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give me the parent function and the way it moves

Let's analyze each of the functions and determine their parent functions along with how they have been transformed.

### Problem 24: \( g(x) = 3|x| - 4 \)
- **Parent Function**: \( f(x) = |x| \)
- **Transformations**:
  - Vertical stretch by a factor of 3.
  - Shifted 4 units downward.

### Problem 25: \( g(x) = 3\sqrt{x} + 8 \)
- **Parent Function**: \( f(x) = \sqrt{x} \)
- **Transformations**:
  - Vertical stretch by a factor of 3.
  - Shifted 8 units upward.

### Problem 26: \( g(x) = \frac{4}{x+1} \)
- **Parent Function**: \( f(x) = \frac{1}{x} \)
- **Transformations**:
  - Vertical stretch by a factor of 4.
  - Shifted 1 unit to the left.

### Problem 27: \( g(x) = 2|x-6| \)
- **Parent Function**: \( f(x) = |x| \)
- **Transformations**:
  - Vertical stretch by a factor of 2.
  - Shifted 6 units to the right.

### Problem 28: \( g(x) = -5[x - 2] \)
- **Parent Function**: \( f(x) = x \) (assuming this is a linear function).
- **Transformations**:
  - Vertical stretch by a factor of 5.
  - Reflected across the x-axis (due to the negative sign).
  - Shifted 2 units to the right.

### Problem 29: \( g(x) = -2|x + 5| \)
- **Parent Function**: \( f(x) = |x| \)
- **Transformations**:
  - Vertical stretch by a factor of 2.
  - Reflected across the x-axis (due to the negative sign).
  - Shifted 5 units to the left.

### Problem 30: \( g(x) = \frac{1}{6x} + 7 \)
- **Parent Function**: \( f(x) = \frac{1}{x} \)
- **Transformations**:
  - Horizontal stretch by a factor of 6 (since \( 6x \) compresses horizontally).
  - Shifted 7 units upward.

### Problem 31: \( g(x) = \frac{\sqrt{x + 3}}{4} \)
- **Parent Function**: \( f(x) = \sqrt{x} \)
- **Transformations**:
  - Shifted 3 units to the left.
  - Vertical shrink by a factor of 4.

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Would you like more details on any of these transformations?

### Related questions:
1. How does a vertical stretch affect the graph of a function?
2. What does it mean when a function is reflected across the x-axis?
3. How do horizontal shifts differ from vertical shifts in terms of transformations?
4. Why does multiplying by a constant affect the slope or curvature of functions differently for linear and nonlinear parent functions?
5. How can we determine if a shift occurs to the left or right by analyzing the equation?

### Tip:
Always check inside the function's parentheses for horizontal shifts and outside for vertical shifts.

Give me the parent function and the way it moves.

Analyze parent functions like absolute value, square root, and rational functions, and explore their transformations including shifts, stretches, and reflections. Suitable for Grades 9-12.

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