1. **State the problem:** Expand and simplify the expression $\frac{1}{2}(T - 4)^2$. 2. **Recall the formula:** The square of a binomial $(a - b)^2$ is expanded as $$ (a - b)^2 = a^2 - 2ab + b^2 $$ 3. **Apply the formula:** Here, $a = T$ and $b = 4$, so $$ (T - 4)^2 = T^2 - 2 \times T \times 4 + 4^2 = T^2 - 8T + 16 $$ 4. **Multiply by $\frac{1}{2}$:** $$ \frac{1}{2}(T^2 - 8T + 16) = \frac{1}{2}T^2 - \frac{1}{2} \times 8T + \frac{1}{2} \times 16 $$ 5. **Simplify each term:** $$ = \frac{1}{2}T^2 - 4T + 8 $$ **Final answer:** $$ \frac{1}{2}T^2 - 4T + 8 $$