To analyze how the graph of [tex]\( g(x) = \frac{1}{x+4} - 8 \)[/tex] compares to the graph of the parent function [tex]\( f(x) = \frac{1}{x} \)[/tex], we should observe the transformations applied to [tex]\( f(x) \)[/tex] to obtain [tex]\( g(x) \)[/tex].
1. Horizontal Shift:
Look at the [tex]\( x+4 \)[/tex] inside the function:
- In [tex]\( f(x) = \frac{1}{x} \)[/tex], [tex]\( x \)[/tex] is the variable.
- In [tex]\( g(x) = \frac{1}{x+4} \)[/tex], the [tex]\( x \)[/tex] has been replaced by [tex]\( x+4 \)[/tex].
- This indicates a horizontal shift of the graph of [tex]\( f(x) \)[/tex] to the left by 4 units.
2. Vertical Shift:
Look at the [tex]\(-8\)[/tex] outside the function:
- In [tex]\( f(x) = \frac{1}{x} \)[/tex], there is no constant term subtracted.
- In [tex]\( g(x) = \frac{1}{x+4} - 8 \)[/tex], the entire term [tex]\( \frac{1}{x+4} \)[/tex] has been shifted downward by 8 units.
Therefore, the graph of [tex]\( g(x) = \frac{1}{x+4} - 8 \)[/tex] is shifted 4 units to the left and 8 units down from the graph of the parent function [tex]\( f(x) = \frac{1}{x} \)[/tex].
Thus, the correct comparison is:
[tex]\[ g(x) \text{ is shifted 4 units left and 8 units down from } f(x). \][/tex]
