To construct an absolute value function that satisfies the given transformations, follow these steps:

1. Reflection across the x-axis: The function is multiplied by $-1$.
2. Vertical shrink by 3/8: Multiply the function by $\frac{3}{8}$.
3. Shift right by 4: Replace $x$ with $x - 4$ inside the absolute value.
4. Shift down by 1: Subtract 1 from the entire function.

Now, starting with the general absolute value function $f(x) = |x|$, applying these transformations gives the following function:

$f(x) = -\frac{3}{8} |x - 4| - 1$

Breakdown of the function:

• The $-$ reflects the graph across the x-axis.
• The $\frac{3}{8}$ shrinks the graph vertically by a factor of 3/8.
• The $x - 4$ shifts the graph 4 units to the right.
• The $-1$ shifts the graph 1 unit down.

Would you like more details on how these transformations work, or do you have any questions?

Related questions:

1. What is the effect of reflecting an absolute value function across the y-axis?
2. How do horizontal stretches affect absolute value functions?
3. What is the role of the absolute value in real-world applications?
4. Can you explain how to graph the function $-\frac{3}{8} |x - 4| - 1$ step by step?
5. What happens if the function is stretched instead of shrunk vertically?

Tip: Always apply horizontal transformations before vertical transformations when graphing functions!