https://file.aiquickdraw.com/mathbot/file/1726869421958cgtdv46f.jpg

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.

Given the function g(x) = sin(2x - π), describe the sequence of transformations from f(x) = sin(x), sketch the graph of g, and write g in terms of f.

Learn how to describe transformations applied to the sine function g(x) = sin(2x - π). This involves a horizontal compression and phase shift, along with graphing the function. Suitable for high school trigonometry students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation