Question

graph the equation. y = 3|x|

Explanation:

Step1: Consider positive - x values

When \(x>0\), \(|x| = x\), so the equation becomes \(y = 3x\). We can find some points. For \(x = 1\), \(y=3\times1 = 3\); for \(x = 2\), \(y = 3\times2=6\).

Step2: Consider negative - x values

When \(x<0\), \(|x|=-x\), so the equation becomes \(y=-3x\). For \(x=-1\), \(y=-3\times(-1) = 3\); for \(x = - 2\), \(y=-3\times(-2)=6\).

Step3: Consider \(x = 0\)

The function \(y = 3|x|\) is not defined in the strict - sense at \(x = 0\) in terms of the non - zero slope behavior, but \(\lim_{x
ightarrow0}3|x|=0\). Plot the points \((1,3)\), \((2,6)\), \((-1,3)\), \((-2,6)\) and connect them with a V - shaped curve. The vertex of the V is at the origin \((0,0)\).

The graph is a V - shaped curve with the vertex at the origin \((0,0)\), opening upwards. For \(x>0\), it is a straight line with a slope of 3, and for \(x<0\), it is a straight line with a slope of 3.

Answer:

Step1: Consider positive - x values

When \(x>0\), \(|x| = x\), so the equation becomes \(y = 3x\). We can find some points. For \(x = 1\), \(y=3\times1 = 3\); for \(x = 2\), \(y = 3\times2=6\).

Step2: Consider negative - x values

When \(x<0\), \(|x|=-x\), so the equation becomes \(y=-3x\). For \(x=-1\), \(y=-3\times(-1) = 3\); for \(x = - 2\), \(y=-3\times(-2)=6\).

Step3: Consider \(x = 0\)

The function \(y = 3|x|\) is not defined in the strict - sense at \(x = 0\) in terms of the non - zero slope behavior, but \(\lim_{x
ightarrow0}3|x|=0\). Plot the points \((1,3)\), \((2,6)\), \((-1,3)\), \((-2,6)\) and connect them with a V - shaped curve. The vertex of the V is at the origin \((0,0)\).

The graph is a V - shaped curve with the vertex at the origin \((0,0)\), opening upwards. For \(x>0\), it is a straight line with a slope of 3, and for \(x<0\), it is a straight line with a slope of 3.