Calculus question

Question

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

Answer 1

Integrating the given function, we get y = -(1/2)x² + x + C, where C is an arbitrary constant.

 

The set of all curves of the above form is called a family of curves, one curve for each value of C.

 

3 members of the family:   y = -(1/2)x² + x   (C = 0)

 

                                       y = -(1/2)x² + x + 1    (C = 1)

 

                                       y = -(1/2)x² + x - 2     (C = -2)

 

Answer 2

The antiderivative of f(x) is -x²/2 + x to which I can add any constant c (since the derivative of a constant is 0).  So three curves could be:

 

y = (-1/2)x² + x

y = (-1/2)x² + x + 1

y = (-1/2)x² + x + 2

 

These are three quadratic equations whose graphs are parabolas.  For help graphing parabolas, click on the following link:

 

https://www.wyzant.com/resources/files/504633/how_to_graph_a_quadratic_equation