1. **Problem statement:** Given the graph of a function $f$, identify intervals where $f$ is increasing or decreasing, and find local and absolute extrema. 2. **Definitions and rules:** - A function is **increasing** on an interval if for any $x_1 < x_2$ in that interval, $f(x_1) < f(x_2)$. - A function is **decreasing** on an interval if for any $x_1 < x_2$ in that interval, $f(x_1) > f(x_2)$. - A **local maximum** is a point where $f$ changes from increasing to decreasing. - A **local minimum** is a point where $f$ changes from decreasing to increasing. - The **absolute maximum** is the highest $y$-value on the entire domain. - The **absolute minimum** is the lowest $y$-value on the entire domain. 3. **From the given points:** - Points: $(1,-8), (2,0), (3,8), (4,3), (5,0), (6,-1), (7,0), (8,3), (9,1)$ 4. **Determine intervals of increase and decrease:** - From $x=1$ to $x=3$: $f$ goes from $-8$ to $8$, so increasing on $(1,3)$. - From $x=3$ to $x=6$: $f$ goes from $8$ down to $-1$, so decreasing on $(3,6)$. - From $x=6$ to $x=8$: $f$ goes from $-1$ up to $3$, so increasing on $(6,8)$. - From $x=8$ to $x=9$: $f$ goes from $3$ down to $1$, so decreasing on $(8,9)$. 5. **Local maxima and minima:** - Local max at $x=3$ with $f(3)=8$ (increasing before, decreasing after). - Local min at $x=6$ with $f(6)=-1$ (decreasing before, increasing after). - Local max at $x=8$ with $f(8)=3$ (increasing before, decreasing after). 6. **Absolute max and min:** - Absolute max is $8$ at $x=3$. - Absolute min is $-8$ at $x=1$ (lowest point). **Final answers:** (a) Increasing intervals: $(1,3), (6,8)$ (b) Decreasing intervals: $(3,6), (8,9)$ (c) Local maximum $y$-values: $8, 3$ (d) Local minimum $y$-values: $-1$ (e) Absolute maximum $y$-value: $8$ (f) Absolute minimum $y$-value: $-8$