Question

translate each graph as specified below.
(a) the graph of ( y = |x| ) is shown. translate it to get the graph of ( y = |x + 1| ).
(b) the graph of ( y = |x| ) is shown. translate it to get the graph of ( y = |x| - 3 ).

Explanation:

Step1: Analyze part (a) translation

For the function \( y = |x| \) to \( y = |x + 1| \), recall the horizontal translation rule: \( y = f(x + h) \) shifts \( y = f(x) \) left by \( h \) units (when \( h>0 \)). Here, \( h = 1 \), so we shift the graph of \( y = |x| \) 1 unit to the left. The vertex of \( y = |x| \) is at \( (0,0) \); after shifting left 1 unit, the new vertex is at \( (-1,0) \). We move each point on \( y = |x| \) left by 1 unit.

Step2: Analyze part (b) translation

For the function \( y = |x| \) to \( y = |x| - 3 \), recall the vertical translation rule: \( y = f(x) - k \) shifts \( y = f(x) \) down by \( k \) units (when \( k>0 \)). Here, \( k = 3 \), so we shift the graph of \( y = |x| \) 3 units down. The vertex of \( y = |x| \) is at \( (0,0) \); after shifting down 3 units, the new vertex is at \( (0, - 3) \). We move each point on \( y = |x| \) down by 3 units.

Answer:

• Part (a): Shift the graph of \( y = |x| \) 1 unit to the left.
• Part (b): Shift the graph of \( y = |x| \) 3 units down.