Question

analyzing tables determining a missing value in a table x -3 -2 -1 0 1 2 f(x) -12 m 4 0 -4 -2 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? -12 < m < 4 m ≥ 4 or m ≤ -12 m ≤ 4 m ≥ -12

Explanation:

Step1: Identify sign changes

We check sign changes of $f(x)$ across intervals:

• $x=-3$ to $x=-2$: $f(-3)=-12$, $f(-2)=m$
• $x=-2$ to $x=-1$: $f(-2)=m$, $f(-1)=4$
• $x=-1$ to $x=0$: $f(-1)=4$, $f(0)=0$ (1 sign change, 1 turning point)
• $x=0$ to $x=1$: $f(0)=0$, $f(1)=-4$ (2nd sign change, 2nd turning point)
• $x=1$ to $x=2$: $f(1)=-4$, $f(2)=-2$ (no sign change)

Step2: Enforce exactly two turning points

To have only 2 turning points, we avoid additional sign changes:

• For $x=-3$ to $x=-2$ and $x=-2$ to $x=-1$: no new sign change. This requires $m$ is between $-12$ and $4$ (so $-12 < m < 4$):
• If $m \geq 4$ or $m \leq -12$, a 3rd sign change (and 3rd turning point) occurs.
• If $m \geq -12$ or $m \leq 4$, it allows values that create extra turning points.

Answer:

$-12 < m < 4$