To find the correct sequence of transformations to graph $g(x) = 2|x + 15| - 8$ based on the parent function $f(x) = |x|$, let's analyze each transformation step-by-step:

Step-by-Step Transformation Analysis

1. Inside the Absolute Value: $x + 15$:

   • The term $x + 15$ inside the absolute value indicates a horizontal shift.
   • Since it is $+15$, this represents a shift to the left by 15 units (the opposite direction of the sign).

2. Vertical Stretch: $2|x + 15|$:

   • The coefficient 2 outside the absolute value indicates a vertical stretch by a factor of 2. This means the graph is stretched away from the x-axis by a factor of 2.

3. Vertical Shift: $-8$:

   • The term $-8$ outside the absolute value indicates a vertical shift.
   • Since it is negative, the graph is shifted down by 8 units.

Correct Answer

Given these transformations:

• Shifted left by 15 units.
• Stretched vertically by a factor of 2.
• Shifted down by 8 units.

This matches Option A:

A.) The graph of $f(x) = |x|$ is:

• Shifted left 15 units,
• Stretched vertically by a factor of 2,
• Shifted down 8 units.

Final Answer

The correct answer is A.

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Would you like further details or have any questions?

Here are some related questions that might interest you:

1. How do you determine horizontal transformations of a function?
2. What is the effect of a negative coefficient outside an absolute value function?
3. How do vertical and horizontal stretches differ in function transformations?
4. How can we graph multiple transformations step-by-step?
5. What are the properties of absolute value functions and their transformations?

Tip: Remember, horizontal shifts inside the function move opposite to the sign (e.g., $x + 15$ moves left).