1. **State the problem:** We are given a piecewise linear graph with points at (-5, -3), (-3, 1), (-1, 1), (1, 3), (3, 4), (4, 0), and (6, 2). We need to identify intervals where the function is increasing, decreasing, constant, and find relative and absolute maxima and minima. 2. **Recall definitions:** - Increasing: function values rise as $x$ increases. - Decreasing: function values fall as $x$ increases. - Constant: function values stay the same as $x$ increases. - Relative max: a point where the function value is higher than nearby points. - Relative min: a point where the function value is lower than nearby points. - Absolute max/min: highest/lowest function value over the entire domain. 3. **Analyze intervals between points:** - From $x=-5$ to $x=-3$: $y$ goes from $-3$ to $1$ (increasing). - From $x=-3$ to $x=-1$: $y$ stays at $1$ (constant). - From $x=-1$ to $x=1$: $y$ goes from $1$ to $3$ (increasing). - From $x=1$ to $x=3$: $y$ goes from $3$ to $4$ (increasing). - From $x=3$ to $x=4$: $y$ goes from $4$ to $0$ (decreasing). - From $x=4$ to $x=6$: $y$ goes from $0$ to $2$ (increasing). 4. **Identify relative maxima and minima:** - Relative max at $x=3$ where $y=4$ (peak before decreasing). - Relative min at $x=4$ where $y=0$ (lowest point before increasing). 5. **Identify absolute max and min:** - Absolute max is $y=4$ at $x=3$. - Absolute min is $y=-3$ at $x=-5$. **Final answers:** - Increasing: $(-5,-3)$, $(-1,1)$, $(1,3)$, $(4,6)$ - Decreasing: $(3,4)$ - Constant: $(-3,-1)$ - Relative max: $(3,4)$ - Relative min: $(4,0)$ - Absolute max: $(3,4)$ - Absolute min: $(-5,-3)$