Question

which of the following cubic functions has a point of inflection of $(2,-5)$? (1 point) $g(x)=-3(x + 2)^{3}-5$ $g(x)=2(x - 2)^{3}-5$ $g(x)=(x + 2)^{3}+5$ $g(x)=-2(x - 2)^{3}+5$

Explanation:

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

The general form of a cubic function \(y = a(x - h)^{3}+k\) has an inflection point at \((h,k)\).

Step2: Identify \(h\) and \(k\) from the given inflection point

Given the inflection point \((2,-5)\), we have \(h = 2\) and \(k=-5\).

Step3: Check each function

• For \(g(x)=-3(x + 2)^{3}-5\), \(h=-2,k = - 5\) (incorrect \(h\)).
• For \(g(x)=2(x - 2)^{3}-5\), \(h = 2,k=-5\) (correct \(h\) and \(k\)).
• For \(g(x)=(x + 2)^{3}+5\), \(h=-2,k = 5\) (incorrect \(h\) and \(k\)).
• For \(g(x)=-2(x - 2)^{3}+5\), \(h = 2,k = 5\) (incorrect \(k\)).

Answer:

\(g(x)=2(x - 2)^{3}-5\)