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The problem asks to create a linear function \( y = f(x) \) subject to two conditions:
- **a)** The function is expanded in the **y-direction by a factor of 4**.
- **b)** The function is **shifted up by 8 units** in the **x-direction**.

### Step-by-Step Solution:

1. **Start with a basic linear function**: 
   The general form of a linear function is \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

2. **Apply the vertical expansion by a factor of 4**:
   Expanding in the y-direction by a factor of 4 means we multiply the entire function by 4. Thus, the modified function becomes:
   \[
   f(x) = 4(mx + b)
   \]

3. **Shift up by 8 units** in the **x-direction**:
   A horizontal shift occurs when we replace \( x \) with \( (x - c) \), where \( c \) is the amount of the shift. In this case, since the shift is **8 units in the x-direction**, the function becomes:
   \[
   f(x) = 4(m(x - 8) + b)
   \]

This is the final form of the linear function based on the given transformations. However, since the problem doesn’t specify the initial slope \( m \) or intercept \( b \), you can assign any values for these, but the transformations are as shown.

Do you want to specify values for \( m \) or \( b \) or further discuss any details?

### 5 Related Questions:
1. How does the expansion factor affect the slope of a linear function?
2. What would happen if the function was shifted down instead of up?
3. Can we apply both vertical and horizontal shifts simultaneously in a function?
4. How does a negative expansion factor change the appearance of a linear function?
5. What transformations occur when we reflect a function across the y-axis?

### Tip:
When transforming functions, always apply stretches or compressions before applying shifts for clarity in your calculations.

Create a linear function defined by y = f(x) subject to the following conditions:
 a) expanded in the y-direction by a factor of 4
 b) shifted up in the x-direction by 8 units

Learn how to transform a linear function by expanding it in the y-direction by a factor of 4 and shifting it up by 8 units in the x-direction. This step-by-step solution explains key concepts such as vertical expansion and horizontal shifting, making it suitable for Grades 9-10.

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