Question

a. $4 - |x + 3|$
b. $4 - |x - 3|$
c. $-|x + 3| - 4$
d. $-|x - 3| - 4$
graph of a v - shaped (absolute value) function with vertex at (-3, -4) (from the grid - like coordinate system shown)

Explanation:

Step1: Identify the vertex of the absolute - value graph

The general form of an absolute - value function is \(y = a|x - h|+k\), where \((h,k)\) is the vertex of the graph. From the given graph, we can see that the vertex of the absolute - value graph (the peak of the "V - shaped" graph, but since it's a downward - opening V, the vertex is the maximum point) is at \((- 3,-4)\)? Wait, no, looking at the graph, the vertex (the highest point of the graph) is at \((-3, - 4)\)? Wait, no, let's re - examine. Wait, the graph is a downward - opening absolute - value graph. The standard form of a downward - opening absolute - value function is \(y=-|x - h|+k\) (or \(y = k-|x - h|\)), where \((h,k)\) is the vertex. From the graph, the vertex (the peak) is at \((-3, - 4)\)? Wait, no, the y - coordinate of the vertex: looking at the grid, the vertex is at \(x=-3\) and \(y = - 4\)? Wait, no, the graph is drawn such that the vertex is at \((-3,-4)\)? Wait, no, let's check the options. The general form of a downward - opening absolute - value function is \(y = k-|x - h|\), where \((h,k)\) is the vertex.

Wait, let's take the vertex. Let's find the vertex of the graph. From the graph, the vertex (the highest point) is at \((-3, - 4)\)? Wait, no, the y - axis: the grid lines, the vertex is at \(x=-3\) and \(y=-4\)? Wait, no, the function is of the form \(y = 4-|x + 3|\) or \(4-|x - 3|\) or \(-|x + 3|-4\) or \(-|x - 3|-4\). Let's test the vertex. For the function \(y = 4-|x + 3|\), when \(x=-3\), \(y = 4-| - 3 + 3|=4-0 = 4\). For \(y = 4-|x - 3|\), when \(x = 3\), \(y=4-|3 - 3| = 4\). For \(y=-|x + 3|-4\), when \(x=-3\), \(y=-|0|-4=-4\). For \(y=-|x - 3|-4\), when \(x = 3\), \(y=-|0|-4=-4\). Wait, the graph's vertex (the highest point) has a y - coordinate of - 4? Wait, no, the graph is drawn with the vertex at \((-3,-4)\)? Wait, the grid: the y - axis has values from - 10 to 4. The vertex is at \(x=-3\) and \(y=-4\). So the function is a downward - opening absolute - value function. The general form of a downward - opening absolute - value function is \(y=-|x - h|+k\), which can be rewritten as \(y = k-|x - h|\). Here, \(h=-3\) and \(k=-4\)? No, that can't be. Wait, no, if the function is \(y=-|x + 3|-4\), it is a downward - opening (since the coefficient of the absolute - value is negative) function with vertex at \((-3,-4)\). Let's check the value at \(x = 0\). For option C: \(y=-|0 + 3|-4=-3 - 4=-7\). For option A: \(y = 4-|0 + 3|=4 - 3 = 1\). For option B: \(y = 4-|0 - 3|=4 - 3 = 1\). For option D: \(y=-|0 - 3|-4=-3 - 4=-7\). Wait, the graph at \(x = 0\) has a y - value of - 7? Let's look at the graph. The graph passes through \((0,-7)\)? Let's check the vertex. The vertex is at \((-3,-4)\). So the function is of the form \(y=-|x + 3|-4\) (option C) or let's re - express. Wait, the standard form of a downward - opening absolute - value function is \(y=k-|x - h|\), where \((h,k)\) is the vertex. If the vertex is \((-3,-4)\), then \(h=-3\) and \(k=-4\), so \(y=-4-|x + 3|\), which is the same as \(y=-|x + 3|-4\) (option C). Wait, but let's check the direction. The absolute - value function \(y = |x|\) opens upwards. \(y=-|x|\) opens downwards. So a function of the form \(y=-|x - h|-k\) (where \(k>0\)) will have a vertex at \((h,-k)\) and open downwards. In our case, the vertex is at \((-3,-4)\), so \(h=-3\) and \(k = 4\)? No, wait, \(y=-|x + 3|-4\) can be written as \(y=-|x-(-3)|-4\), so the vertex is at \((-3,-4)\), and it opens downwards. Let's check the options:

- Option A: \(y = 4-|x + 3|\). When \(x=-3\), \(y = 4\). The vertex is at \((-3,4)\), which is not our graph.
- Option B: \(y = 4-|x - 3|\). When \(x = 3\), \(y = 4\). Vertex at \((3,4)\), not our graph.
- Option C: \(y=-|x + 3|-4\). When \(x=-3\), \(y=-4\). Vertex at \((-3,-4)\), which matches our graph.
- Option D: \(y=-|x - 3|-4\). When \(x = 3\), \(y=-4\). Vertex at \((3,-4)\), not our graph.

Answer:

C. \(-|x + 3|-4\)