In the graph, we have four points: A=(-2,2), B=(-1,1), C=(2,0) and D=(7,-1)
At the segment AB, we have the function f(x). Its slope is (1 - 2)/[-1 -(-2)] = -1
Then, it can be expressed in the form f(x) = -x + b
Replacing x with -1, we got: 1 = 1 + b, which implies b = 0
Therefore, f(x) = -x
At the segment BC, we have the function f'(x). Its slope is (0 -1)/[2 -(-1)] = -1/3.
From this, we have: f'(x) = -x/3 + b
Replacing x with 2, we got -2/3 + b = 0, which implies b = 2/3
Therefore, function f'(x) = -x/3 + 2/3 is the function f(x) turned left with an angle of arctan(1/3) and turned up by 2/3 units.
From the segment CD, we have the function f''(x). Its slope is -1/(7-2) = -1/5
Then, it can be expressed in the form f''(x) = -x/5 + b
Replacing x with 2, we got -2/5 + b = 0, which implies b = 2/5
Therefore, the function f''(x) = -x/5 + 2/5 is the function f(x) turned left with an angle of arctan(1/