Question

what kind of transformation converts the graph of $f(x) = 10|x + 5| + 8$ into the graph of $g(x) = 10|x + 3| + 8$?
translation 2 units left
translation 2 units right
translation 2 units down
translation 2 units up

Explanation:

Step1: Recall horizontal translation rule

For a function \( y = f(x - h) \), if \( h>0 \), it's a translation \( h \) units right; if \( h<0 \), it's \( |h| \) units left. Here, we can rewrite \( f(x) \) and \( g(x) \) in terms of horizontal shifts. Let's look at the argument of the absolute value.

Step2: Analyze the argument change

\( f(x)=10|x + 5|+8=10|x-(- 5)| + 8 \) and \( g(x)=10|x + 3|+8=10|x-(-3)| + 8 \). The change in the horizontal shift parameter: from \( -5 \) to \( -3 \), the difference is \( -3-(-5)=2 \). Since \( h = 2>0 \) (in the form \( y = f(x - h) \), here \( h = 2 \) because \( x-(-3)=x + 3 \) and \( x-(-5)=x + 5 \), so moving from \( x + 5 \) to \( x + 3 \) is like replacing \( x \) with \( x+2 \) in \( f(x) \)? Wait, no, better way: the vertex of \( f(x) \) is at \( x=-5 \), vertex of \( g(x) \) is at \( x = - 3 \). The change in \( x \)-coordinate of the vertex: \( -3-(-5)=2 \), so moving from \( x=-5 \) to \( x=-3 \) is a shift of 2 units to the right (since we go from a smaller \( x \) to a larger \( x \), right direction).

Answer:

B. translation 2 units right