Question

what is the value of ( k ) for the function? (fill in the blank with an integer.)

Explanation:

Step1: Identify the function type

The graph is a V-shaped absolute value function, which has the form $y = |x + h| + k$, where $(h,k)$ is the vertex.

Step2: Locate the vertex

From the graph, the vertex (the lowest point) is at $(-2, 0)$. Wait, correction: rechecking, the vertex is at $(0,2)$? No, wait, no—wait, the vertex is at $(-2, 0)$? No, no, looking at the axes: the vertex sits at $x=-2$, $y=0$? No, no, the standard absolute value function shifted horizontally: $y = |x + 2|$. Wait, no, when $x=-2$, $y=0$; when $x=0$, $y=2$; when $x=2$, $y=4$. Wait, the function is $y = |x + 2|$? No, no, the right side: when $x=0$, $y=2$; $x=2$, $y=4$, so slope 1, so equation $y = x + 2$ for $x \geq -2$. Left side: $x=-4$, $y=2$; $x=-2$, $y=0$, slope -1, equation $y = -x -2$ for $x \leq -2$. Combined, this is $y = |x + 2|$, which can be written as $y = |x - (-2)| + 0$. Wait, no, the question asks for $k$—wait, maybe the function is given as $y = |x| + k$? No, no, the vertex is at $(0,2)$? No, no, the x-axis: the vertex is at $x=-2$, $y=0$. Wait, no, the grid: each square is 1 unit. The vertex is 2 units left of the origin, on the x-axis. Wait, maybe the function is $y = |x| - k$? No, no, when $x=0$, $y=2$—wait, I misread the graph. The vertex is at $(0,2)$? No, the y-axis: the vertex is at y=2, x=0? No, the right line goes from (0,2) to (2,4), left line from (0,2) to (-2,4). Oh! I see, I flipped it. The vertex is at $(0, 2)$. So the function is $y = |x| + 2$. So $k=2$.

Step3: Confirm the vertex form

For the absolute value function $y = |x| + k$, the vertex is at $(0, k)$. From the graph, the vertex (minimum point) is at $(0, 2)$, so $k=2$.

Answer:

2