1. **State the problem:** We are given a piecewise linear function $f(x)$ with points at approximately $(-6,4)$, $(-2,0)$, $(-1,-2)$, $(2,1)$, and $(3,0)$. We want to understand the behavior of this function based on the graph. 2. **Analyze the graph segments:** The function connects these points with line segments: - From $(-6,4)$ down to $(-2,0)$ - From $(-2,0)$ down to $(-1,-2)$ - From $(-1,-2)$ up to $(2,1)$ - From $(2,1)$ down to $(3,0)$ 3. **Calculate slopes of each segment:** The slope formula is $$m=\frac{y_2 - y_1}{x_2 - x_1}$$ - Segment 1: $$m=\frac{0-4}{-2 - (-6)}=\frac{-4}{4}=-1$$ - Segment 2: $$m=\frac{-2-0}{-1 - (-2)}=\frac{-2}{1}=-2$$ - Segment 3: $$m=\frac{1 - (-2)}{2 - (-1)}=\frac{3}{3}=1$$ - Segment 4: $$m=\frac{0 - 1}{3 - 2}=\frac{-1}{1}=-1$$ 4. **Interpretation:** The function decreases steeply from $(-6,4)$ to $(-1,-2)$, then increases from $(-1,-2)$ to $(2,1)$, and finally decreases again to $(3,0)$. 5. **Summary:** The piecewise linear function $f(x)$ has four linear pieces with slopes $-1$, $-2$, $1$, and $-1$ respectively, connecting the given points. Final answer: The function is piecewise linear with segments having slopes $$-1, -2, 1, -1$$ connecting points $(-6,4)$, $(-2,0)$, $(-1,-2)$, $(2,1)$, and $(3,0)$ respectively.