### Instruction: Find the phase shift of the graph of $y = 2 \sin \left( 2x + \frac{\pi}{3} \right).$.### Response: I recall that the phase shift of a sine or cosine graph is the horizontal displacement from the standard position, which is when the graph passes through the origin.
To find the phase shift, I need to identify the value of $x$ that makes the argument of the sine function equal to zero, since that is when the sine function is also zero.
So I set $2x + \frac{\pi}{3} = 0$ and solve for $x$.
Subtracting $\frac{\pi}{3}$ from both sides, I get $2x = -\frac{\pi}{3}$.
Dividing both sides by 2, I get $x = -\frac{\pi}{6}$.
This means that the graph of $y = 2 \sin \left( 2x + \frac{\pi}{3} \right)$ is shifted to the left by $\frac{\pi}{6}$ units from the standard position.
Therefore, the phase shift is $-\frac{\pi}{6}$.