$$ \left\lvert\,

Question

$$ \left\lvert\,<\begin{array}{l}	ext { Use transformations of the absolute value function, } f(x)=|x| 	ext {, to graph the function } \ g(x)=-|x-2|-4 	ext {. } \ 	ext { What transformations are needed in order to obtain the graph of } g(x) 	ext { from the graph } \ 	ext { of } f(x) 	ext { ? Select all that apply. } \ \square 	ext { A. Vertical stretch/shrink } \ \square 	ext { B. Horizontal shift } \ \square 	ext { C. Horizontal stretch/shrink } \ \square 	ext { D. Reflection about the } y 	ext {-axis } \ \square 	ext { E. Vertical shift } \ \square 	ext { F. Reflection about the } x 	ext {-axis } \ 	ext { Use the graphing tool to graph the equation. } \ 	ext { B... }\end{array}ight. $$.

UpStudy AI · Accepted Answer

1. Begin with the parent function:  
   $$
   f(x)=|x|
   $$

2. Apply a horizontal shift by replacing $$ x $$ with $$ x-2 $$:  
   $$
   f(x) \rightarrow |x-2|
   $$  
   This shifts the graph to the right by 2 units.

3. Apply a reflection about the $$ x $$-axis by multiplying the function by $$-1$$:  
   $$
   |x-2| \rightarrow -|x-2|
   $$

4. Apply a vertical shift downward by subtracting 4:  
   $$
   -|x-2| \rightarrow -|x-2| - 4
   $$

Thus, the graph of $$ g(x) $$ is obtained from the graph of $$ f(x) $$ using the following transformations:
- Horizontal shift (Option B)
- Vertical shift (Option E)
- Reflection about the $$ x $$-axis (Option F)
To graph $$ g(x) = -|x-2| - 4 $$ from $$ f(x) = |x| $$, apply the following transformations: - **Horizontal Shift (B):** Move the graph right by 2 units. - **Vertical Shift (E):** Move the graph down by 4 units. - **Reflection (F):** Flip the graph over the x-axis. So, select options B, E, and F.

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