To determine the correct transformation of $f(x) = |x|$ given the transformations:

1. Vertical stretch of 3 → This modifies the coefficient $a$ to 3.
2. Horizontal shift to the right by 4 units → Replace $x$ with $x - 4$.
3. Vertical shift up 2 units → Add 2 to the equation.

Using the general form for transformations: $y = a |x - h| + k$ Where:

• $a$ is the vertical stretch/compression factor,
• $h$ is the horizontal shift ($h > 0$ shifts right, $h < 0$ shifts left),
• $k$ is the vertical shift ($k > 0$ shifts up, $k < 0$ shifts down).

Substituting the given transformations:

• $a = 3$ (vertical stretch),
• $h = 4$ (right shift),
• $k = 2$ (upward shift).

The transformed equation becomes: $g(x) = 3|x - 4| + 2$

Correct Answer:

The third option: $g(x) = 3|x - 4| + 2$.

If you have more questions or want me to explain further, let me know!

Here are 5 related questions you might explore:

1. What happens if $a = -3$ instead of $a = 3$? How would the graph reflect?
2. How do horizontal compressions or stretches affect $|x|$?
3. How can you identify transformations directly from a graph of $|x|$?
4. What happens if both horizontal and vertical shifts are negative?
5. How do vertical stretches compare to horizontal stretches mathematically?

Tip: Always substitute transformations systematically into the general equation $y = a |x - h| + k$ to avoid errors.