Question

below is the graph of $y = |x|$. translate it to make it the graph of $y = |x - 2| - 5$.

Explanation:

Step1: Analyze vertex of \( y = |x| \)

The vertex of \( y = |x| \) is at \( (0, 0) \).

Step2: Determine vertex of \( y = |x - 2| - 5 \)

For a function \( y = |x - h| + k \), the vertex is \( (h, k) \). Here, \( h = 2 \), \( k = -5 \), so vertex is \( (2, -5) \).

Step3: Find translation from \( (0, 0) \) to \( (2, -5) \)

To get from \( (0, 0) \) to \( (2, -5) \), move the graph 2 units to the right (since \( x \)-coordinate changes from 0 to 2) and 5 units down (since \( y \)-coordinate changes from 0 to -5).

Answer:

Translate the graph of \( y = |x| \) 2 units to the right and 5 units down to obtain the graph of \( y = |x - 2| - 5 \). The vertex moves from \( (0, 0) \) to \( (2, -5) \), and the rest of the graph follows this translation.