Question

graph the equation.
y = -2|x| + 4

Explanation:

Step1: Identify the vertex

The general form of an absolute - value function is \(y = a|x - h|+k\), where \((h,k)\) is the vertex. For the function \(y=-2|x| + 4\), we can rewrite it as \(y=-2|x - 0|+4\). So, the vertex \((h,k)=(0,4)\). We plot the point \((0,4)\) on the coordinate plane.

Step2: Determine the slope for \(x\geq0\) and \(x < 0\)

For an absolute - value function \(y=a|x - h|+k\), when \(x\geq h\), the function is \(y=a(x - h)+k\), and when \(x < h\), the function is \(y=-a(x - h)+k\).
For our function \(y=-2|x| + 4\), when \(x\geq0\), \(y=-2x + 4\). The slope \(m=-2\) (the coefficient of \(x\)). This means that for every 1 unit we move to the right (increase \(x\) by 1) from the vertex, we move down 2 units (decrease \(y\) by 2). So, from \((0,4)\), when \(x = 1\), \(y=-2(1)+4=2\), so we get the point \((1,2)\). When \(x = 2\), \(y=-2(2)+4 = 0\), so we get the point \((2,0)\).
When \(x<0\), \(y=-2(-x)+4=2x + 4\). The slope \(m = 2\). This means that for every 1 unit we move to the left (decrease \(x\) by 1) from the vertex, we move down 2 units (since the slope for \(x<0\) is 2, but the direction is such that when \(x=- 1\), \(y=2(-1)+4 = 2\), so we get the point \((-1,2)\). When \(x=-2\), \(y=2(-2)+4=0\), so we get the point \((-2,0)\).

Step3: Draw the graph

We connect the points for \(x\geq0\) (using the slope - 2) and the points for \(x < 0\) (using the slope 2) to form a V - shaped graph (since it's an absolute - value function) with the vertex at \((0,4)\), passing through \((1,2)\), \((2,0)\), \((-1,2)\), \((-2,0)\) etc.

The graph of \(y =- 2|x|+4\) is a V - shaped graph opening downwards with vertex at \((0,4)\), passing through points like \((1,2)\), \((2,0)\), \((-1,2)\), \((-2,0)\) and so on. To sketch it, plot the vertex \((0,4)\), then use the slopes for \(x\geq0\) (slope - 2) and \(x < 0\) (slope 2) to find additional points and draw the two line segments that form the V.

Answer:

The graph is a V - shaped graph (absolute - value graph) with vertex at \((0,4)\), passing through \((1,2)\), \((2,0)\), \((-1,2)\), \((-2,0)\) etc., and opening downwards. (To plot it, follow the steps above to mark the points and draw the lines.)