1. **Problem Statement:** Given that $y$ varies inversely as $x$, and $y=8$ when $x=3$, find the constant of variation $k$, and the value of $x$ when $y=12$. Also, find the variation table for given $x$ values. 2. **Formula for Inverse Variation:** When $y$ varies inversely as $x$, the relationship is given by: $$y = \frac{k}{x}$$ where $k$ is the constant of variation. 3. **Find the constant $k$:** Using the given values $y=8$ and $x=3$: $$8 = \frac{k}{3} \implies k = 8 \times 3 = 24$$ 4. **Find $x$ when $y=12$:** Using $k=24$: $$12 = \frac{24}{x} \implies x = \frac{24}{12} = 2$$ 5. **Find $y$ values for given $x$ values:** Using $y = \frac{24}{x}$, calculate $y$ for each $x$: - For $x=-24$, $y=\frac{24}{-24} = -1$ - For $x=-12$, $y=\frac{24}{-12} = -2$ - For $x=-6$, $y=\frac{24}{-6} = -4$ - For $x=-4$, $y=\frac{24}{-4} = -6$ - For $x=-3$, $y=\frac{24}{-3} = -8$ - For $x=-2$, $y=\frac{24}{-2} = -12$ - For $x=-1$, $y=\frac{24}{-1} = -24$ - For $x=1$, $y=\frac{24}{1} = 24$ - For $x=2$, $y=\frac{24}{2} = 12$ - For $x=3$, $y=\frac{24}{3} = 8$ - For $x=4$, $y=\frac{24}{4} = 6$ - For $x=6$, $y=\frac{24}{6} = 4$ - For $x=12$, $y=\frac{24}{12} = 2$ - For $x=24$, $y=\frac{24}{24} = 1$ 6. **Summary:** The constant of variation is $k=24$. When $y=12$, $x=2$. The inverse variation function is: $$y = \frac{24}{x}$$ This function can be graphed to show the inverse relationship between $x$ and $y$.