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To analyze the transformations of the function \( m(x) \) that result in the function \( n(x) = 5 - m(-x) \), we can break down the expression into steps to understand the sequence of transformations:

### 1. Reflected over the y-axis:
The function \( m(-x) \) represents a reflection of the original function \( m(x) \) over the y-axis. This means that every point on the graph of \( m(x) \) is reflected across the y-axis to produce the graph of \( m(-x) \).

### 2. Vertical Reflection and Vertical Shift:
The expression \( 5 - m(-x) \) further transforms \( m(-x) \) by first applying a vertical reflection and then a vertical shift:
- **Vertical reflection:** The term \( -m(-x) \) indicates a reflection over the x-axis. It changes the sign of all y-values of \( m(-x) \).
- **Vertical shift:** Adding 5, as in \( 5 - m(-x) \), shifts the entire graph upward by 5 units.

### Combined Effect:
- **Reflection over the y-axis** followed by **reflection over the x-axis** and a **vertical shift upwards by 5 units**.

Thus, the function \( n(x) \) is obtained by reflecting the function \( m(x) \) over the y-axis, then reflecting it over the x-axis, and finally shifting it upward by 5 units.

### Would you like more details on any of these steps or have other questions?

Here are some additional related questions you might find useful:

1. How does reflecting a function over the y-axis affect its graph?
2. What happens to the graph of a function when it is shifted vertically?
3. What are other common transformations of functions?
4. How can we describe a function's transformation algebraically?
5. How do multiple transformations affect the final shape of a graph?

**Tip:** Always apply transformations in the correct order to understand how the graph changes step-by-step.

Let n be a function that is a transformation of the function m such that n(x) = 5 - m(-x). Describe the transformations of the function m that result with the function n.

Explore the transformation of the function m to the function n where n(x) = 5 - m(-x). This includes a reflection over the y-axis, a reflection over the x-axis, and a vertical shift. Suitable for high school students in Grades 9-12.

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