Question

which of the following is the graph of this absolute value function?

\\y = -\frac{1}{2}|x|\\

Explanation:

Identify the vertex of the function

The given absolute value function is:

$$y = -\frac{1}{2}|x|$$

The general form of an absolute value function is \(y = a|x - h| + k\), where \((h, k)\) is the vertex. Here, \(h = 0\) and \(k = 0\), so the vertex is at the origin:

$$(0, 0)$$

Determine the direction of opening

The coefficient \(a\) in front of the absolute value is:

$$a = -\frac{1}{2}$$

Since \(a < 0\), the graph opens downwards. This eliminates the first graph, which opens upwards.

Calculate key points on the graph

We can find points on the graph by substituting \(x\)-values into the equation:

• For \(x = 2\):

$$y = -\frac{1}{2}|2| = -1$$

So, the point \((2, -1)\) must be on the graph.

• For \(x = -2\):

$$y = -\frac{1}{2}|-2| = -1$$

So, the point \((-2, -1)\) must be on the graph.

Compare with the given options

Let's analyze the three graphs from left to right:

• First graph: Vertex is at \((0,0)\), opens upwards, and passes through \((1, 2)\). This represents \(y = 2|x|\).
• Second graph: Vertex is at \((0,0)\), opens downwards, and passes through \((2, -1)\) and \((-2, -1)\). This matches our calculated points.
• Third graph: Vertex is at \((0,0)\), opens downwards, and passes through \((1, -2)\) and \((-1, -2)\). This represents \(y = -2|x|\).

Therefore, the second graph is the correct representation.

Answer:

• (A) The first graph (opens upwards with vertex at \((0,0)\) passing through \((1,2)\))
• (B) The second graph (opens downwards with vertex at \((0,0)\) passing through \((2,-1)\)) (Correct answer)
• (C) The third graph (opens downwards with vertex at \((0,0)\) passing through \((1,-2)\))