Question

what is the correct equation for this graph?
$y \leq \frac{1}{2}|x-1| + 4$
$y \geq -2|x+1| - 2$
$y \leq 2|x-1| - 1$
$y \geq -2|x-1|$

Explanation:

Step1: Identify vertex of the graph

The vertex of the V-shaped graph is at $(1, 4)$.

Step2: Match vertex to absolute value form

The vertex form of an absolute value function is $y = a|x-h|+k$, where $(h,k)$ is the vertex. Here $h=1$, $k=4$, so the base function is $y = a|x-1|+4$.

Step3: Check inequality direction

The shaded region is below the graph, so we use $\leq$. Also, the slope $a = \frac{1}{2}$ matches the shallow slope of the graph.

Step4: Verify the option

The option $y \leq \frac{1}{2}|x-1| + 4$ matches all the above characteristics.

Answer:

y ≤ 1/2|x-1| + 4