Question

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

Explanation:

Step1: Recall inflection point of \(y = a(x - h)^3 + k\)

The general form of a cubic function is \(y = a(x - h)^3 + k\), and its point of inflection is \((h, k)\). We need to find the function where \(h=-4\) and \(k = 2\).

Step2: Analyze each option

• For \(h(x)=2(x + 4)^3-2\): Rewrite as \(h(x)=2(x-(-4))^3+(-2)\). Here, \(h=-4\), \(k=-2\). Not matching \((-4,2)\).
• For \(h(x)=3(x - 4)^3-2\): Here, \(h = 4\), \(k=-2\). Not matching.
• For \(h(x)=-2(x + 4)^3+2\): Rewrite as \(h(x)=-2(x-(-4))^3+2\). Here, \(h=-4\), \(k = 2\). This matches the point of inflection \((-4,2)\).
• For \(h(x)=(x - 4)^3+2\): Here, \(h = 4\), \(k = 2\). Not matching.

Answer:

\(h(x)=-2(x + 4)^3+2\) (the third option)