Question

use transformations of the standard cubic function, ( f(x)=x^{3} ), to graph the function ( h(x)=3x^{3} ).

Explanation:

Step1: Recall the transformation rule

For a function \(y = af(x)\), if \(|a|> 1\), it is a vertical stretch of the function \(y = f(x)\) by a factor of \(a\). Here \(f(x)=x^{3}\) and \(a = 3\) for \(h(x)=3x^{3}\).

Step2: Analyze key points of the standard cubic function

The standard cubic function \(y = f(x)=x^{3}\) has key points: when \(x=-2\), \(y=(-2)^{3}=-8\); when \(x = - 1\), \(y=(-1)^{3}=-1\); when \(x=0\), \(y = 0^{3}=0\); when \(x = 1\), \(y=1^{3}=1\); when \(x=2\), \(y=2^{3}=8\).

Step3: Apply the vertical - stretch transformation

For the function \(h(x)=3x^{3}\), if \(x=-2\), \(h(-2)=3\times(-2)^{3}=3\times(-8)=-24\); if \(x=-1\), \(h(-1)=3\times(-1)^{3}=3\times(-1)=-3\); if \(x = 0\), \(h(0)=3\times0^{3}=0\); if \(x=1\), \(h(1)=3\times1^{3}=3\); if \(x = 2\), \(h(2)=3\times2^{3}=3\times8 = 24\).

Step4: Graph the function

Plot the points \((-2,-24)\), \((-1,-3)\), \((0,0)\), \((1,3)\), \((2,24)\) and connect them with a smooth curve. The graph of \(h(x)=3x^{3}\) is a vertical stretch (by a factor of \(3\)) of the graph of \(y=x^{3}\).

Answer:

The graph of \(h(x)=3x^{3}\) is a vertical stretch (by a factor of \(3\)) of the graph of the standard cubic function \(y = x^{3}\).