1. **State the problem:** We need to find the ticket price that maximizes revenue for a minor league hockey team. 2. **Define variables:** Let $x$ be the number of $1$ dollar increases in ticket price above $12$. 3. **Write expressions:** - Ticket price: $12 + x$ - Attendance: $800 - 50x$ 4. **Revenue function:** Revenue $R(x)$ is price times attendance: $$R(x) = (12 + x)(800 - 50x)$$ 5. **Expand the revenue function:** $$R(x) = 12 \times 800 - 12 \times 50x + x \times 800 - 50x^2 = 9600 - 600x + 800x - 50x^2$$ 6. **Simplify:** $$R(x) = 9600 + 200x - 50x^2$$ 7. **Find the vertex of the parabola:** Since $R(x)$ is a quadratic with a negative leading coefficient, the maximum revenue occurs at $$x = -\frac{b}{2a} = -\frac{200}{2 \times (-50)} = -\frac{200}{-100} = 2$$ 8. **Calculate the ticket price for maximum revenue:** $$12 + x = 12 + 2 = 14$$ **Final answer:** The ticket price that maximizes revenue is $14$ dollars.