Question

analyzing tables which is a possible turning point for the continuous function f(x)? x: -4, -3, -2, -1, 0, 1, 2 f(x): -6, -4, -1, -2, -5, -8, -15 options: (-3, -4), (1, -8), (-2, -1), (0, -5)

Explanation:

Step1: Identify sign changes in slope

First, calculate the average rate of change (slope) between consecutive points:

• Between $x=-4$ and $x=-3$: $\frac{-4 - (-6)}{-3 - (-4)} = \frac{2}{1}=2$
• Between $x=-3$ and $x=-2$: $\frac{-1 - (-4)}{-2 - (-3)} = \frac{3}{1}=3$
• Between $x=-2$ and $x=-1$: $\frac{-2 - (-1)}{-1 - (-2)} = \frac{-1}{1}=-1$
• Between $x=-1$ and $x=0$: $\frac{-5 - (-2)}{0 - (-1)} = \frac{-3}{1}=-3$
• Between $x=0$ and $x=1$: $\frac{-8 - (-5)}{1 - 0} = \frac{-3}{1}=-3$
• Between $x=1$ and $x=2$: $\frac{-16 - (-8)}{2 - 1} = \frac{-8}{1}=-8$

Step2: Locate slope sign shift

A turning point occurs where the slope changes from positive to negative (or vice versa). The slope changes from positive (3) to negative (-1) between $x=-3$ and $x=-1$, so the turning point lies in this interval. The only point in the options within this interval is $(-2, -1)$.

Answer:

$(-2, -1)$