Question

on each coordinate plane, the parent function f(x) = |x| is represented by a dashed line and a translation is represented by a solid line. which graph represents the translation g(x) = |x + 2| as a solid line?

Explanation:

Step1: Recall translation rules for absolute value functions

For a function \( y = |x - h| \), the graph of \( y = |x| \) is shifted \( h \) units horizontally. If \( h < 0 \), it's a shift to the left; if \( h > 0 \), it's a shift to the right. For \( g(x)=|x + 2|=|x-(-2)| \), this means a horizontal shift of 2 units to the left.

Step2: Find the vertex of the parent and translated function

The parent function \( f(x)=|x| \) has its vertex at \( (0,0) \). After a horizontal shift of 2 units to the left, the vertex of \( g(x)=|x + 2| \) should be at \( (-2,0) \).

Step3: Analyze the graphs

We check each graph to see which solid line (translation) has its vertex at \( (-2,0) \) while the dashed line (parent \( f(x)=|x| \)) has vertex at \( (0,0) \). Looking at the options, the graph where the solid V - shaped graph has its vertex at \( (-2,0) \) (and the dashed one at \( (0,0) \)) is the correct one. From the given graphs, the third graph (the one with the solid line vertex at \( (-2,0) \)) matches this description.

Answer:

The Middle - Right Graph (or the third graph in the set) where the solid line (translation \( g(x)=|x + 2| \)) has its vertex at \( (-2,0) \) and the dashed line (parent \( f(x)=|x| \)) has its vertex at \( (0,0) \).