Question

given the function $f(x) = (x + 8)^3$, which of the following is the point of inflection of the horizontally shifted function $g(x) = f(x - 3)$? (1 point)
○ (0, -5)
○ (-5, 0)
○ (0, 5)
○ (5, 0)

Explanation:

Step1: Find the inflection point of \( f(x) \)

The function \( f(x)=(x + 8)^3 \) is a cubic function. For \( y = x^3 \), the inflection point is at \( (0,0) \). For \( f(x)=(x + 8)^3 \), it is a horizontal shift of \( y=x^3 \) left by 8 units, so the inflection point of \( f(x) \) is \( (-8,0) \).

Step2: Analyze the horizontal shift for \( g(x) \)

The function \( g(x)=f(x - 3) \) means we shift \( f(x) \) horizontally to the right by 3 units. To find the inflection point of \( g(x) \), we shift the inflection point of \( f(x) \) right by 3 units. The x - coordinate of the inflection point of \( f(x) \) is \( x=-8 \), after shifting right by 3 units, the new x - coordinate is \( x=-8 + 3=-5 \), and the y - coordinate remains 0 (since horizontal shifts do not affect the y - coordinate of the inflection point for this cubic function). So the inflection point of \( g(x) \) is \( (-5,0) \).

Answer:

B. (-5, 0) (assuming the options are labeled as A. (0, -5), B. (-5, 0), C. (0, 5), D. (5, 0))