Question

which absolute value equation creates the lower function in the graph? (1 point) o $y = \frac{1}{2}|x|+3$ o $y = |x + 1|+2$ o $y = |x + 2|$ o $y = 2|x|+3$

Explanation:

Step1: Recall absolute - value function form

The general form of an absolute - value function is $y=a|x - h|+k$, where $(h,k)$ is the vertex of the absolute - value function. The vertex of the lower function in the graph is at $(0,3)$.

Step2: Analyze each option

For $y = \frac{1}{2}|x|+3$, the vertex is $(0,3)$ and the slope of the rays is $\pm\frac{1}{2}$.
For $y=|x + 1|+2$, the vertex is $(-1,2)$.
For $y=|x + 2|$, the vertex is $(-2,0)$.
For $y = 2|x|+3$, the vertex is $(0,3)$ but the slope of the rays is $\pm2$.

Step3: Match with the graph

The lower function in the graph has a vertex at $(0,3)$ and relatively flat rays (slope of $\pm\frac{1}{2}$). The function $y=\frac{1}{2}|x|+3$ satisfies these conditions.

Answer:

$y=\frac{1}{2}|x|+3$