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### 1. Fill in the blanks for the following statements:

#### a) Given a function \( f(x) \):
- A new function \( g(x) = Af(x) \), where \( A \) is a constant, is a **vertical stretch** or **compression** of the function \( f(x) \).

- If \( A > 1 \), then the graph will be **vertically stretched** by a factor of \( A \).

- If \( 0 < A < 1 \), then the graph will be **vertically compressed** by a factor of \( A \).

- If \( A < 0 \), then there will be a **reflection** of a vertical stretch or compression, and either a **vertical stretch** or **compression**.

#### b) Given a function \( f(x) \):
- A new function \( g(x) = f(Bx) \), where \( B \) is a constant, is a **horizontal stretch** or **compression** of the function \( f(x) \).

- If \( 0 < B < 1 \), then the graph will be **horizontally stretched** by a factor of \( \frac{1}{B} \).

- If \( B > 1 \), then the graph will be **horizontally compressed** by a factor of \( \frac{1}{B} \).

- If \( B < 0 \), then there will be a **reflection** of a horizontal stretch or compression, and either a **horizontal stretch** or **compression**.

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### 2. Graph transformations:

You are asked to sketch and describe the transformations for two functions \( g(x) \) and \( h(x) \) based on the graph of \( f(x) \).

- **For \( g(x) = \frac{1}{2} f(x) \):**
  - This represents a **vertical compression** by a factor of 2 (since \( \frac{1}{2} \) is less than 1).
  
  - **Transformation:**
    - The graph of \( f(x) \) will be vertically compressed, meaning the y-values of the original graph will be multiplied by \( \frac{1}{2} \).
    - Sketch: The new graph will look like the original graph, but "flatter," with all points scaled down to half their original y-value.

- **For \( h(x) = f\left(\frac{1}{2} x\right) \):**
  - This represents a **horizontal stretch** by a factor of 2 (since the coefficient \( \frac{1}{2} \) inside the function causes a horizontal stretch).
  
  - **Transformation:**
    - The graph of \( f(x) \) will be horizontally stretched, meaning the x-values of the original graph will be multiplied by 2.
    - Sketch: The new graph will look like the original, but stretched horizontally, making it wider.

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Do you have any questions or want more details? 

Here are 5 related questions to further explore this topic:

1. How do vertical shifts affect the graph of \( f(x) \)?
2. What happens if \( A \) or \( B \) is negative in the transformations?
3. How can you determine if a graph has undergone a reflection?
4. What is the difference between vertical and horizontal transformations in terms of their impact on the graph's shape?
5. How would you describe the transformation \( g(x) = 2f(3x) \)?

**Tip:** When working with transformations, always look at how constants inside or outside the function affect the scaling, shifting, or reflection of the graph.

Fill in the blanks for the following transformations related to the graph of functions and apply these transformations to sketch graphs of given functions.

Explore vertical and horizontal transformations of functions, including stretches, compressions, and reflections. Learn how to apply transformations such as g(x) = Af(x) and g(x) = f(Bx) to sketch graphs. Suitable for students in Grades 9-12.

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