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

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Answer 1 · 2017-08-18T20:02:23

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]}