Let's analyze each transformation one by one, starting with $1 - f(-x)$.

1. $1 - f(-x)$:

This transformation involves two key components:

• Horizontal Reflection: The term $f(-x)$ indicates a reflection of the function $f(x)$ across the y-axis (a horizontal reflection).
• Vertical Shift: The term $1 - f(-x)$ means the entire function is shifted upwards by 1 unit (since we subtract $f(-x)$ from 1).

So, the transformation consists of:

1. Reflecting the graph of $f(x)$ across the y-axis.
2. Shifting the resulting graph upward by 1 unit.

2. $5 - \frac{1}{6}f(x)$:

This transformation involves a vertical stretch and a vertical shift:

• Vertical Compression: The term $\frac{1}{6}f(x)$ compresses the graph of $f(x)$ vertically by a factor of $\frac{1}{6}$ (making it 6 times shorter in height).
• Vertical Shift: The term $5 - \frac{1}{6}f(x)$ indicates a vertical shift of the function upward by 5 units.

So, the transformation consists of:

1. Compressing the graph of $f(x)$ vertically by a factor of $\frac{1}{6}$.
2. Shifting the resulting graph upward by 5 units.

Summary:

• $1 - f(-x)$: Reflect the graph across the y-axis, then shift up by 1.
• $5 - \frac{1}{6}f(x)$: Compress the graph vertically by $\frac{1}{6}$, then shift up by 5.

Would you like further details on any part of this explanation or examples with graphs?

Here are 5 related questions to explore:

1. What happens to the domain and range of $f(x)$ under these transformations?
2. How would the transformation $-f(x)$ affect the graph of $f(x)$?
3. Can horizontal shifts be combined with these transformations? If so, how?
4. How does a vertical stretch by a factor of 2 differ from a compression by $\frac{1}{6}$?
5. What are the combined effects of reflections and shifts on the symmetry of $f(x)$?

Tip: Always apply horizontal transformations (like reflections or shifts) before vertical ones when graphing functions.