below is the graph of $y = |x|$. transform it to make it the graph of $y = -|x - 2| + 1$.

Explanation:

Step1: Reflect over the x - axis

For the function (y = |x|), when we change it to (y=-|x|), we reflect the graph of (y = |x|) over the (x) - axis. The general rule for reflecting a function (y = f(x)) over the (x) - axis is (y=-f(x)).

Step2: Translate horizontally

For the function (y=-|x|), when we change it to (y =-|x - 2|), we use the rule (y = f(x - h)) (horizontal translation). Here (h = 2), so we shift the graph of (y=-|x|) to the right by (2) units.

Step3: Translate vertically

For the function (y=-|x - 2|), when we change it to (y=-|x - 2|+1), we use the rule (y=f(x)+k) (vertical translation). Here (k = 1), so we shift the graph of (y=-|x - 2|) up by (1) unit.

Answer:

First, reflect the graph of (y = |x|) over the (x) - axis. Then, shift the resulting graph (2) units to the right. Finally, shift the graph (1) unit up.