The problem asks us to determine whether the following statement is true or false:

The graph of $y = \cos \left( x + \frac{3\pi}{8} \right)$ lies $\frac{3\pi}{8}$ units to the left of the graph of $y = \cos(x)$.

Explanation:

The equation $y = \cos \left( x + \frac{3\pi}{8} \right)$ represents a phase shift of the standard cosine function, $y = \cos(x)$.

• A phase shift to the left happens when we have $y = \cos(x + c)$, where $c > 0$.
• A phase shift to the right occurs when $y = \cos(x - c)$.

In this case, $x$ is replaced with $x + \frac{3\pi}{8}$, which indeed shifts the graph to the left by $\frac{3\pi}{8}$.

Therefore, the original statement is true.

Answer:

• The correct answer is True.

Would you like more details on phase shifts or a deeper breakdown of this concept? Here are some follow-up questions to help reinforce your understanding:

1. What is the general rule for phase shifts in trigonometric functions?
2. How would the graph change if the equation was $y = \cos \left( x - \frac{\pi}{4} \right)$?
3. Can you describe how vertical shifts differ from phase shifts in trigonometric graphs?
4. What happens to the amplitude of the cosine function if we modify the equation to $y = 2\cos(x + \frac{\pi}{4})$?
5. How would the graph of $y = \cos(x)$ change if we applied a horizontal stretch instead of a phase shift?

Tip: Always remember that adding a constant inside the function argument causes horizontal shifts, while adding a constant outside affects vertical shifts.