Question

which graph represents the function $g(x) = |x + 4| + 2?$

Explanation:

Step1: Identify parent function

The parent function is $f(x) = |x|$, which has a vertex at $(0,0)$.

Step2: Analyze horizontal shift

For $g(x) = |x + 4| + 2$, the $+4$ inside the absolute value shifts the parent function left 4 units. The vertex $x$-coordinate becomes $0 - 4 = -4$.

Step3: Analyze vertical shift

The $+2$ outside the absolute value shifts the function up 2 units. The vertex $y$-coordinate becomes $0 + 2 = 2$.

Step4: Locate correct vertex

The vertex of $g(x)$ is $(-4, 2)$. Match this to the graph with vertex at $(-4, 2)$.

Answer:

The first graph (leftmost graph, with vertex at $(-4, 2)$)