Question

the graph shows the function f(x) = |x − h| + k. what is the value of k?
○ k = −2.5
○ k = −1
○ k = 1
○ k = 2.5

Explanation:

Step1: Recall the vertex form of absolute value function

The function \( f(x) = |x - h| + k \) has its vertex at \( (h, k) \). So we need to find the vertex of the given graph.

Step2: Identify the vertex from the graph

Looking at the graph, the lowest point (vertex) of the V - shaped graph is at \( (1, -2.5) \)? Wait, no, let's check the grid. Wait, the x - coordinate of the vertex: looking at the graph, the vertex is at \( x = 1 \) (since it's the point where the two lines meet). The y - coordinate: from the grid, the vertex is at \( y=-2.5 \)? Wait, no, let's count the grid lines. Wait, the vertical axis: each grid line is 1 unit. Wait, the vertex is at \( (1, -2.5) \)? Wait, no, maybe I made a mistake. Wait, the function is \( f(x)=|x - h|+k \), so the vertex is \( (h,k) \). From the graph, the vertex (the minimum point) is at \( (1, -2.5) \)? Wait, no, let's look again. Wait, the graph: when x = 1, what's the y - value? Let's see the grid. The vertex is at (1, -2.5)? Wait, the options are \( k=-2.5, - 1,1,2.5 \). Wait, the vertex of \( y = |x - h|+k \) is \( (h,k) \). So the y - coordinate of the vertex is k. So we need to find the y - coordinate of the vertex. Looking at the graph, the vertex is at (1, -2.5)? Wait, no, maybe the vertex is at (1, -2.5)? Wait, the graph: let's see the points. The left side: when x=-2, y = 0 (since it crosses the x - axis at x=-2). When x = 0, y=-2? Wait, no, the vertex is at (1, -2.5). Wait, the options include \( k = - 2.5 \). So since the vertex is \( (h,k) \), and the vertex's y - coordinate is k, so k is the y - coordinate of the vertex. From the graph, the vertex is at (1, -2.5), so \( k=-2.5 \). Wait, but let's confirm. The absolute value function \( y=|x - h|+k \) has its minimum (or maximum) at \( (h,k) \). The graph shown is a V - shape opening upwards, so it has a minimum at the vertex. The vertex's y - coordinate is k. So by looking at the graph, the vertex is at (1, -2.5), so \( k=-2.5 \).

Answer:

\( k=-2.5 \) (Option: \( k = - 2.5 \))