1. Statement of the problem: We are given triangle $\triangle STU$ with vertices $S=(-8,-4)$, $T=(-8,8)$, and $U=(4,-8)$ and asked to find the image after a dilation centered at the origin with scale factor $k=\frac{1}{4}$. 2. Formula and rules: The coordinate rule for a dilation centered at the origin is $$ (x',y') = (k x, k y) $$. Apply this rule by multiplying each coordinate of each vertex by $k$; maintain the order of coordinates (x then y). 3. Compute the scale factor: Here $k=\frac{1}{4}$. 4. Compute the image of $S$: Start with $S=(-8,-4)$. Compute $S'=(-8\cdot k, -4\cdot k)$. Substitute $k$: $S'=(-8\cdot\frac{1}{4}, -4\cdot\frac{1}{4})$. Simplify: $S'=(-2,-1)$. 5. Compute the image of $T$: Start with $T=(-8,8)$. Compute $T'=(-8\cdot k, 8\cdot k)$. Substitute $k$: $T'=(-8\cdot\frac{1}{4}, 8\cdot\frac{1}{4})$. Simplify: $T'=(-2,2)$. 6. Compute the image of $U$: Start with $U=(4,-8)$. Compute $U'=(4\cdot k, -8\cdot k)$. Substitute $k$: $U'=(4\cdot\frac{1}{4}, -8\cdot\frac{1}{4})$. Simplify: $U'=(1,-2)$. 7. Final answer and plotting: The dilated image has vertices $S'=(-2,-1)$, $T'=(-2,2)$, and $U'=(1,-2)$. To graph the image, plot these three points and connect them in the same order to form the image triangle $\triangle S'T'U'$.