Question

which graph represents the equation ( y = |x - 2| + 1 )?
a. graph with a v - shaped graph on a coordinate grid

Explanation:

Step1: Recall absolute value graph form

The general form of an absolute value function is \( y = |x - h| + k \), where \((h, k)\) is the vertex. For \( y = |x - 2| + 1 \), the vertex is \((2, 1)\)? Wait, no, wait: \( y = |x - (-2)| + 1 \)? Wait, no, the equation is \( y = |x + 2| + 1 \)? Wait, the user wrote \( y = |x - 2| + 1 \)? Wait, the graph shown: let's check the vertex. Wait, maybe a typo? Wait, the graph in option A: the vertex is at \( x = -2 \), \( y = 1 \)? Wait, let's re-express the equation. Wait, the standard form is \( y = |x - h| + k \), vertex at \((h, k)\). If the equation is \( y = |x + 2| + 1 \), then \( h = -2 \), \( k = 1 \), so vertex at \((-2, 1)\). Let's check the graph: the vertex is at \( x = -2 \), \( y = 1 \), which matches the graph in option A. Wait, maybe the original equation was \( y = |x + 2| + 1 \). Let's confirm: the graph has a vertex at \( x = -2 \), \( y = 1 \), and it's a V - shape opening upwards. So the equation \( y = |x + 2| + 1 \) (which is \( y = |x - (-2)| + 1 \)) has vertex \((-2, 1)\), which matches the graph in option A.

Step2: Analyze the graph's vertex

Looking at the graph in option A, the vertex (the point of the V) is at \( x = -2 \), \( y = 1 \). For the function \( y = |x + 2| + 1 \) (equivalent to \( y = |x - (-2)| + 1 \)), the vertex is \((-2, 1)\), which matches the graph's vertex. Also, the slope of the right side (for \( x > -2 \)) is 1, and left side is -1, which is consistent with the absolute value function's graph (V - shaped, opening upwards, slope 1 and -1 on either side of the vertex).

Answer:

A. The graph with vertex at (-2, 1) (the one shown in option A)