Question

Graph 3 members of the family of curves that have f(x) = -x+1 as a common derivative. Help!?

Answer 1

If `f(x) = -x + 1` is the derivative of three different equations, then the easiest way to accomplish this is to have them all differ by a constant.

So, try integrating the function and setting the arbitrary constant to something arbitrary.

`int f(x)dx = int -x + 1 dx`

`= -x^2/2 + x + C`
(power rule)

This constant `C` can differ, so we can write an infinite number of members of the quadratic family of curves:

`F(x) = -x^2/2 + x + 1`
`G(x) = -x^2/2 + x + 2`
`H(x) = -x^2/2 + x + 3`
`vdots`
`" "" "= -x^2/2 + x + C`

These will be offset from each other as a result of vertical shifts. Here are the graphs for `C = 1` through `C = 3`:

graph{(-x^2/2 + x + 1 - y)(-x^2/2 + x + 2 - y)(-x^2/2 + x + 3 - y) = 0 [-5, 7, -2.75, 6]}