Question

below is the graph of y = |x|. transform it to make it the graph of y = |x - 2| - 4.

Explanation:

Step1: Analyze horizontal shift

For the function \( y = |x - h| + k \), the horizontal shift is determined by \( h \). Here, \( h = 2 \), so we shift the graph of \( y = |x| \) 2 units to the right. The vertex of \( y = |x| \) is at \( (0,0) \); after shifting right 2 units, the x - coordinate of the vertex becomes \( 0 + 2=2 \), so the vertex is now at \( (2,0) \) (temporarily, before vertical shift).

Step2: Analyze vertical shift

The vertical shift is determined by \( k \). Here, \( k=- 4 \), so we shift the graph (with the vertex now at \( (2,0) \)) 4 units down. The y - coordinate of the vertex becomes \( 0-4 = - 4 \). So the new vertex is at \( (2,-4) \). To perform the translation, we move the original graph (starting from the vertex \( (0,0) \)) 2 units to the right and 4 units down.

Answer:

To graph \( y = |x - 2|-4 \) from \( y = |x| \), first shift the graph of \( y = |x| \) 2 units to the right (so the vertex moves from \( (0,0) \) to \( (2,0) \)) and then 4 units down (so the vertex moves from \( (2,0) \) to \( (2,-4) \)).