x-6` (for `x != -5`)

For `(-5+h, f(-5+h))`, we find slope:

`m=(f(-5+h)-(-12))/((-5+h)-(-5))`

`= ((-5+h)^2-(-5+h)-42+12)/(-5+h+5)`

`= (25-10h+h^2+5-h-42+12)/h`

`= (-10h+h^2-h)/h = -11+h` (for `h != 0`)

The general equation of a secant line to the curve that includes the point `(7, 0)` is found by similar methods.