Question

graph this function:
$y = |2x|$
click to plot the vertex first.

Explanation:

Step1: Find the vertex of \( y = |2x| \)

The general form of an absolute - value function is \( y=a|x - h|+k \), where \((h,k)\) is the vertex. For the function \( y = |2x|=2|x - 0|+0 \), by comparing with the general form, we can see that \( h = 0 \) and \( k = 0 \). So the vertex of the function \( y = |2x| \) is at the point \((0,0)\).

Step2: Analyze the slope of the two parts of the absolute - value function

We know that for \( y = |2x| \), we can split it into two cases:

• When \( x\geq0 \), \( y = 2x \). The slope of this linear function is \( m = 2 \). So, for example, when \( x = 1 \), \( y=2\times1 = 2 \), so the point \((1,2)\) is on the graph. When \( x = 2 \), \( y = 2\times2=4 \), so the point \((2,4)\) is on the graph.
• When \( x<0 \), \( y=- 2x \). The slope of this linear function is \( m=- 2 \). For example, when \( x=- 1 \), \( y=-2\times(-1) = 2 \), so the point \((- 1,2)\) is on the graph. When \( x = - 2 \), \( y=-2\times(-2)=4 \), so the point \((-2,4)\) is on the graph.

To graph the function, first plot the vertex \((0,0)\). Then, use the slope information to plot other points on both sides of the vertex and draw the two rays of the absolute - value graph.

Answer:

The vertex of the function \( y = |2x| \) is at \((0,0)\). To complete the graph, plot points like \((1,2)\), \((2,4)\) (for \( x\geq0 \)) and \((-1,2)\), \((-2,4)\) (for \( x < 0 \)) and draw the two linear segments from the vertex through these points.