Question

given the function ( f(x)=(x + 8)^3 ), which of the following is the point of inflection of the horizontally shifted function ( g(x)=f(x - 3) )? (1 point)
( (0,-5) )
( (-5,0) )
( (0,5) )
( (5,0) )

Explanation:

Step1: Find the function \(g(x)\)

Given \(f(x)=(x + 8)^{3}\), then \(g(x)=f(x - 3)=((x - 3)+8)^{3}=(x + 5)^{3}\).

Step2: Recall the inflection point of \(y = x^{3}\)

The function \(y=x^{3}\) has an inflection point at \((0,0)\) since \(y' = 3x^{2}\) and \(y''=6x\), and \(y'' = 0\) when \(x = 0\).

Step3: Determine the inflection point of \(g(x)=(x + 5)^{3}\)

For a function \(y=(x - h)^{3}+k\), the inflection - point is \((h,k)\). For \(g(x)=(x+5)^{3}=(x-(- 5))^{3}+0\), when \(x=-5\), \(g(-5)=0\).

Answer:

\((-5,0)\)