Let \( f(x)=|x| \). Find a formula for a function \( g \) whose graph is obtained from \( f \) after the following sequence of t;onsformations: (1) Shift left 3 units. (2) Stretch vertically by a facto of 2 . (3) Reflect across the \( x \)-axis. (4) Shift down 1 unit. \( g(x)=-2|x+3|-1 \) \( g(x)=-2|x-3|-1 \) \( g(x)=2|-x+3|-1 \) \( g(x)=-\frac{1}{2}|x+3|-1 \)

To find the function \( g \) based on the transformations applied to \( f(x) = |x| \): 1. **Shift left 3 units**: This changes \( f(x) \) to \( f(x + 3) = |x + 3| \). 2. **Stretch vertically by a factor of 2**: This transforms it to \( 2f(x + 3) = 2|x + 3| \). 3. **Reflect across the \( x \)-axis**: Now we have \( -2|x + 3| \). 4. **Shift down 1 unit**: Finally, this leads to \( g(x) = -2|x + 3| - 1 \). Hence, the correct formula for \( g \) is \( g(x) = -2|x + 3| - 1 \). One might wonder why these transformations work as they do. The absolute value graph, with its characteristic V-shape, responds predictably to such shifts and stretches. It's like giving the graph a "makeover" that makes it reflect its new position and orientation in the coordinate plane! If you want to experiment further with transformations, consider graphing \( f(x) = |x| \) and applying transformations step by step using graphing tools. It’s a fun way to visualize how each change affects the final graph!