f(x) = |2x|+6 and is translated 4 units down and 3 units left  FOLLOWED BY a reflection over the x axis

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.

f(x) = |2x|+6 and is translated 4 units down and 3 units left FOLLOWED BY a reflection over the x axis

Discover how to apply transformations to the absolute value function f(x) = |2x| + 6. This detailed guide walks through translating the function 3 units left, 4 units down, and reflecting it over the x-axis, with step-by-step explanations suitable for students in Grades 9-11.

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