Question

the point of inflection of the cubic function ( f(x) ) is ( (-5,0) ). what is the equation of the new function ( h(x) ) that is obtained by shifting ( f(x) ) 6 units to the right? (1 point)

( \bigcirc h(x)=(x + 11)^{3} )

( \bigcirc h(x)=(x + 1)^{3} )

( \bigcirc h(x)=(x - 1)^{3} )

( \bigcirc h(x)=(x - 11)^{3} )

Explanation:

Step1: Recall the form of a cubic function

The general form of a cubic function with inflection point \((a,0)\) is \(y=(x - a)^3\). For \(f(x)\) with inflection point \((- 5,0)\), \(f(x)=(x+5)^3\).

Step2: Apply the horizontal - shift rule

The rule for shifting a function \(y = f(x)\) \(k\) units to the right is \(y=f(x - k)\). Here \(k = 6\), so \(h(x)=f(x - 6)\).
Substitute \(x\) with \(x - 6\) in \(f(x)=(x + 5)^3\). Then \(h(x)=((x - 6)+5)^3=(x-1)^3\).

Answer:

\(h(x)=(x - 1)^3\) (the third option)