Let's properly match each description with the corresponding transformation of the function [tex]\( f(x) \)[/tex].
### 1. [tex]\( f(x) - 4 \)[/tex]:
- This expression represents the function [tex]\( f(x) \)[/tex] translated 4 units down. When subtracting a value from the function, it shifts the graph down by that amount.
Description: [tex]\( f(x) \)[/tex] is translated 4 units down.
### 2. [tex]\( f(x-4) \)[/tex]:
- This represents the function [tex]\( f(x) \)[/tex] translated 4 units to the right. When a function's input is replaced with [tex]\( x - a \)[/tex], it results in the function shifting to the right by [tex]\( a \)[/tex] units.
Description: [tex]\( f(x) \)[/tex] is translated 4 units right.
### 3. [tex]\( f(x) + 4 \)[/tex]:
- This represents the function [tex]\( f(x) \)[/tex] translated 4 units up. Adding a value to the function shifts the graph up by that amount.
Description: [tex]\( f(x) \)[/tex] is translated 4 units up.
### 4. [tex]\( 4 \cdot f(x) \)[/tex]:
- This represents the function [tex]\( f(x) \)[/tex] vertically stretched by a factor of 4. Multiplying the function by a constant greater than 1 stretches the graph vertically.
Description: [tex]\( f(x) \)[/tex] is vertically stretched by a factor of 4.
### 5. [tex]\( \frac{1}{4} f(x) \)[/tex]:
- This represents the function [tex]\( f(x) \)[/tex] vertically compressed by a factor of 4. Multiplying the function by a constant between 0 and 1 compresses the graph vertically.
Description: [tex]\( f(x) \)[/tex] is vertically compressed by a factor of 4.
So, the correct matching is:
- [tex]\( f(x) - 4 \)[/tex]: [tex]\( f(x) \)[/tex] is translated 4 units down.
- [tex]\( f(x-4) \)[/tex]: [tex]\( f(x) \)[/tex] is translated 4 units right.
- [tex]\( f(x) + 4 \)[/tex]: [tex]\( f(x) \)[/tex] is translated 4 units up.
- [tex]\( 4 \cdot f(x) \)[/tex]: [tex]\( f(x) \)[/tex] is vertically stretched by a factor of 4.
- [tex]\( \frac{1}{4} f(x) \)[/tex]: [tex]\( f(x) \)[/tex] is vertically compressed by a factor of 4.
