Describe the sequence of transformations that are required to graph g left parenthesis x right parenthesis equals 2 open vertical bar x plus 15 close vertical bar minus 8 based on f left parenthesis x right parenthesis equals open vertical bar x close vertical bar.


A.)
The graph of f left parenthesis x right parenthesis equals open vertical bar x close vertical bar is:

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

B.)
The graph of f left parenthesis x right parenthesis equals open vertical bar x close vertical bar is:

Shifted right 8 units
Stretched vertically by a factor of 2
Shifted up 15 units

C.)
The graph of f left parenthesis x right parenthesis equals open vertical bar x close vertical bar is:

Shifted left 15 units
Stretched vertically by a factor of 2
Shifted up 8 units

D.)
The graph of f left parenthesis x right parenthesis equals open vertical bar x close vertical bar is:

Shifted right 15 units
Stretched vertically by a factor of 2
Shifted down 8 units

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).

Describe the sequence of transformations that are required to graph g(x) = 2|x + 15| - 8 based on f(x) = |x|.

Learn how to graph the function g(x) = 2|x + 15| - 8 by applying a sequence of transformations based on the parent function f(x) = |x|. This involves horizontal shifts, vertical stretches, and vertical shifts. Suitable for high school students studying function transformations.

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