Question

what is the correct equation for this graph?
$y \leq |x-4| -1$
$y \leq |x+4| + 3$
$y < |x+4| -3$
$y < \frac{1}{2}|x+4| + \frac{1}{2}$

Explanation:

Step1: Identify vertex of the V-graph

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

Step2: Check line style and shading

The graph has a solid line (so $\leq$ or $\geq$) and shading above the line (matches $y \leq |x-4| -1$, as the inequality represents all points below/on the line, which is the shaded red region).

Step3: Eliminate other options

• Option 2: Vertex at $(-4,3)$ does not match.
• Option 3: Vertex at $(-4,-3)$ does not match, and it uses a dashed line ($<$).
• Option 4: Vertex at $(-4, 0.5)$ does not match, and has a slope of $\frac{1}{2}$.

Answer:

A. $y \leq |x-4| -1$