Question

graph the following function by moving the green and blue dots (if necessary).
$y = 2x^{3}$

Explanation:

Step1: Find points on the function

Choose some \(x\) - values and calculate the corresponding \(y\) - values using \(y = 2x^{3}\).
When \(x=-2\), \(y = 2\times(-2)^{3}=2\times(-8)=-16\).
When \(x = - 1\), \(y=2\times(-1)^{3}=2\times(-1)=-2\).
When \(x = 0\), \(y = 2\times0^{3}=0\).
When \(x = 1\), \(y=2\times1^{3}=2\).
When \(x = 2\), \(y=2\times2^{3}=2\times8 = 16\).

Step2: Plot the points

Plot the points \((-2,-16)\), \((-1,-2)\), \((0,0)\), \((1,2)\), \((2,16)\) on the coordinate - plane.

Step3: Draw the curve

Connect the points with a smooth curve. Since it is a cubic function \(y = 2x^{3}\), the curve will pass through the origin \((0,0)\) and have a general shape of a cubic function. As \(x\to-\infty\), \(y\to-\infty\) and as \(x\to+\infty\), \(y\to+\infty\).

Answer:

The graph of the function \(y = 2x^{3}\) is a cubic - shaped curve passing through the points \((-2,-16)\), \((-1,-2)\), \((0,0)\), \((1,2)\), \((2,16)\) and other points calculated from the function, with a smooth connection and the appropriate end - behavior.