Question

which one best describes the graph following function: $y = -2(x + 3)^3 - 1$?○ cubic functionpoint of inflection: (-3,-1)increasing○ cubic functionpoint of inflection: (-3,-1)decreasing○ quadratic functionvertex: (-3,-1)opens downwards○ cubic functionpoint of inflection: (3,-1)decreasing

Explanation:

Step1: Identify function type

The function $y = -2(x + 3)^3 - 1$ has the highest power of $x$ as 3, so it is a cubic function (eliminates the quadratic option).

Step2: Find inflection point

For a transformed cubic function $y = a(x - h)^3 + k$, the inflection point is $(h, k)$. Rewrite the given function as $y = -2(x - (-3))^3 + (-1)$, so $h=-3$, $k=-1$. Inflection point is $(-3, -1)$ (eliminates the option with inflection point (3,-1)).

Step3: Determine increasing/decreasing

Take the first derivative: $y' = -6(x + 3)^2$. Since $(x + 3)^2 \geq 0$ for all real $x$, multiplying by -6 gives $y' \leq 0$ for all real $x$. The function is non-increasing, and since $y'=0$ only at $x=-3$, the function is decreasing everywhere else, so overall it is a decreasing function.

Answer:

Cubic Function
Point of Inflection: (-3,-1)
Decreasing