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.