Question

Which of the following cubic functions has a point of inflection of (6,−8)
?(1 point)
Responses

y=2(x−6)3−8
y equals 2 left parenthesis x minus 6 right parenthesis cubed minus 8

y=−3(x+6)3−8
y equals negative 3 left parenthesis x plus 6 right parenthesis cubed minus 8

y=5(x−6)3+8
y equals 5 left parenthesis x minus 6 right parenthesis cubed plus 8

y=2(x+6)3+8
y equals 2 left parenthesis x plus 6 right parenthesis cubed plus 8

Answer 1

To determine which cubic function has a point of inflection at (6, -8), we need to confirm that the function passes through this point and check the criteria for an inflection point.

1. **Check if each function passes through (6, -8)**:
- \( y = 2(x - 6)^3 - 8 \)
- When \( x = 6 \):
\[ y = 2(6 - 6)^3 - 8 = 2(0)^3 - 8 = -8 \] (passes through)
- \( y = -3(x + 6)^3 - 8 \)
- When \( x = 6 \):
\[ y = -3(6 + 6)^3 - 8 = -3(12)^3 - 8 \neq -8 \] (does not pass through)
- \( y = 5(x - 6)^3 + 8 \)
- When \( x = 6 \):
\[ y = 5(6 - 6)^3 + 8 = 5(0)^3 + 8 = 8 \neq -8 \] (does not pass through)
- \( y = 2(x + 6)^3 + 8 \)
- When \( x = 6 \):
\[ y = 2(6 + 6)^3 + 8 = 2(12)^3 + 8 \neq -8 \] (does not pass through)

2. **Identify function with a point of inflection**:
- A point of inflection occurs where the second derivative changes sign. For cubic functions, the point of inflection can typically be determined from the form of the function itself. In general, for a cubic of the form \( y = a(x - h)^3 + k \), the inflection point is located at the vertex \( (h, k) \).

For the valid function \( y = 2(x - 6)^3 - 8 \), the vertex is at \( (6, -8) \), which is indeed the inflection point.

Thus, **the cubic function that has a point of inflection at (6, -8) is**:

\[
\boxed{y = 2(x - 6)^3 - 8}
\]