Which statement fully describes $g(x)=-2\tan(3(x-\pi/6))+4$ as a transformation of $f(x)=\tan x$?

Answer and derivation

Correct answer: A

Here $a=-2$, $b=3$, $h=\pi/6$, and $d=4$, giving the reflection, dilations, and right/up shifts in choice A.

1. The factor -2 reflects the tangent graph across the x-axis and vertically dilates it by 2.
2. The factor 3 inside the argument horizontally compresses the graph by 3, so its period is $\pi/3$.
3. The factored input $x-\pi/6$ shifts right $\pi/6$, and +4 shifts the inflection line up to $y=4$.

Why the other options fail

A

Correct interpretation of the outside factor, inside factor, factored shift, and vertical shift.

B

This reverses the effects and directions of every parameter.

C

The inside factor divides the parent period, and the factored shift is already π/6.

D

This ignores the negative outside factor and reverses the vertical shift.

Common misconception: For tangent transformations, the inside multiplier changes horizontal distances and period reciprocally; it is not itself the period.

China/AP bridge: China-course bridge: function transformations are familiar; AP newly expects students to coordinate additive and multiplicative tangent transformations; unlike reading signs term by term, use factored a·tan(b(x-h))+d; a common pitfall is treating b as the period or reversing h.