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To determine the sequence of transformations from the parent function (f(x) = |x|) to (g(x) = -2|x - 3| + 1), we follow the standard order of applying transformations (internal to external changes):

Step 1: Horizontal Shift

The (x) in (|x|) is replaced by ((x - 3)). This is a horizontal shift of 3 units to the right.
Resulting function: (f_1(x) = |x - 3|).

Step 2: Vertical Stretch

Multiply by 2: this is a vertical stretch by a factor of 2.
Resulting function: (f_2(x) = 2|x - 3|).

Step 3: Reflection Over the X-axis

Multiply by (-1): this is a reflection across the x-axis.
Resulting function: (f_3(x) = -2|x - 3|).

Step 4: Vertical Shift

Add 1: this is a vertical shift upward by 1 unit.
Final function: (g(x) = -2|x - 3| + 1).

Note: Steps 2 and 3 can be swapped (reflection first, then stretch) since multiplication is commutative, but the overall effect remains the same.

Answer:
The transformations applied to (f(x)=|x|) to get (g(x)=-2|x-3|+1) are, in order:

1. Shift 3 units to the right.
2. Stretch vertically by a factor of 2.
3. Reflect over the x-axis.
4. Shift up by 1 unit.

(Alternatively, steps 2 and 3 can be reversed: reflect over x-axis then stretch vertically by factor 2.)

(\boxed{1. \text{Shift right 3 units; 2. Vertical stretch by 2; 3. Reflect over x-axis; 4. Shift up 1 unit}})
(Or equivalent sequence with steps 2 and 3 swapped.)

But since the problem says to list them in correct order and describe clearly, the boxed answer should present the sequence concisely. However, the problem might expect the sequence as a list, but following the instructions, the final answer should be in a box. Let me check the problem statement again: it says "list the transformations in the correct order and describe each one clearly". So the answer should be the sequence as above, but formatted properly. Maybe:

Final Answer:

1. Shift the graph of (f(x)=|x|) 3 units to the right.
2. Stretch the resulting graph vertically by a factor of 2.
3. Reflect the graph over the x-axis.
4. Shift the graph upward by 1 unit.

But since the problem says to put the answer in a box, perhaps we can write it as a list separated by semicolons:

(\boxed{Shift\ right\ 3\ units;\ Vertical\ stretch\ by\ factor\ 2;\ Reflect\ over\ x-axis;\ Shift\ up\ 1\ unit})

Yes, that should work.

(\boxed{Shift\ right\ 3\ units;\ Vertical\ stretch\ by\ factor\ 2;\ Reflect\ over\ x-axis;\ Shift\ up\ 1\ unit})

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