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### 1. Problem Breakdown:

#### 21. \( g(x) = 12 - x^2 \)
- **a. Identify the parent function**: 
  The parent function here is \( f(x) = x^2 \) (a basic quadratic function).
  
- **b. Describe the sequence of transformations from \( f \) to \( g \)**: 
  1. Reflection across the x-axis because of the negative sign in front of \( x^2 \).
  2. Vertical translation upwards by 12 units due to the "+12".
  
- **c. Sketch the graph of \( g \)**:
  The graph is an upside-down parabola (due to the reflection) shifted 12 units upwards.

- **d. Use function notation to write \( g \) in terms of \( f \)**: 
  If \( f(x) = x^2 \), then \( g(x) = 12 - f(x) \).

#### 23. \( g(x) = x^3 + 7 \)
- **a. Identify the parent function**: 
  The parent function is \( f(x) = x^3 \) (a cubic function).
  
- **b. Describe the sequence of transformations**:
  1. Vertical translation upwards by 7 units.
  
- **c. Sketch the graph of \( g \)**:
  The graph looks like a cubic curve shifted 7 units upwards.

- **d. Use function notation to write \( g \) in terms of \( f \)**: 
  If \( f(x) = x^3 \), then \( g(x) = f(x) + 7 \).

#### 27. \( g(x) = 2 - (x + 5)^2 \)
- **a. Identify the parent function**: 
  The parent function is \( f(x) = x^2 \) (quadratic).
  
- **b. Describe the sequence of transformations**:
  1. Horizontal shift left by 5 units because of \( (x + 5) \).
  2. Reflection across the x-axis because of the negative in front of \( (x + 5)^2 \).
  3. Vertical translation upwards by 2 units.
  
- **c. Sketch the graph of \( g \)**:
  An upside-down parabola (due to the reflection) shifted 5 units to the left and 2 units upwards.

- **d. Use function notation to write \( g \) in terms of \( f \)**: 
  If \( f(x) = x^2 \), then \( g(x) = 2 - f(x+5) \).

#### 31. \( g(x) = (x - 1)^3 + 2 \)
- **a. Identify the parent function**: 
  The parent function is \( f(x) = x^3 \) (cubic).
  
- **b. Describe the sequence of transformations**:
  1. Horizontal shift right by 1 unit because of \( (x - 1) \).
  2. Vertical translation upwards by 2 units.
  
- **c. Sketch the graph of \( g \)**:
  A cubic curve shifted 1 unit to the right and 2 units upwards.

- **d. Use function notation to write \( g \) in terms of \( f \)**: 
  If \( f(x) = x^3 \), then \( g(x) = f(x - 1) + 2 \).

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Do you want more details or have any questions? 

Here are 5 related questions:
1. What are the effects of horizontal and vertical shifts on a function’s graph?
2. How does a reflection across the x-axis affect the graph of a function?
3. Can you describe the transformations for the function \( g(x) = -x^2 - 3x + 5 \)?
4. What happens when a cubic function is reflected across both the x-axis and y-axis?
5. How does a vertical stretch or compression affect a quadratic function?

**Tip**: When analyzing transformations, always handle reflections and shifts carefully to avoid confusion.

Identify the parent function. Describe the sequence of transformations from f to g. Sketch the graph of g. Use function notation to write g in terms of f. Problems: 21. g(x) = 12 - x^2, 23. g(x) = x^3 + 7, 27. g(x) = 2 - (x + 5)^2, 31. g(x) = (x - 1)^3 + 2.

Step-by-step analysis of function transformations involving quadratic and cubic functions, including vertical shifts, horizontal shifts, and reflections. Understand how to transform f(x) = x^2 and f(x) = x^3 into g(x) for each problem.

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