Let's analyze the problem displayed in the image.

The given function is:

$g(x) = \sin(2x - \pi)$

Part (a): Describe the sequence of transformations from $f(x) = \sin(x)$ to $g(x)$.

1. Horizontal Stretch/Compression: The coefficient of $2$ inside the sine function causes a horizontal compression by a factor of $\frac{1}{2}$. This means the period of the sine wave is halved.

2. Phase Shift: The term $-\pi$ inside the argument $2x - \pi$ represents a phase shift. To determine how much the graph is shifted: $2x - \pi = 0 \quad \text{when} \quad x = \frac{\pi}{2}$ This results in a shift of $\frac{\pi}{2}$ units to the right.

So, the sequence of transformations is:

• A horizontal compression by a factor of 2.
• A phase shift of $\frac{\pi}{2}$ units to the right.

Part (b): Sketch the graph of $g(x)$.

The graph shown depicts a sine wave with a compressed period (horizontal compression by a factor of 2) and a shift to the right by $\frac{\pi}{2}$. The period of the graph is $\frac{2\pi}{2} = \pi$, and it is shifted accordingly.

Part (c): Write $g(x)$ in terms of $f(x)$.

We know that $f(x) = \sin(x)$, and the transformation can be written using function notation.

To express $g(x)$ in terms of $f(x)$: $g(x) = f(2x - \pi)$ Since $f(x) = \sin(x)$, the transformation applies directly to the argument inside the sine function.

Would you like further clarification or additional details? Here are some related questions:

1. What is the effect of a vertical stretch on a sine function graph?
2. How do you calculate the period of a sine function after applying transformations?
3. What are the steps for determining phase shifts in trigonometric functions?
4. How does a horizontal stretch differ from a horizontal compression in trigonometric graphs?
5. Can you describe the difference between vertical and horizontal transformations?

Tip: When working with sine and cosine transformations, always check for both horizontal (period and phase shift) and vertical (amplitude and vertical shift) changes.