Question

the function $f(x) = 2|x - 1| + 3$ is shifted left 4 units and down 5 units to form function $g(x)$. select the equation for the transformed function $g(x)$.\
\
a. $g(x) = 2|x + 3| - 2$\
b. $g(x) = 2|x - 5| - 2$\
c. $g(x) = 2|x + 3| + 8$\
d. $g(x) = 2|x - 5| + 8$

Explanation:

Step1: Recall horizontal shift rule

For a function \( y = f(x) \), shifting left \( h \) units (where \( h>0 \)) gives \( y = f(x + h) \). Here, shift left 4 units, so replace \( x \) with \( x + 4 \) in \( f(x) \).
Original \( f(x)=2|x - 1|+3 \), after left shift: \( f(x + 4)=2|(x + 4)-1|+3 = 2|x + 3|+3 \).

Step2: Recall vertical shift rule

For a function \( y = f(x) \), shifting down \( k \) units (where \( k>0 \)) gives \( y = f(x)-k \). Here, shift down 5 units, so subtract 5 from the function after horizontal shift.
After vertical shift: \( g(x)=2|x + 3|+3 - 5=2|x + 3|-2 \).

Answer:

A. \( g(x) = 2|x + 3| - 2 \)