To find the transformed function based on the transformations applied to $f(x) = |x|$, we need to apply each transformation step by step:

1. Vertical Compression by a Factor of $\frac{2}{3}$:

• A vertical compression by a factor of $\frac{2}{3}$ modifies the function by multiplying the entire function by $\frac{2}{3}$.
• So, the function becomes: $f(x) = \frac{2}{3}|x|$

2. Horizontal Translation 4 Units to the Left:

• A horizontal translation 4 units to the left shifts the input $x$ by adding 4 inside the function.
• The function now becomes: $f(x) = \frac{2}{3}|x + 4|$

3. Reflection Across the $x$-Axis:

• A reflection across the $x$-axis changes the sign of the entire function.
• The final transformed function is: $f(x) = -\frac{2}{3}|x + 4|$

Final Answer:

The transformed function is: $f(x) = -\frac{2}{3}|x + 4|$

Would you like further details or have any questions about this transformation process?

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Here are 5 related questions to consider:

1. What happens to the graph of $f(x) = |x|$ when it's vertically stretched instead of compressed?
2. How would you express a horizontal translation 4 units to the right instead of to the left?
3. How does a reflection across the y-axis change the equation of $f(x) = |x|$?
4. Can you describe the effect of combining both vertical and horizontal transformations at the same time?
5. How would the function change if we applied a horizontal compression instead of a vertical one?

Tip: Always apply horizontal transformations inside the function argument and vertical transformations outside the function to ensure the correct sequence of changes.