Let's work through each part step by step:

(a) Describe the sequence of transformations from $f(x) = \sin(x)$ to $g(x) = \sin(2x - \pi)$

We can break down the transformation process into two parts:

1. Horizontal compression by a factor of 2: The coefficient of $2$ in front of $x$ inside the sine function indicates a horizontal compression by a factor of $\frac{1}{2}$. This affects the period of the sine function, making it half of the original period.

2. Horizontal shift to the right by $\frac{\pi}{2}$: The $- \pi$ inside the argument of the sine function results in a horizontal shift. Specifically, since $g(x) = \sin(2x - \pi)$, this corresponds to a shift to the right by $\frac{\pi}{2}$.

Thus, the transformation is:

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

(b) Sketch the graph of $g(x)$

Based on the transformations in part (a):

• The horizontal compression by 2 means the period of the sine wave is now $\frac{2\pi}{2} = \pi$ instead of $2\pi$.
• The horizontal shift to the right by $\frac{\pi}{2}$ means the entire sine curve starts $\frac{\pi}{2}$ units later along the x-axis.

The key points on the graph:

• One cycle of the sine curve begins at $x = \frac{\pi}{2}$, goes down, reaches a minimum at $x = \frac{\pi}{2} + \frac{\pi}{4} = \frac{3\pi}{4}$, goes up through the x-axis at $x = \frac{\pi}{2} + \frac{\pi}{2} = \pi$, and continues similarly.

The new period is $\pi$, and the amplitude remains 1 (since there is no vertical scaling).

(c) Use function notation to write $g(x)$ in terms of $f(x)$

The parent function is $f(x) = \sin(x)$, and we need to express $g(x)$ in terms of $f$.

Since $g(x) = \sin(2x - \pi)$, and $f(x) = \sin(x)$, we can express $g(x)$ as: $g(x) = f(2x - \pi)$

This shows that $g(x)$ is simply the sine function $f(x)$, with the argument $x$ replaced by $2x - \pi$.

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Do you want further details or have any questions?

Here are five related questions to deepen your understanding:

1. How does a horizontal compression affect the period of any trigonometric function?
2. What is the difference between a horizontal shift and a vertical shift in trigonometric graphs?
3. How would the graph change if we added a vertical shift to $g(x)$?
4. What would be the period of $g(x) = \sin(3x - \pi)$?
5. How would you express $g(x) = \cos(2x - \pi)$ in terms of $f(x) = \cos(x)$?

Tip:

For horizontal transformations, remember that a multiplier inside the function (like $2x$) affects the period, while a constant shift (like $-\pi$) affects the phase.