Compare g(x) = 12
|x + 3| − 1 to the graph of f.

To compare the graph of \( g(x) = 12|x + 3| - 1 \) to the graph of \( f(x) = |x| \), we'll break down the transformations step by step. 

### 1. **Starting Function \( f(x) = |x| \)**
   - This is the basic absolute value function, which has a "V" shape centered at the origin \((0, 0)\), opening upwards.

### 2. **Horizontal Shift**
   - The function \( g(x) = |x + 3| \) indicates a horizontal shift of \( f(x) \) by 3 units to the left. This is because the expression inside the absolute value, \( x + 3 \), means that every \( x \) value is effectively reduced by 3 before applying the absolute value.

### 3. **Vertical Stretch**
   - The function \( g(x) = 12|x + 3| \) introduces a vertical stretch by a factor of 12. This makes the "V" shape of the graph steeper. Every y-value of \( |x + 3| \) is multiplied by 12, stretching the graph vertically.

### 4. **Vertical Shift**
   - Finally, \( g(x) = 12|x + 3| - 1 \) involves a vertical shift downwards by 1 unit. This means the entire graph is moved down by 1 unit.

### **Summary of Transformations:**
- Start with \( f(x) = |x| \).
- Shift the graph 3 units to the left.
- Apply a vertical stretch by a factor of 12.
- Move the graph 1 unit downwards.

### **Visual Comparison:**
- **\( f(x) = |x| \)**: "V" shape centered at \((0,0)\).
- **\( g(x) = 12|x + 3| - 1 \)**: Much steeper "V" shape centered at \((-3,-1)\).

### Key Points:
- The vertex of \( g(x) \) is at \((-3, -1)\).
- The graph is 12 times steeper than \( f(x) \).

Would you like to see the graphs or need any further details?

Here are 5 related questions:
1. How would the graph change if the coefficient 12 were different?
2. What happens to the graph if we shift \( f(x) = |x| \) by 3 units to the right instead?
3. How do you determine the vertex of a transformed absolute value function?
4. What effect does multiplying by a negative number have on the graph?
5. How would the graph look if we added a vertical stretch instead of compression?

**Tip:** Remember, horizontal shifts inside the function's argument \( |x + h| \) always move the graph in the opposite direction of the sign.

Compare g(x) = 12|x + 3| − 1 to the graph of f.

Explore how the graph of g(x) = 12|x + 3| − 1 compares to the basic absolute value function f(x) = |x|. This problem covers key concepts in function transformations, including horizontal shifts, vertical stretches, and vertical shifts. Suitable for students in Grades 9-11.

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