Question

graph this function:
y = -3|x|
click to plot the vertex first.
(graph with x-axis from -10 to 10, y-axis from -10 to 10, grid lines, and a submit button at the bottom left)

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 \(y=-3|x|\), we can rewrite it as \(y = - 3|x - 0|+0\). So, the vertex \((h,k)\) is \((0,0)\). We plot the point \((0,0)\) on the coordinate plane.

Step2: Determine the slope of the two branches

For the absolute - value function \(y = a|x|\), when \(x\geq0\), the function becomes \(y = ax\), and when \(x<0\), the function becomes \(y=-ax\). In our function \(y=-3|x|\), when \(x\geq0\), \(y=-3x\), so the slope of the line for \(x\geq0\) is \(- 3\). When \(x = 1\) (since \(x\geq0\)), \(y=-3\times1=-3\), so we can plot the point \((1, - 3)\). When \(x<0\), \(y=-3(-x)=3x\), so the slope of the line for \(x < 0\) is \(3\). When \(x=-1\) (since \(x < 0\)), \(y = 3\times(-1)=-3\), so we can plot the point \((-1,-3)\). Then we draw two lines: one passing through \((0,0)\) and \((1, - 3)\) (for \(x\geq0\)) and the other passing through \((0,0)\) and \((-1,-3)\) (for \(x < 0\)).

Answer:

The vertex is at \((0,0)\), and the graph is a "V - shaped" graph opening downwards with the left - hand branch having a slope of \(3\) and the right - hand branch having a slope of \(-3\). To graph it, first plot \((0,0)\), then plot points like \((1, - 3)\) and \((-1,-3)\) and draw the two lines connecting the vertex to these points.