Question

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

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

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

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

y=2(x+6)3+8

Answer 1

To determine which of the cubic functions has a point of inflection at \((6, -8)\), we need to check two conditions:

1. The point \((6, -8)\) should lie on the curve defined by the function, meaning that when we substitute \(x = 6\), we should obtain \(y = -8\).
2. A cubic function has a point of inflection where its second derivative changes sign, which typically occurs at the critical points of the first derivative.

Let’s substitute \(x = 6\) into each function and check if we get \(y = -8\):

1. For \(y = 2(x - 6)^3 - 8\):
\[
y = 2(6 - 6)^3 - 8 = 2(0) - 8 = -8 \quad \text{(Valid)}
\]

2. For \(y = 5(x - 6)^3 + 8\):
\[
y = 5(6 - 6)^3 + 8 = 5(0) + 8 = 8 \quad \text{(Not valid)}
\]

3. For \(y = -3(x + 6)^3 - 8\):
\[
y = -3(6 + 6)^3 - 8 = -3(12)^3 - 8 = -3(1728) - 8 = -5184 - 8 = -5192 \quad \text{(Not valid)}
\]

4. For \(y = 2(x + 6)^3 + 8\):
\[
y = 2(6 + 6)^3 + 8 = 2(12)^3 + 8 = 2(1728) + 8 = 3456 + 8 = 3464 \quad \text{(Not valid)}
\]

From the calculations, only the first option \(y = 2(x - 6)^3 - 8\) yields the point \((6, -8)\).

Therefore, the cubic function that has a point of inflection at \((6, -8)\) is:
\[
\boxed{y = 2(x - 6)^3 - 8}
\]