graph this function:\n\n$y = 4|x|$\n\nclick to plot the vertex first.

Answer:

The vertex is at ((0,0)). To graph the function (y = 4|x|), we can find additional points:

• When (x = 1), (y=4|1| = 4), so the point ((1,4)) is on the graph.
• When (x=- 1), (y = 4|-1|=4), so the point ((-1,4)) is on the graph.
• When (x = 2), (y=4|2| = 8), so the point ((2,8)) is on the graph.
• When (x=-2), (y = 4|-2|=8), so the point ((-2,8)) is on the graph.

Plot the vertex ((0,0)) and the additional points ((1,4)), ((-1,4)), ((2,8)), ((-2,8)) and then connect them with straight - line segments on either side of the (y) - axis (since the absolute - value function (y = a|x|) has a "V" shape with vertex at the origin when (a>0)).

Explanation:

Step1: Find the vertex

For the absolute - value function (y=a|x - h|+k), in the function (y = 4|x|), (h = 0) and (k = 0). So the vertex ((h,k)=(0,0)).

Step2: Find points for (x\geq0)

Use the equation (y = 4x) (since when (x\geq0), (|x|=x)). When (x = 1), (y=4\times1 = 4); when (x = 2), (y=4\times2=8).

Step3: Use the property of absolute - value function

Since (y = 4|x|) is an even function ((y(-x)=y(x))), for (x=-1), (y = 4|-1|=4) and for (x=-2), (y = 4|-2|=8).

Step4: Plot the points and draw the graph

Plot the vertex ((0,0)) and the points ((1,4)), ((-1,4)), ((2,8)), ((-2,8)). Then connect the points ((0,0)) to ((1,4)) to ((2,8)) (for (x\geq0)) and ((0,0)) to ((-1,4)) to ((-2,8)) (for (x<0)) with straight lines. The graph is a "V" - shaped graph opening upwards.