Cubic Transformations - Graph shown is best represented by the equation:

Question

The answer is out of B or D:

$f(x) = -(x-a)^3 + b$ because turning point at $(0,0)$ is now $(a,b)$

OR

$f(x+a)^3$ because "$a$" is implied as negative... $(0,0)$ is now $(-a,b)$

The textbook answer is D, but my teachers argue B

Answer 1

Based on the way the picture is drawn and the options give, teacher is correct. What is usually seen in graph transformation problems is actual numbers. So if the turning point had been written out as $(-3,5)$, then the resulting function would have been $f(x) = -(x-(-3))^3+5 = -(x+3)^3+5$.

Answer 2

Take the function $y=x^3$.Transformations:- 1) The graph is inverted.So equation is $y=-x^3$ 2)Shift the graph to the left by $a$ units.So equation is $y=-(x-a)^3$ 3)Shift the graph up by $b$ units.So equation is $y-b=-(x-a)^3 \implies y=-(x-a)^3+b$ which is option B