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

Question

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

UpStudy AI · Accepted Answer

1. Start with the basic absolute value function:  
   $$
   f(x) = |x|
   $$

2. Replace $$ x $$ with $$ x + 4 $$ to get:  
   $$
   |x+4|
   $$  
   This transformation shifts the graph horizontally 4 units to the left.

3. Multiply the absolute value expression by $$-1$$ to obtain:  
   $$
   -|x+4|
   $$  
   This reflects the graph across the $$ x $$-axis.

4. Subtract 2 to get the final form:  
   $$
   -|x+4| - 2
   $$  
   This shifts the graph vertically downward by 2 units.

Thus, the needed transformations are:  
- A. Horizontal shift  
- B. Reflection about the $$ x $$-axis  
- C. Vertical shift
The transformations needed are: - A. Horizontal shift - B. Reflection about the $$ x $$-axis - C. Vertical shift

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