坐标轴上已标出点 $\left(\frac{1}{2},\frac{1}{2}\right)$。请画出 $y = 2x^3$ 的函数图像草图。

1. Identify key points on the graph

For $y=2x^3$,

when $x=0$, $y=2(0)^3=0$, so the graph passes through $O(0,0)$.

Also,

$y=2\left(\frac12\right)^3=2\left(\frac18\right)=\frac14.$

So the point $\left(\frac12,\frac12\right)$ shown is not on the exact graph of $y=2x^3$; it is only a reference point on the axes.

2. Sketch the correct shape of the cubic graph

The graph of $y=2x^3$ is an increasing cubic curve:

• it passes through $O(0,0)$,
• it lies in the first quadrant for $x>0$ and the third quadrant for $x<0$,
• it is steeper than $y=x^3$ because of the factor $2$,
• at $x=\frac12$, the graph is below the shown point since $y=\frac14<\frac12$.

Sketch a smooth S-shaped curve through the origin, below the given point $\left(\frac12,\frac12\right)$, rising to the right and falling to the left.