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The problem asks you to transform the function \( y = \cos(x) \) through a series of steps. Let's break it down step by step:

1. **Start with the graph of** \( y = \cos(x) \).

2. **Expand the function horizontally by a factor of 8**: 
   - When you stretch a function horizontally by a factor of \( a \), you replace \( x \) with \( \frac{x}{a} \). 
   - Here, \( a = 8 \), so the function becomes:
   \[
   y = \cos\left(\frac{x}{8}\right)
   \]

3. **Shift the graph right by 2 units**: 
   - To shift a function horizontally to the right by \( b \) units, you replace \( x \) with \( x - b \). 
   - So, shift by 2 units to the right means the function becomes:
   \[
   y = \cos\left(\frac{x - 2}{8}\right)
   \]

4. **Expand the graph vertically by a factor of 4**: 
   - Vertical expansion by a factor of \( c \) means multiplying the entire function by \( c \). 
   - Here, \( c = 4 \), so the function becomes:
   \[
   y = 4\cos\left(\frac{x - 2}{8}\right)
   \]

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

Do you want any more details or further explanation on this problem?

Here are 5 related questions you might explore:
1. How would the transformation change if the horizontal expansion factor was \( \frac{1}{8} \) instead of 8?
2. What happens if you shift the graph left instead of right?
3. How would the graph change with a vertical compression rather than expansion?
4. What is the effect of reflecting the graph over the x-axis?
5. How would you express this transformation using sine instead of cosine?

**Tip:** When transforming functions, always remember that changes inside the function argument (with \( x \)) affect the horizontal transformations, while changes outside affect vertical transformations.

Start with the graph of y = cos(x). Expand the function horizontally by a factor of 8, and then shift the graph you obtained right by 2 units. Finally, expand this graph vertically by a factor of 4. Enter the resulting function into the answer box below.

Learn how to transform the cosine function y = cos(x) with a horizontal expansion by a factor of 8, a shift to the right by 2 units, and a vertical expansion by a factor of 4. This problem involves key concepts in trigonometric graph transformations. Suitable for students in Grades 10-12.

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