Question: The graph of $y = r(x)$ is given below. Answer each of the questions below the graph. A) What is the domain of $r(x)$? Give your answer using interval notation. B) What is the range of $r(x)$? Give your answer using interval notation. C) What is the $x$-intercepts of $r(x)$? Give your answer in coordinate form (i.e. as ordered pairs), separating multiple answers with commas as needed. D) The zeros of $r(x)$ are E) $r(8) =$ F) $r(1) =$ G) Is $r(-5)$ positive, negative, or 0? H) How many times does the line $y = 0$ intersect $y = r(x)$? I) For what value(s) of $x$ does $r(x) = 5$? J) Give the interval(s) of $x$ such that $r(x) > 0$. Use the union symbol $\cup$ between multiple intervals. K) Give the interval(s) of $x$ such that $r(x) < 0$. Use the union symbol $\cup$ between multiple intervals.

1. **State the problem:** We analyze the graph of $y = r(x)$ to find domain, range, intercepts, zeros, function values, sign at a point, intersections with $y=0$, and intervals where $r(x)$ is positive or negative. 2. **Domain:** The graph starts at a filled point near $x = -6$ and extends to an open circle near $x = 12$. A filled point means the function is defined at $-6$, and an open circle means it is not defined at $12$. So the domain is $$[-6, 12).$$ 3. **Range:** The minimum value is near $0$ at $x = -3$, and the maximum value is near $6$ at $x = 3$. The graph goes down to about $-2$ near $x=12$ (open circle), so the range is $$[-2, 6].$$ 4. **X-intercepts:** The graph crosses the $x$-axis at $x = -3$ (minimum point) and at $x = 10$ (line crosses $x$-axis). So the $x$-intercepts are $$(-3, 0), (10, 0).$$ 5. **Zeros of $r(x)$:** Zeros are the $x$-values where $r(x) = 0$, which are $$-3, 10.$$ 6. **Evaluate $r(8)$:** At $x=8$, the graph is on the decreasing line between $(3,6)$ and $(10,0)$. By linear interpolation, $r(8)$ is approximately halfway between $6$ and $0$, closer to $2$. So $$r(8) ext{ is about } 2.$$ 7. **Evaluate $r(1)$:** At $x=1$, the graph is between $(-3,0)$ and $(3,6)$, rising. Approximate $r(1)$ near $2$. 8. **Sign of $r(-5)$:** At $x=-5$, the graph is near $3$ (starting point), so $r(-5)$ is positive. 9. **Number of intersections with $y=0$:** The graph crosses $y=0$ twice (at $x=-3$ and $x=10$). 10. **Values where $r(x) = 5$:** The graph hits $5$ twice: once going up between $x=0$ and $x=3$, and once going down between $x=3$ and $x=8$. Approximate values are near $x=2$ and $x=5$. 11. **Intervals where $r(x) > 0$:** From $x=-6$ to $x=-3$ (above $0$), from $x=-3$ to $x=10$ (mostly above $0$ except near $x=10$), so $$[-6, 10).$$ 12. **Intervals where $r(x) < 0$:** From $x=10$ to $x=12$ (open circle), the graph is below $0$, so $$(10, 12).$$ **Final answers:** - Domain: $$[-6, 12)$$ - Range: $$[-2, 6]$$ - $x$-intercepts: $$(-3, 0), (10, 0)$$ - Zeros: $$-3, 10$$ - $r(8) \approx 2$ - $r(1) \approx 2$ - $r(-5)$ is positive - Number of intersections with $y=0$: 2 - $r(x) = 5$ at approximately $x=2$ and $x=5$ - $r(x) > 0$ on $$[-6, 10)$$ - $r(x) < 0$ on $$(10, 12)$$