We begin by analyzing the original function and applying each transformation step-by-step.

Original function:

$f(x) = |2x| + 6$

Step 1: Translation 3 units left

To move a function 3 units to the left, we replace $x$ with $x + 3$: $f(x) = |2(x + 3)| + 6 = |2x + 6| + 6$

Step 2: Translation 4 units down

To translate a function 4 units down, subtract 4 from the entire function: $f(x) = |2x + 6| + 6 - 4 = |2x + 6| + 2$

Step 3: Reflection over the x-axis

To reflect a function over the x-axis, multiply the entire function by $-1$: $f(x) = -(|2x + 6| + 2) = -|2x + 6| - 2$

Final transformed function:

$f(x) = -|2x + 6| - 2$

This is the equation after applying all the transformations: 3 units left, 4 units down, and reflection over the x-axis.

Would you like more details on any part of this? Here are some related questions:

1. How do we translate functions vertically or horizontally?
2. What is the effect of a reflection over the x-axis on a graph?
3. How do absolute value functions behave under transformations?
4. Can we generalize these transformations for any absolute value function?
5. How do translations affect the vertex of a function graph?

Tip: Always apply transformations in the correct order: horizontal shifts first, followed by vertical shifts, and finally reflections or stretches/compressions.