True or false f(x) and f ‘ (x) have the same vertical translation

Answer 1

False.

The derivative of a constant is zero, hence any information about vertical translation is lost when you differentiate a function. For example: f(x)=x² + 4 (vertically shifted upwards by four units) however, f'(x)=2x (no vertical shift)

If you have covered integrals yet, this is why you need to include a +C at the end of a indefinite integral, since derivatives eliminate any constants within the function.

Additionally you can conceptualize this with the idea that the derivative is just the slope of the tangent line. When a function is vertically translated, the slope of the tangent line at any point has not changed, and hence the derivative will not be affected by any vertical translations.