The function $g(x) = \sin(4x - \pi)$ is a transformation of the parent function $f(x) = \sin(x)$. We can express $g(x)$ in terms of $f$ using function notation as follows:

$g(x) = f(4x - \pi)$

Explanation:

• The transformation inside the sine function, $4x - \pi$, represents a horizontal compression by a factor of 4 and a phase shift of $\frac{\pi}{4}$ to the right.
• Since the original function is $f(x) = \sin(x)$, replacing $x$ with $4x - \pi$ gives the transformed function.

Would you like a deeper explanation of these transformations?

Here are five related questions for practice:

1. How does the coefficient 4 inside the sine function affect the graph of $f(x)$?
2. What is the phase shift of $g(x) = \sin(4x - \pi)$ and how is it calculated?
3. How would you express $g(x) = \sin(4x - \pi)$ in terms of cosine?
4. What is the period of $g(x)$, and how does it compare to $f(x) = \sin(x)$?
5. If $h(x) = \sin(2x + \frac{\pi}{3})$, how would you express $h(x)$ in terms of $f(x)$?

Tip:

To determine phase shifts in trigonometric functions, solve $4x - \pi = 0$ for $x$. The solution gives the shift in the positive $x$-direction.