if the table of the function contains exactly two potential turning points, one with an input value of -1…

if the table of the function contains exactly two potential turning points, one with an input value of -1, which statement best describes all possible values of m? m ≥ -12 -12 < m < 4 m ≤ 4 m ≥ 4 or m ≤ -12

if the table of the function contains exactly two potential turning points, one with an input value of -1, which statement best describes all possible values of m? m ≥ -12 -12 < m < 4 m ≤ 4 m ≥ 4 or m ≤ -12

Answer

Explanation:

Step1: Understand turning - point concept

A turning point of a function is a point where the function changes from increasing to decreasing or vice - versa.

Step2: Analyze the given points

We know one turning - point is at $x = - 1$ with $f(-1)=4$. The points are $(-3,-12),(-2,m),(-1,4),(0,0),(1, - 4)$.

Step3: Consider the behavior around the turning - point

For there to be exactly two turning points, the value of $m$ must be such that the function changes its increasing/decreasing behavior appropriately. Since $f(-3)=-12$ and $f(-1) = 4$, for the correct behavior around the known turning - point at $x=-1$, either $m\geq4$ (so the function first increases from $x=-3$ to $x = - 2$ and then turns at $x=-1$) or $m\leq-12$ (so the function first decreases from $x=-3$ to $x=-2$ and then turns at $x = - 1$).

Answer:

$m\geq4$ or $m\leq-12$