Question

which of the following cubic functions has a point of inflection of (6, -8)? (1 point)
○ $y = 2(x + 6)^3 + 8$
○ $y = 5(x - 6)^3 + 8$
○ $y = -3(x + 6)^3 - 8$
○ $y = 2(x - 6)^3 - 8$

Explanation:

Step1: Recall Inflection Point of Cubic

The general form of a cubic function with a point of inflection at \((h, k)\) is \(y = a(x - h)^3 + k\). The inflection point occurs at \(x = h\), and substituting \(x = h\) gives \(y = k\), so the point is \((h, k)\).

Step2: Match Given Point \((6, -8)\)

We need \(h = 6\) and \(k = -8\) in the form \(y = a(x - h)^3 + k\). Let's check each option:

• Option 1: \(y = 2(x + 6)^3 + 8\) has \(h = -6\), \(k = 8\) (not matching \((6, -8)\)).
• Option 2: \(y = 5(x - 6)^3 + 8\) has \(h = 6\), \(k = 8\) ( \(k = 8\) does not match \(-8\)).
• Option 3: \(y = -3(x + 6)^3 - 8\) has \(h = -6\), \(k = -8\) ( \(h = -6\) does not match \(6\)).
• Option 4: \(y = 2(x - 6)^3 - 8\) has \(h = 6\), \(k = -8\) (matches the point \((6, -8)\)).

Answer:

\(y = 2(x - 6)^3 - 8\) (the fourth option)