The graph of y = sin(2x + pi/6) is translated pi/12 units to the left. Write the equation of the new graph.

Step-by-step solution

1. 1

   Recall the rule for a horizontal shift. Translating a graph $h$ units to the left means replacing $x$ by $x+h$ throughout; shifting right would use $x-h$. The direction feels backwards precisely because a leftward move must make the function reach each output at a smaller $x$.

2. 2

   Substitute $x\to x+\dfrac{\pi}{12}$. With $h=\tfrac{\pi}{12}$:

   $y=\sin\!\left(2\left(x+\frac{\pi}{12}\right)+\frac{\pi}{6}\right).$

   The substitution goes inside the entire argument, so the coefficient $2$ multiplies the shift as well — this is where the answer stops being obvious.

3. 3

   Expand the argument. Distributing the $2$:

   $2\left(x+\frac{\pi}{12}\right)=2x+\frac{2\pi}{12}=2x+\frac{\pi}{6},$

   so the argument becomes

   $2x+\frac{\pi}{6}+\frac{\pi}{6}=2x+\frac{\pi}{3}.$

   A shift of $\tfrac{\pi}{12}$ therefore changes the phase by $\tfrac{\pi}{6}$ — twice the shift, because the angular frequency is $2$.

4. 4

   Write the new equation.

   $y=\sin\!\left(2x+\frac{\pi}{3}\right).$

   The amplitude ($1$) and the period ($\tfrac{2\pi}{2}=\pi$) are unchanged; only the phase moves, as a pure translation must.

5. 5

   Check at two points. At $x=0.3$: the shifted original gives $\sin\!\left(2(0.3+\tfrac{\pi}{12})+\tfrac{\pi}{6}\right)=0.9970828$ and the answer gives $\sin\!\left(0.6+\tfrac{\pi}{3}\right)=0.9970828$ ✓. At $x=1.1$ both give $-0.1054087$ ✓. A useful cross-check on the direction: writing the original as $\sin\!\left(2\left(x+\tfrac{\pi}{12}\right)\right)$ shows its own left-shift is $\tfrac{\pi}{12}$, so the new graph is shifted $\tfrac{\pi}{6}$ left of $\sin 2x$.