https://file.aiquickdraw.com/mathbot/file/17264584320512vldht9t.jpg

The problem provided asks to modify the graph of \( y = \cos(x) \) in three steps:

1. **Horizontal Expansion by a Factor of 8:**  
   This affects the period of the cosine function. To horizontally stretch a function by a factor of \( k \), replace \( x \) with \( \frac{x}{k} \) in the function. Here, \( k = 8 \), so we replace \( x \) with \( \frac{x}{8} \).  
   The equation becomes:  
   \[
   y = \cos\left(\frac{x}{8}\right)
   \]

2. **Shift Right by 2 Units:**  
   A horizontal shift is achieved by replacing \( x \) with \( x - c \), where \( c \) is the number of units to shift. Since we're shifting the graph right by 2 units, we replace \( x \) with \( x - 2 \).  
   The equation becomes:  
   \[
   y = \cos\left(\frac{x - 2}{8}\right)
   \]

3. **Vertical Expansion by a Factor of 4:**  
   Vertical expansion is achieved by multiplying the entire function by a factor of \( a \). Since the vertical expansion is by a factor of 4, we multiply the function by 4.  
   The final equation becomes:  
   \[
   y = 4 \cos\left(\frac{x - 2}{8}\right)
   \]

So, the resulting function is:
\[
y = 4 \cos\left(\frac{x - 2}{8}\right)
\]

Would you like a detailed explanation of any step? Here are 5 related questions to deepen your understanding:

1. How does a horizontal stretch affect the period of a trigonometric function?
2. What happens to the amplitude when you apply a vertical stretch?
3. How would you shift the graph of \( \sin(x) \) instead of \( \cos(x) \)?
4. What effect does a horizontal shift have on the phase of a cosine function?
5. What are the differences between horizontal and vertical transformations?

**Tip:** When applying transformations, it's important to work one step at a time and ensure you correctly follow the order of operations!

Start with the graph of y = cos(x). Expand the function horizontally by a factor of 8, shift the graph right by 2 units, and expand vertically by a factor of 4. Enter the resulting function.

Learn how to transform the graph of y = cos(x) by expanding it horizontally by a factor of 8, shifting it right by 2 units, and expanding it vertically by a factor of 4. Understand key concepts in graph transformations for trigonometric functions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation