The function `f` is defined by `f(x) = (x - 5)^2 + 8`. Which of the following is an equivalent form of the function `f`?

Correct answer: A. `f(x) = x^2 - 10x + 33`

The correct answer is A. The given function is `f(x) = (x - 5)^2 + 8`. To find an equivalent form, first expand the `(x - 5)^2` term. This is equivalent to `(x - 5)(x - 5)`, which expands to `x^2 - 5x - 5x + 25`, or `x^2 - 10x + 25`. Now substitute this back into the original function definition: `f(x) = (x^2 - 10x + 25) + 8`. Combining the constant terms `25 + 8` gives `33`. Therefore, the equivalent form is `f(x) = x^2 - 10x + 33`. B is incorrect. This results from incorrectly expanding `(x - 5)^2` as `x^2 + 25` and then adding 8. C is incorrect. This is the result of correctly expanding `(x - 5)^2` but forgetting to add the constant term 8 from the original function. D is incorrect. This results from incorrectly expanding `(x - 5)^2` as `x^2 - 25` and then adding 8.