Question

which of the following cubic functions has a point of inflection of $(-4,2)$? (1 point)
$h(x)=3(x - 4)^3-2$
$h(x)=(x - 4)^3+2$
$h(x)=2(x + 4)^3-2$
$h(x)=-2(x + 4)^3+2$

Explanation:

Step1: Recall the form of a cubic function with inflection point

The general form of a cubic function with inflection point \((a,b)\) is \(y = k(x - a)^{3}+b\). Here \(a=-4\) and \(b = 2\). So the function should be of the form \(y=k(x+4)^{3}+2\) (since \(x - a=x-(-4)=x + 4\)).

Step2: Check each option

• For \(h(x)=3(x - 4)^{3}-2\), \(a = 4\) and \(b=-2\), not \((-4,2)\).
• For \(h(x)=(x - 4)^{3}+2\), \(a = 4\) and \(b = 2\), not \((-4,2)\).
• For \(h(x)=2(x + 4)^{3}-2\), \(a=-4\) but \(b=-2\), not \((-4,2)\).
• For \(h(x)=-2(x + 4)^{3}+2\), \(a=-4\) and \(b = 2\) which matches the inflection - point \((-4,2)\).

Answer:

\(h(x)=-2(x + 4)^{3}+2\) (the fourth option)