https://file.aiquickdraw.com/mathbot/file/17293497158078udgyqfu.jpg

The text in the image asks to **transform the graph of** \( y = \sin(x) \) to show a new function with:

1. **Smaller amplitude**
2. **Larger period**
3. **Phase shift** of \( -\pi \)

To achieve this transformation, we can modify the general form of the sine function:

\[
y = A \sin(B(x - C)) + D
\]

Where:
- \( A \) affects the **amplitude**.
- \( B \) affects the **period**, which is calculated as \( \frac{2\pi}{B} \).
- \( C \) is the **phase shift** (horizontal shift).
- \( D \) affects the **vertical shift** (not relevant here).

Given the conditions:
1. **Smaller amplitude**: \( A < 1 \). If the original amplitude is 1 (for \( \sin(x) \)), we choose \( A \) to be smaller, such as \( A = \frac{1}{2} \).
2. **Larger period**: The period is given by \( \frac{2\pi}{B} \), so for a larger period, \( B \) should be less than 1. For example, if \( B = \frac{1}{2} \), the period would be \( 2\pi \times 2 = 4\pi \), which is larger than the original \( 2\pi \).
3. **Phase shift** of \( -\pi \): The phase shift means \( C = \pi \), since the function shifts to the right by \( C \).

Thus, the transformed equation would be:

\[
y = \frac{1}{2} \sin\left(\frac{1}{2}(x + \pi)\right)
\]

This satisfies:
- **Amplitude**: \( \frac{1}{2} \) (smaller than 1),
- **Period**: \( 4\pi \) (larger than the original \( 2\pi \)),
- **Phase shift**: \( -\pi \) (shifted to the left by \( \pi \)).

Would you like further details or have any other questions?

Here are 5 related questions:
1. How does changing the amplitude affect the shape of the sine wave?
2. How do you calculate the period of a sine function?
3. What is the difference between a phase shift and a vertical shift in trigonometric functions?
4. Can the amplitude of a sine function ever be negative?
5. How would a vertical shift change the equation?

**Tip**: Always remember that for a sine function, the period is inversely related to the coefficient \( B \) in front of \( x \). A smaller \( B \) gives a larger period.

The graph of y = sin(x) is shown below. Transform the graph to show a new function with a smaller amplitude, a larger period, and a phase shift of -π.

Learn how to transform the sine function y = sin(x) by changing its amplitude, period, and phase shift. This problem involves key trigonometric concepts and graph transformations. Suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation