To determine the transformations that map the standard absolute value function $f(x) = |x|$ to the function $g(x) = 12|x + 3| - 1$, we analyze the changes step-by-step:

Step 1: Translating horizontally

The absolute value argument changes from $|x|$ to $|x + 3|$. A shift of $+3$ inside the absolute value moves the graph 3 units to the left, as the opposite sign affects the direction.

Step 2: Translating vertically

The entire function is adjusted by subtracting 1. This moves the graph 1 unit down.

Step 3: Vertical stretch/compression

The coefficient $12$ in front of the absolute value represents a vertical stretch. It increases the steepness of the graph, as it multiplies the output values by 12.

Summary of transformations:

• The graph is translated 3 units left.
• The graph is translated 1 unit down.
• The graph is vertically stretched.

Correct Answer: B. translated 3 units left and 1 unit down; vertically stretched

Would you like to go through the graphing process or see more examples? Let me know!

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Follow-Up Questions:

1. What happens if the coefficient of $|x + 3|$ is less than 1?
2. How do translations affect the vertex of an absolute value graph?
3. How can you identify a vertical stretch versus a vertical compression?
4. What is the domain and range of $g(x) = 12|x + 3| - 1$?
5. How would the graph change if $g(x)$ included a horizontal stretch?

Tip:

Always remember that changes inside the absolute value affect the horizontal direction, while changes outside affect the vertical direction.