Question

compare the equation to the absolute value parent function ($y = |x|$). select all the transformations that apply. $y = |x + 1| + 3$
a reflected over the x - axis
b horizontal shift left
c horizontal shift right
d vertical shift up
e vertical shift down
f graph is narrower
g graph is wider

Explanation:

Step1: Recall transformation rules

For absolute value function \( y = |x - h| + k \):

• \( h \) determines horizontal shift: \( h>0 \) shift right, \( h<0 \) shift left.
• \( k \) determines vertical shift: \( k>0 \) shift up, \( k<0 \) shift down.
• Reflection over x - axis: \( y=-|x| \)
• Vertical stretch/compression: \( y = a|x| \), \( |a|>1 \) narrower, \( 0<|a|<1 \) wider.

Step2: Analyze \( y = |x + 1|+3 \)

Rewrite as \( y=|x-(- 1)| + 3 \).

• Horizontal shift: \( h=-1<0 \), so horizontal shift left (matches option b).
• Vertical shift: \( k = 3>0 \), so vertical shift up (matches option d).
• No reflection (since no negative sign in front of absolute value), so a is out.
• No vertical stretch/compression (a = 1), so f and g are out.
• Horizontal shift is left, not right (c out), vertical shift is up, not down (e out).

Answer:

b. Horizontal Shift Left
d. Vertical Shift Up