1. **Problem 4:** Find the equation of the line passing through (0, -3) parallel to the line given by options A-D. 2. The given lines have the form $2x - 3y = c$. To find the slope, rewrite in slope-intercept form: $$2x - 3y = c \implies -3y = -2x + c \implies y = \frac{2}{3}x - \frac{c}{3}$$ The slope is $\frac{2}{3}$. 3. Parallel lines have the same slope. So the new line passing through $(0, -3)$ has slope $\frac{2}{3}$. 4. Use point-slope form: $$y - y_1 = m(x - x_1)$$ $$y - (-3) = \frac{2}{3}(x - 0) \implies y + 3 = \frac{2}{3}x \implies y = \frac{2}{3}x - 3$$ 5. Convert back to standard form: $$y = \frac{2}{3}x - 3 \implies 3y = 2x - 9 \implies 2x - 3y = 9$$ 6. Check which option matches $2x - 3y = 9$. Option A matches. --- 7. **Problem 5:** Find center and radius of circle with diameter endpoints $(5,0)$ and $(-1,-6)$. 8. Center is midpoint: $$\left(\frac{5 + (-1)}{2}, \frac{0 + (-6)}{2}\right) = (2, -3)$$ 9. Radius is half the distance between endpoints: $$d = \sqrt{(5 - (-1))^2 + (0 - (-6))^2} = \sqrt{6^2 + 6^2} = \sqrt{36 + 36} = \sqrt{72}$$ $$r = \frac{d}{2} = \frac{\sqrt{72}}{2} = \sqrt{18}$$ 10. So center $(2, -3)$ and radius $\sqrt{18}$ matches option D. --- 11. **Problem 6:** Analyze properties of function $f$ from graph. 12. Domain: From graph, $f$ is defined for $x \in [-6,3)$ and $(3, \infty)$ because of open circle at $x=3$. Option A matches domain. 13. Range: From graph, $f$ takes values from about $-5$ to $-1$ and from $0$ upwards. Option B matches range. 14. $f(3)$ is not defined (open circle), so C is false. 15. $f(x) = -1$ at $x=-3$ and $x=3$ is false because $f(3)$ undefined. 16. So correct is A. --- 17. **Problem 7:** Analyze increasing/decreasing intervals from graph. 18. From graph, $f$ decreases on $(-3,0)$, so A is true. 19. $f$ is not increasing on $(0, \infty)$ because of open circle and behavior. 20. $f$ is not decreasing on $[-6,0]$ because it increases from $-6$ to $-5$. 21. $f$ is not increasing on $(0,3]$ because of discontinuity. 22. So correct is A. **Final answers:** 4: A 5: D 6: A 7: A